diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzdgf" "b/data_all_eng_slimpj/shuffled/split2/finalzdgf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzdgf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn condensed matter research there is a strong desire for new experimental tools that can measure the local electrical properties of materials and buried layers at high frequencies and with high spatial resolution \\cite{basov2011electrodynamics}.\nFor this goal, in recent years a variety of powerful scanning probe techniques have emerged which cover different parts of the electromagnetic spectrum: on the one hand, scanning near-field optical microscopy (SNOM) has enabled imaging with\ninfrared and far-infrared \\cite{qazilbash2007mott,liu2016nanoscale,huber2008terahertz} frequencies down to a few THz by utilizing the optical toolbox, i.e. free-space radiation, lasers and fiber technology; on the other hand, at GHz frequencies (typically 1-20 GHz) co-axial probes and shielded cantilevers have made possible quantitative local imaging by making use of commercially available microwave electronics for high performance signal processing (scanning microwave impedance microscopy, SMIM) \\cite{lai2010mesoscopic,de2016spatial,gramse2017nondestructive,anlage2007principles,steinhauer1998quantitative,buchter2018scanning,tselev2007broadband,huber2012calibrated}. \nThe frequency band in between, however, ranging approximately from 100 GHz to a few THz (also referred to as \\textit{sub-mm waves}), is a technological challenge. In the field of astronomy detection major progress has been made in sub-mm technology, for instance in the development of phase preserving instruments based on superconducting tunnel junctions such as for the Herschel Space Telescope, the Atacama Pathfinder Experiment (APEX) and the Atacama Large Millimeter array (ALMA)\\cite{eht2019black,siegel2002terahertz}.\nThese advances are being picked up to promote technological progress also in other research fields. In condensed matter physics this is expected to have a strong impact on measurement instrument development which will help understanding of a variety of important problems, in particular for disordered and unconventional superconductors, as well as for the so-called quantum materials where strong electron-electron interactions in the THz energy range give rise to a number of puzzling, unconventional and often spatially inhomogeneous electrical properties \\cite{liu2016nanoscale,basov2011electrodynamics,dordevic2006electrodynamics}. \nRealizing an experimental tool for probing these properties, however, remains challenging. Only recently first scattering SNOM measurements below 1 THz have been reported \\cite{liewald2018all}. \nIn the sub-mm band on-chip electronic circuits suffer from high losses (at room temperature) which strongly complicates the fabrication of more complex circuitry, required for signal control and processing. An alternative technology, much less prone to losses, consists of quasi-optical and metallic waveguide components. However, this technology increases the size of the measurement instrument compared with on-chip circuitry and thus also imposes certain boundaries when more complex signal handling is needed.\nIn order to overcome these technological hurdles, there is an ongoing development to combine quasi optical and on-chip electronics in hybrid devices \\cite{westig2011balanced}, but also to push the performance of microwave electronics into the sub-mm-band \\cite{zmuidzinas2004superconducting}. Picking up on this development, we have recently reported on a microstrip (MS) fabrication technology based on PECVD SiN$_x$, that is compatible with thin film membranes. For this technology losses are sufficiently well controlled at frequencies around 0.3 THz, such that the realization of room-temperature THz on-chip components is feasible \\cite{finkel2017performance}. \nHere, we use this technology to extend scanning impedance microscopy from microwaves into the THz frequency range. We present a shielded THz cantilever suitable for scanning probe microscopy that enables quantitative measurement of the impedance of the cantilever tip at around 0.3 THz. A key ingredient is a branchline coupler which is patterned on the cantilever and which acts as an interferometer for the THz signal, thereby providing high sensitivity of the circuit to small impedance changes at the cantilever tip. \n\nFirst, we will revisit the concept of scanning near field microscopy with shielded cantilevers as it is currently being used in microwave microscopy and we will identify the key features a THz cantilever should comprise. We then present the concept of our THz on-chip interferometer. Finally, we demonstrate how this concept enables impedance measurements with a THz cantilever that is compatible with conventional atomic force microscopy (AFM).\n\n\n\n\\section{Principle of shielded cantilever microscopy}\n\n \n The principle of scanning near field microscopy with a shielded microstrip (MS) cantilever is illustrated in Fig.~\\ref{fig:principle}a) \\cite{anlage2007principles,lai2011nanoscale}. The cantilever consists of a dielectric membrane of which the bottom side is covered with a thin metal layer, serving as a transmission line ground plane. \n The signal line of width \\textit{w} is patterned on the top side of the same dielectric. A cross section of the resulting MS transmission line geometry is sketched in Fig.~\\ref{fig:principle}b). Because the high frequency fields are mostly confined within the dielectric, such a transmission line geometry allows for delivering the signal to the cantilever tip in a controlled way while the ground plane screens the environment and prevents radiation losses. At the end of the cantilever the signal line terminates in a metallic tip. When a high frequency tone is launched to the MS, the tip acts as a capacitive termination, reflecting the signal back into the cantilever. This is quantified by the reflection coefficient $\\Gamma$, which is given by the mismatch between the generally complex valued tip impedance $Z$ and the characteristic MS line impedance $Z_0 = 50~\\Omega$: $\\Gamma={(Z-Z_0)}\/{(Z+Z_0)}$. When the cantilever is lifted far away from the sample surface, $Z$ is given by the capacitance $C_t$ between the tip and the cantilever ground plane (see Fig. \\ref{fig:principle} a). When the tip is on a sample, $Z$ is modified by contributions from the tip-sample capacitance $C_{s,tip}$, the capacitance between the sample and the ground plane $C_{s,gnd}$ and from resistive losses inside the sample, $R_s$. Measuring changes in $\\Gamma$ by detecting phase and amplitude of the reflected signal while scanning the tip over the sample provides a quantitative image of the spatial distribution of the conducting and dielectric properties of the sample \\cite{lai2010mesoscopic}. Since the electric field becomes strongly enhanced at the sharp tip, these local contributions dominate the total response, thus enabling spatial resolution down to 100 nm, i.e. three orders of magnitude below the signal wavelength \\cite{,lai2010mesoscopic,gramse2017nondestructive}. \n \n The tip-ground plane capacitance C$_t$ is generally given by the size and geometry of the cantilever close to the tip, which results in values of the order of $C_t \\sim 10^{-15}F$ . Since $Z\\propto 1\/i\\omega C$, at GHz frequencies ($\\omega = 2\\pi f$, $f \\sim 10^9$ Hz) the terminating impedance is large $Z \\sim 10^6 \\Omega$ $ \\gg Z_0$ and therefore the reflection coefficient becomes $\\Gamma \\simeq 1$. This means that most of the signal is reflected back into the cantilever when the tip is floating over the sample. We will refer to this part of the signal as scattered signal because it does not carry information about the sample itself. When the tip is in contact with the sample, the desired contributions from the tip-sample interaction thus only lead to small variations on top of an otherwise large $\\Gamma$, which is obviously difficult to detect. It is therefore highly desirable to minimize the scattered signal in the detector line and to become sensitive to those contributions only, which originate from the tip-sample interaction. At microwave frequencies this problem has been addressed by making the cantilever and the tip part of a resonator \\cite{cui2016quartz,gramse2017nondestructive,huber2012calibrated,anlage2007principles,tselev2007broadband} or by adding an impedance matching circuit which matches the open tip impedance to $Z_0$ \\cite{lai2010mesoscopic}. Both solutions create a narrow band resonance condition which enhances the sensitivity of the circuit to changes in the tip impedance. Furthermore, a common mode cancellation loop is typically included into the microwave readout circuit \\cite{huber2012calibrated,lai2010mesoscopic} which further reduces the scattered signal level at the detector.\n \n \\begin{figure}\n \t\\includegraphics[width=1\\linewidth]{Fig1.pdf}\n \t\\caption{\\textbf{a)} Sketch of a shielded cantilever and the corresponding lumped element circuit for measuring the complex reflection coefficient $\\Gamma$ of the cantilever tip. The cantilever consists of a metallic signal line and groundplane, separated by a dielectric, which determines the mechanical properties of the cantilever. The probe-cantilever interaction can be described with a lumped element circuit. \\textbf{b)} Cross section of the microstrip shielded cantilever. $w$ and $h$ denote the width of the signal line and the height of dielectric layer, respectively. \\textbf{c)} Equivalent lumped element circtuit of the situation depicted in a). \\textbf{d)} Distributed element circuit diagram for measuring the cantilever impedance using a balanced branchline coupler. The reflection coefficient $\\Gamma$ at the cantilever tip is determined by measuring the scattering parameter $S_{41}$ of the branchline coupler. This is achieved by intereference of the reflected signal from the tip with an unknown phase shift $\\Delta \\Phi_2$ with that of a balanced cancellation arm with known phase shift $\\Delta \\Phi_{3}$. $Z_0$ denotes the characteristic line impedance and $\\lambda$ is the signal wavelength. }\n \t\\label{fig:principle}\n \\end{figure}\n \n\n While at microwave frequencies such circuitry can be incorporated rather easily, this is not straight forward at THz frequencies because many required technologies are not readily available. In order to realize a shielded impedance microscope cantilever at THz frequencies it is therefore plausible to aim for an on-chip THz circuit solution that can be patterned close to the tip with lithographical means. We identify the following key properties such a circuit should provide: 1) separation of the in-going and reflected signal to facilitate signal processing. 2) Cancellation of the scattered signal in the detector line. 3) Sensitive response when the tip is brought into contact with a dielectric or a metallic sample. 4) Short signal lines to minimize losses. \n In the following section we will present and demonstrate a circuit that fulfills all of these requirements.\n \n\n\\section{Balanced branchline coupler as on-chip interferometer} \nFigure \\ref{fig:principle}d) depicts the diagram of a circuit designed to accomplish the above criteria. \nA key component is the balanced branchline coupler. It consists of 4 ports (labelled 1 - 4) which are connected through transmission line segments of a quarter wavelength $\\lambda\/4$ of the aimed for measurement frequency. By properly designing the impedance of each branch of the coupler (i.e. by choosing the appropriate signal line width $w$ for a constant thickness of the dielectric layer, cf. Fig.\\ref{fig:principle}b), one can control the transmission coefficients between the ports. \n The key idea of the concept we introduce here derives from analogies between a branchline coupler and an optical beam splitter: When the branch impedances $Z$ are chosen such that for two opposite branches $Z=Z_0$ ($w = 3.75~\\mu$m), while for the other two $Z = Z_0\/\\sqrt{2}$ ($w = 7.5~\\mu$m), an incoming signal at, for example, port 1, is split in equal parts between ports 2 and 3, and it acquires an additional phase shift of $-\\pi\/2$ between the these ports, while no signal arrives at port 4 \\cite{pozar2009microwave}. Since the coupler is designed symmetrically, the signal is split in the same fashion when injected at any other port.\n\n\n\n We can now use these properties, signal splitting and phase delay, to build an on-chip interferometer that is highly sensitive to impedance changes at the cantilever tip: We attach transmission lines of finite length $L_2$ and $L_3 = L_2 + \\Delta L$ at ports 2 and 3, respectively, as shown in Fig. \\ref{fig:principle}d (which we will refer to as \\textit{arms}, in analogy to an optical Michelson-interferometer). As the signal gets reflected at the end of each arm, it picks up a phase shift and gets re-injected into the coupler. For simplicity, assuming an ideal coupler with perfect isolation \\cite{pozar2009microwave} and neglecting losses, the signal at port 4 (detector line) is then given by the sum of the reflected signals re-injected at port 2 and 3,\n \n \\begin{equation}\n \tS_{41} = \\frac{A}{2} e^{i (\\Phi_c + \\Phi_{L2} + \\Delta \\Phi_2)} + \\frac{A}{2} e^{i (\\Phi_c + \\Phi_{L3} +\\Delta \\Phi_3)},\n \t\\label{eq:coupler} \n \\end{equation}\n where $A$ corresponds to the total signal amplitude, $\\Phi_c = 3\\pi\/2$ is the total phase accumulated in the coupler, $\\Phi _{L2,3}$ refer to the phase picked up due to the signal traveling down the respective arms and $\\Delta \\Phi_{2,3}$ is the phase picked up due to reflection at the terminations of arms 2 and 3, respectively. For our purposes it is convenient to express Eq.\\ref{eq:coupler} as \n \n \\begin{equation}\n S_{41} = \\frac{A}{2} e^{i (\\Phi_c + \\Phi_{L2})} (e^{i \\Delta \\Phi_2} + e^{i (\\Delta \\Phi_3 + \\Phi_{\\Delta L})}).\n \\label{eq:coupler_DeltaLeq0} \n \\end{equation}\n \n which indicates that signal cancellation in the detector line is achieved for \n \\begin{equation}\n \\Delta \\Phi_2 = \\Delta \\Phi_{3} + \\Phi_{\\Delta L} - \\pi .\n \\label{eq:phase}\n \\end{equation}\n \n\n\n As shown in Fig.~\\ref{fig:principle}d), in our case arm 2 terminates in the cantilever tip, which, in a first approximation ($C_t \\rightarrow 0$), acts as an \\textit{open} termination ($\\Delta \\Phi_2 \\rightarrow -\\pi$) when the tip is lifted off the sample. It is therefore convenient to terminate arm 3 with a \\textit{short} ($\\Delta \\Phi_3 = 0$) and to choose $\\Phi_{\\Delta L} = 0$ to achieve good signal cancellation at port 4. \n When scanning, changes in the dielectric or metallic environment of the tip lead to a phase mismatch at the detector line due to an enhanced capacitance at the tip, according to Fig.~\\ref{fig:principle}c. This results in a measurable signal which can be directly related to the phase change due to modified reflection conditions at the tip, using Eqs. \\ref{eq:coupler_DeltaLeq0} and \\ref{eq:phase}. When the tip is landed on a fully metallic sample, $(1\/C_{s,tip} + 1\/C_{s,gnd})^{-1} \\gg C_t$. As can be seen from the circuit in Fig.~\\ref{fig:principle}c) this corresponds to arm 2 being effectively shorted. In this case the reflected signals will interfere constructively and the full signal is detected at port 4. \n We note that in a real cantilever $C_t$ is finite ($\\Delta \\Phi_2 \\gtrsim -\\pi$) in which case $\\Delta L$ can be used as an additional phase matching parameter to achieve cancellation of the scattered signal. \n\n\\begin{figure}\n\t\\includegraphics[width=0.7\\linewidth]{Fig2.pdf}\n\t\\caption{\\textbf{a)} Optical image of the on-chip interferometer (device A), containing a balanced branchline coupler with branch lengths $\\lambda\/4$. The ports of the coupler are denoted 1-4. At a distance $L_2$ and $L_3$ from port 2 and 3 the transmission lines terminate in an open and short circuit, respectively. The signal (indicated with blue arrows) is injected at port 1 and detected at port 4. The scale bar corresponds to $100~\\mu$m. \\textbf{b)} S-parameter magnitude of a transmission measurement (symbols) and the corresponding analytic calculation (solid line) of device A (blue) and B (black). $L_2 = L_3 = 49 ~\\mu$m.}\n\t\\label{fig:branchline}\n\\end{figure}\n \n In order to test this concept, we have realized a series of balanced branchline couplers on a Si substrate using the technology described by Finkel \\textit{et al.}\\cite{finkel2017performance} All structures consist of $3~\\mu$m thin SiN$_x$ ($\\epsilon_r = 5.9$) serving as a MS dielectric and 2\/300 nm Ti\/Au as ground plane and stripline. As THz source and detector we use a vector network analyser together with frequency multipliers that cover the WR-03 band (220 to 325 GHz) and a GSG landing probe setup (for details see Finkel et al, Ref.\\cite{finkel2017performance}). Note however, that our concept is also compatible with other THz sources and detectors, for instance photomixers \\cite{mayorga2012first,westig2019waveguide}. We first demonstrate conceptually the basic idea. For this we have fabricated two samples (device A and B) for which $\\Delta L = 0$ and which realize two different arm terminations \\textit{open\/short} and \\textit{open\/open} at ports 2 and 3, respectively. An optical image of device A is shown in Fig.~\\ref{fig:branchline}a. The signal is launched and picked up from the circuit via the ports labeled 1 and 4 in Fig.~\\ref{fig:branchline}a, which consist of co-planar waveguide type fixtures\\cite{finkel2017performance} (not visible) that enable coupling of THz signals into the circuit with the landing probes. \n The $\\lambda\/4$-branches of the coupler have a length of 130 $\\mu$m, corresponding to a branchline coupler center frequency of $f_c = 270$ GHz. For the arm lengths we choose $L_2= L_3 = 49~\\mu$m. \n \n\nFigure \\ref{fig:branchline} b shows the measured scattering parameter $S_{41}$ obtained for device A and B (blue and black symbols, respectively). As expected, for device A we observe a low transmission ($\\sim -30$~dB) between port 1 and 4 with a minimum at $f =280$ GHz, which is close to the branchline coupler's center frequency $f_c = 270$ GHz. For device B both arms terminate in an open, i.e. $\\Delta \\Phi_2 = \\Delta \\Phi_3$. As a result constructive interference leads to a high transmission ($\\sim -5$~dB) over the full frequency range. \n\nNext we demonstrate how, owing to the sharp interferometer cancellation conditions, the circuit is highly sensitive to contributions from $\\Phi_{\\Delta L}$. Figure \\ref{fig:varyL} shows the measured $S_{41}$ parameter obtained from a series of devices for which we have varied $L_\\text{3} = 49~\\mu$m$ + \\Delta L$ by $\\Delta L = (1, 0, -1...-5)~ \\mu$m, while leaving $L_\\text{2} = 49~\\mu$m fixed. This leads to a small phase imbalance for the signal paths along arms 2 and 3. The experimental data reveal that indeed the position of the dip in frequency as well as its depth sensitively depend on $\\Delta L$ (dotted line). In Fig.~\\ref{fig:Hyb_IQ}a we have extracted magnitude and phase (symbols) for each $\\Delta L$ at fixed frequency $f = 280$ GHz (dashed line in Fig.\\ref{fig:varyL}). The data show that signal cancellation improves for small $\\Delta L$ with an optimal configuration at $\\Delta L = -1~\\mu m$. For even larger length difference it levels off. As discussed above this behavior reflects the termination of arm 2 with a finite capacitance, leading to phase shift slightly different from $-\\pi$, which gets compensated for by a slightly shorter $L_3$.\nThis has been confirmed quantitatively within a textbook analytical model of the circuit \\cite{pozar2009microwave} (for details see Appendix and Supplementary Material) that nicely reproduces all of our experimental data consistently (solid lines in Fig.~\\ref{fig:varyL} and Fig.~\\ref{fig:Hyb_IQ}a). In addition to a small dissipative contribution in the via, $R_\\text{short} = 1.6~\\Omega$, we have taken into account a finite terminating capacitance $C_{t} = 0.163$~fF, consistent with a standard text book approximation for an open MS line (see Appendix).\n\n\\begin{figure}\n\t\\includegraphics[width=0.6\\linewidth]{Fig3.pdf}\n\t\\caption{ Transmission measurements (symbols) and calculation (solid line) for a set of \\textit{open\/short} circuits with $L_3 = L_2 + \\Delta L$ and $\\Delta L = +1,0,...,-5~\\mu$m and $L_2 = 49~ \\mu$m. The curves are offset by -30 dB for clarity. }\n\t\\label{fig:varyL}\n\\end{figure}\n\n\\begin{figure}\n\t\\includegraphics[width=0.8\\linewidth]{Fig4.pdf}\n\t\\caption{\\textbf{a)} Magnitude (left) and phase (right) as function of $\\Delta L$ (bottom axis). Symbols: measurements. solid line: calculation. Dashed line and top axis: Calculated phase and magnitude for $L_2=L_3 = 49~\\mu$m and with varying $C_{load}$ connected in parallel to the termination of interferometer arm 2. \\textbf{b)} In-phase (I, red) and quadrature (Q, black) representation of the measured (circles and squares) and calculated (solid and dashed line) response versus $\\Delta L$. Dashed lines and top axis: Calculated I (red) and Q (black) as a function of $C_\\text{load}.$}\n\t\\label{fig:Hyb_IQ}\n\\end{figure}\n\nWe can further use the analytical model to analyse theoretically the circuit's response to a load capacitance $C_\\text{load}$ connected in parallel to $C_t$, representing a sample in a scanning probe experiment (cf. lumped element diagram in Fig. \\ref{fig:principle}c). The resulting amplitude and phase are plotted in Fig.~\\ref{fig:Hyb_IQ}a) as dashed lines. The corresponding $C_\\text{load}$ is given in the top axis. As expected this yields a fairly similar behaviour as a variation of $\\Delta L$. \nFigure ~\\ref{fig:Hyb_IQ}b) plots the data as in-phase (I) and quadrature (Q) amplitudes, for a variation of $\\Delta L$ (bottom axis, solid lines) and $C_\\text{load}$ (top axis, dashed lines), respectively. In this representation $I$ can be directly related to dissipative contributions to the signal while $Q$ represents the imaginary part of the reflection coefficient which is related to capacitive (and, in principle, also inductive) contributions. This is consistent with the observed linear behaviour of $Q$ and a constant $I$. From these plots we estimate our circuit to be sensitive to a capacitance change smaller than 0.25 fF. \n\n \n \\section{Cantilever implementation}\n \n We will now describe how this detection scheme can be implemented and used in a scanning probe cantilever to detect impedance changes at the probe tip.\n Figure \\ref{fig:cantilever}b shows an optical microscope image of the shielded cantilever containing the THz circuit, patterned on its top side. The \\textit{signal in} and \\textit{signal out} lines (corresponding to ports 1 and 4 in Fig.~\\ref{fig:branchline}a) are connected via landing probes with the source and detector (not visible). Since the dimensions of the cantilever (300 $\\mu$m long, 75 $ \\mu$m wide) are too small to host a circuit as shown in Fig.~\\ref{fig:branchline}a, we have re-designed the branchline coupler such that the cross-branches are now folded inwards to fit the lateral dimensions of the cantilever. This slightly modifies the coupler properties. However, it does not change its basic functionality. \n As discussed previously for the branchline coupler devices, one of the interferometer arms terminates in a \\textit{short}. The other one, previously terminating in an \\textit{open}, is now connected to the tip. We will keep the notation of the arms as introduced above, referring to the arm terminating in the tip as arm 2 with length $L_2$, and to the arm terminating in a short to ground as arm 3 with length $L_3$. In order to balance the coupler such that scattered signal cancellation is achieved, we have to take into account the finite capacitance of the open tip ($C_t \\sim 2$ fF, obtained from finite element (FE) simulations) and adjust $L_3$ by $\\Delta L$ accordingly. However, due to the folded geometry of the coupler and a resulting unwanted cross-coupling between the branches, significant leakage currents within the coupler result in a non-trivial relation between $\\Delta L$ and signal cancellation at the detector line. Therefore, we use FE simulations to empirically determine a well-balanced configuration for the given $C_t$, for which we obtain $L_2 = 44~ \\mu m$ and $L_3 = 54~ \\mu m$, i.e. $\\Delta L = 8~\\mu$m . \n \n \\begin{figure}\n \t\\includegraphics[width=1\\linewidth]{Fig5.pdf}\n \t\\caption{\\textbf{a)} Main fabrication steps of the THz cantilever. \\textbf{b)} Optical image of the final, released cantilever. The scale bar corresponds to $75~\\mu$m. \\textit{Signal in} and \\textit{signal out} denote the transmission lines connected to THz source and detector, respectively. The folded hybrid coupler patterned on the back of the cantilever is indicated. One arm of the coupler terminates in a short circuit, the other one terminates in the tip. \\textbf{c)} Transmission amplitude from \\textit{signal in} to \\textit{signal out}. The transmission dip at 250 GHz indicates cancellation of the signal reflected from the open tip. The dashed curve shows the result obtained from finite element (FE) modeling.}\n \t\\label{fig:cantilever}\n \\end{figure}\n\n \\subsection{Fabrication}\nIn fig. \\ref{fig:cantilever}a the fabrication flow for the cantilever is sketched.\nIn a first step (1) a pyramid shaped pit ($5~\\mu$m deep) is etched into the Si wafer using KOH etching. This defines the position and shape of the tip. Next (2) we deposit (10+300)nm Ti\/Au which serves as a ground plane. During this step also the pit is filled with a Ti\/Au layer, which will become the metallic tip. The area around the pit is protected with an optical mask. The wafer is then (3) covered with 3 $\\mu$m of PECVD SiNx that is subsequently etched with a Bosch process to define the geometry of the cantilever, $300~\\mu$m long and $75~\\mu$m wide. In a separate step $5\\times 5~\\mu$m$^2$ sized \\textit{vias} are etched into the SiN$_x$ layer. These \\textit{vias} will serve as an electrical connection between the MS top layer and the ground plane (to form a short) or with the cantilever tip, respectively. \nAfter a cleaning step, we pattern the strip lines with electron beam lithography and lift off techniques (5) and, in a separate step, connection of the\\textit{ via} is established through angled deposition of Au. We use (2+300)nm Ti\/Au bilayers. This step concludes the patterning of the transmission lines on the cantilever. Next, we release the cantilevers. In order to avoid exposure of the striplines to chemicals, we protect the surface of the wafer by gluing a Sp wafer on top of it with \"black wax\" (Apiezon W100).\nWe take particular care that no air bubbles remain in the wax to ensure a complete and efficient protection. The release step is prepared by patterning a SiN$_x$ mask on the backside of the wafer which contains windows at those positions where the cantilevers have been patterned on the front side of the wafer. The two wafers are then subjected to a KOH etch which etches through the windows on the backside of the wafer until the Ti\/Au ground layer is reached. At this point the cantilever gets released from the Si wafer. Note, however, that on its front side it is still glued to the protection wafer. When the KOH etch is complete, the wafer is carefully immersed in Toluene to dissolve the black wax and to fully release the cantilever chips. \nSubsequently the cantilever is mounted on the landing probe setup. \n \n \\subsection{Experimental results}\n \n\nThe measured THz response of the cantilever is shown in Fig.\\ref{fig:cantilever}c. We clearly observe a dip in transmission ($\\sim -30$ dB) indicating a suppression of the scattered signal that gets reflected from the open tip into the detector line. We note that compared to the previously discussed branchline couplers without the tip (Fig.~\\ref{fig:branchline}), the position of the dip is slightly shifted towards lower frequencies ($f = 250$ GHz). This is most likely a result of the folded geometry of the branchline coupler, consistent with a cross-capacitive coupling between neighbouring parts of the circuit. Moreover, the frequency shift as well as the relatively low transmission at higher and lower frequencies, suggest that the terminating capacitance of the tip slightly differs form the assumed value ($C_t = 2$ fF) such that the chosen $\\Delta L = 8~\\mu$m turns out to be not yet the best match. The FE model for the THz response (dashed line) yields good agreement with the experiment if we assume $C_t = 2.9~$fF and $R_{short} = 5~\\Omega$. \n\nOur cantilever can be used to detect changes in the tip impedance when landed on a dielectric or metallic sample. This is demonstrated in Fig.~\\ref{fig:TIM}. We have mounted the cantilever on the landing probe setup and we get the tip in contact with different materials, approached from below via a mechanical height control. Fig.~\\ref{fig:TIM}a compares the measured response for the tip floating near the sample surface (red) and landed on 3 different materials, Au (green), Si (black) and SrTiO$_3$ (blue). \nWe clearly observe distinct responses for each material. When brought into contact with a dielectric (Si, $\\epsilon_r = 11.9$ and SrTiO$_3$, $\\epsilon_r = 300$) the dip shifts towards lower frequencies by $\\Delta f_\\text{(Si)} = 2$ GHz for Si and $\\Delta f_\\text{(STO)} = 10$ GHz for SrTiO$_3$. Notably, the overall line shape remains fairly similar. In contrast, upon contact with highly conductive Au ($\\rho = 2\\mu \\Omega $cm), the dip vanishes and transmission is high over the full frequency range, as expected for a shorted tip. \n\n\n \\begin{figure}\n \t\\includegraphics[width=1\\linewidth]{Fig6.pdf}\n \t\\caption{ \\textbf{a)} THz response of the cantilever when open (red) and landed on Si (black), SrTiO$_3$ (blue) and Au (green) samples. Solid lines correspond to measurements, dashed lines indicate finite element (FE) simulations with the following parameters: $R_\\text{via} = 5 \\Omega$, Tip capacitance: $C_t= 2.9~$fF; $L_2 = 44~\\mu$m, $L_3 = 52~\\mu$m, conductivity signal line and gnd plane: $\\sigma_{au} = 19.2\\times 10^{-6}$~S\/m, $\\epsilon_r(\\text{SiN}_x) = 5.9$. $C_\\text{load}$ (corresponding to $(1\/C_{s,tip} + 1\/C_{s,gnd})^{-1}$ in Fig. 1, $R_s = 0$) for Off: 0 fF; Si: 0.25fF; SrTiO$_3$: 0.75fF; Au: 15fF. \\textbf{b)} I (dashed line, red) and Q (solid line, black) of the cantilever response at 260 GHz for different $C_{load}$ obtained from FE simulations. Symbols: experimental data extracted from a). \\textbf{c)} $20~\\mu$m$\\times 20~\\mu$m topography image of a scratched Si wafer, obtained with a cantilever similar to the one used in a) utilizing the beam deflection mode in a commercial AFM. Inset: SEM image of the cantilever tip. } \n \t\\label{fig:TIM}\n \\end{figure}\n \t \n\t \n\n\n\\subsection{Discussion}\n\nIn order to quantitatively understand the cantilever response we use FE modelling of the full circuit and we include a load capacitance $C_{load}$ in parallel to $C_{t}$ to take into account contributions from the sample materials (cf. fig. \\ref{fig:principle}a, neglecting Ohmic dissipation in the sample, $R_s = 0$). We find that the curves can be reproduced very well if we use $C_{Si} = 0.25 ~$fF, $C_{STO} = 0.75 ~$fF, and $C_{Au} = 15 ~$fF as the only adjustable parameter for each material. \nIn Fig.~\\ref{fig:TIM}b) we compare $I$ and $Q$ of the FE response for various $C_{load}$ with the experimental values obtained at $f = 260$ GHz for each sample. Since the response of the folded branchline coupler connected to the tip is slightly off resonance ($\\sim 250$ GHz) we do not expect a simple linear behaviour as for the branchline coupler without the tip discussed previously. Figure \\ref{fig:TIM}b shows that the response becomes more sensitive, i.e. the slope of the curves for $I$ and $Q$ becomes steep, for larger $C_{load}$ ($\\sim 10$ fF). Since this is in the range of capacitance which we obtained for the Au sample, this indicates that our THz cantilever becomes more sensitive for metallic samples. In contrast to shielded microwave cantilevers, where sensitivity is highest around the metal-insulator transition \\cite{lai2010mesoscopic}, for our cantilever the working point is shifted towards samples with higher conductivity. It may thus be used to detect electronic variations at high frequencies within a metal or even in superconductors in future scanning experiments.\n\nOur THz cantilever is also compatible with AFM topography imaging. This is shown in Fig.\\ref{fig:TIM} c) where a topography image of a scratched Si wafer surface is displayed, obtained using a THz cantilever mounted on a commercial Asylum Cypher AFM with a laser deflection read-out. Using a de-convoluting tip geometry modeling algorithm (\\textit{Gwyddion} blind tip estimation algorithm \\cite{nevcas2012gwyddion}) we estimate the tip apex to be $\\approx$100 nm. An SEM image of a cantilever tip is shown in the inset in Fig. \\ref{fig:TIM} c). \n\nFinally, we like to point out some aspects that we aim to improve for future THz cantilever generations. Firstly, even though our fabrication technique provides useful devices, the current yield is rather low ($\\approx 10 \\%$). This is mostly related to the use of the black wax, which is needed to protect the THz circuitry from chemicals during the release step, but which also induces mechanical stress, resulting in loss of a large number of cantilevers. Secondly, in its current design the substrate, which serves as a handling wafer, faces in the same direction as the tip. This limits the surface region on the sample, that can be reached by the cantilever to approximately the cantilever length ($\\sim 300 ~\\mu$m). In order to lift this constraint, developing a flip-chip technology may provide the most suitable means to bond a handling wafer to the top side of the cantilever chip. At the same time, however, it will be important to maintain access to the circuitry with landing probes. Thirdly, our experiments have shown that due to the folded geometry of the balanced branchline coupler the circuit response deviates slightly from that of the un-folded geometry tested on a substrate (Fig.~\\ref{fig:cantilever}b in comparison to Fig.~\\ref{fig:branchline}a). Therefore, FE simulations are required for a quantitative analysis of the measurement signal while a simple analytical equation would be more desirable. This may motivate a re-design of the cantilever such that it can host the balanced coupler without the need to modify its layout. In this case the measurements could be modeled within a textbook analytical description, which will strongly facilitate a quantitative interpretation of the measurement signal.\n\n\n\n\\section{Conclusion}\nWe have presented a shielded THz probe suitable for impedance microscopy in the sub-mm band (0.3 THz). As a key challenge for the realization of such a device we have identified the necessity to carry out common mode cancellation and impedance matching at THz frequencies close to the cantilever tip in order to enable sensitive detection of small changes of the tip impedance. We have addressed these challenges by developing an on-chip circuit that can be patterned on the cantilever which comprises a balanced branchline coupler. The coupler functions as an on-chip interferometer and in this manner achieves the required common mode suppression as well as high sensitivity to small impedance changes. To demonstrate the basic functionality of this concept, we have realized a set of devices on substrates and we have characterized their response at THz frequencies. The results can be directly modeled within an analytical model of the circuit. Furthermore, a fabrication technology has been developed that allows for patterning the circuit on a free standing cantilever including the tip. When the released cantilever is landed on different dielectric (Si, SrTiO$_3$) and metallic (Au) samples we observe distinct THz responses which enable us to determine the corresponding capacitive load at the cantilever tip using finite element modelling. Our cantilever removes several critical technological challenges towards scanning impedance microscopy at THz frequencies. \n \n\\section*{Appendix: Analytical Model for the balanced branchline coupler}\n\nIn order to compute the response of the balanced hybrid coupler we describe the signal evolution in the coupler in terms of forward and backward travelling waves in the transmission lines and the reflection coefficents $\\Gamma_\\text{ms}$ and $\\Gamma_\\text{s}$ at the open and shorted transmission line, respectively. Using Kirchoff's rules for the voltage and current at each node of the hybrid, we can construct a system of equations that allows us to determine the voltage measured at the detector line at port 4 upon signal injection at port 1. \n(The full set of equations is provided in the Supplementary material).\nTo describe wave propagation along each transmission line segment with length $L$ and impedance $Z$ we use the frequency dependent wave propagation factor $e^{\\gamma L}$ and\n\n\\begin{align*}\n&\\gamma = \\alpha \\frac{Z}{Z_0} + i\\beta\\\\\n&\\alpha = (1.1f \\times 10^{-9} + 86.9)~\\text{Np\/m} \\\\\n&\\beta = \\frac{2 \\pi}{c} f \\sqrt{\\epsilon_\\text{eff}} \n\\end{align*}\n\nwith $Z_0 = 55.5~\\Omega$, $c = 3\\times 10^8$ m(s)$^{-1}$, $\\epsilon_{eff} = 4.47$ and $\\alpha$ extracted from a direct measurement of a $130~\\mu$m transmission line. For the low impedance lines we have used Z =Z$_l$ = 37 $\\Omega$. The branch lengths of the coupler are $L = 130 \\mu $m. \n\nTo calculate the open terminating capacitance $C_t$ of the open MS line we have used \n\n\\begin{align*}\nC_{t} = G \\frac{\\sqrt{\\epsilon_{eff}}}{c Z_0}, \\\\\nG = \\frac{\\xi_1 \\xi_3 \\xi_5 h}{\\xi_4}\n\\end{align*}\n\ntogether with the following closed form expression:\n\\begin{align*}\n&\\xi_1 = 0.434907\\frac{(\\epsilon_{eff}^{0.81} + 0.26(w\/h)^{0.8544} + 0.236)}{(\\epsilon_{eff}^{0.81} - 0.189(w\/h)^{0.8544} + 0.87)}\\\\\n&\\xi_2 = 1+\\frac{(w\/h)^{0.371}}{2.35\\epsilon_r + 1}\\\\\n&\\xi_3 = 1 + \\frac{0.5274~\\tan^{-1}[0.084(w\/h)^{1.9413\/\\xi_2}]}{\\epsilon_{eff}^{0.9236}}\\\\\n&\\xi_4 = 1+ 0.037\\tan^{-1}[0.067(w\/h)^{1.456}] \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times (6-5\\exp(0.036(1-\\epsilon_r)))\\\\\n&\\xi_5 = 1-0.218 \\exp(-7.5(w\/h))\n\\end{align*}\n\n\n\n\nfor which we have used the stripline width $w = 3.75~\\mu$m, dielectric thickness $h = 3~\\mu$m, $\\epsilon_{eff} = 4.47$, PECVD SiN$_x$ dielectric constant $\\epsilon_r = 5.9$ \\cite{finkel2017performance}, characteristic impedance $Z_0 = 55.5~\\Omega$ and the vacuum speed of light $ c=3\\times10^{8}$ m\/s. \n\n\n\\section{Acknowledgements}\nWe would like to thank Carmine de Martino and Luca Galatro for help with the THz characterization setup and David Thoen for support during the fabrication process development.\nWe further acknowledge scientific discussions with Andrea Neto, Nuria Llombart, Akira Endo, Jochem Baselmans, Ronald Hesper and Ivan Camara Mayorga. \nThis research has been funded by the European Research Council Advanced Grant No.~339306 (METIQUM).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLow grade gliomas (LGGs) are slowly growing tumours which arise from supporting glial cells in the brain. Although they proliferate slowly, the ultimate behaviour of these tumours is not benign. They are almost invariably incurable due to their diffusive infiltrative nature and often lead to patient death due to malignant transformation, \\emph{i.e.} transformation of the tumour into more aggressive anaplastic form.\n\nTreatment of~LGGs is controversial among clinicians as LGGs usually occur in young and otherwise healthy patients. Besides seizures, patients with LGGs appear neurologically asymptomatic. Thus, life-prolonging treatment should not come at the cost of compromising the quality of life of these otherwise healthy and mostly young patients. Management decisions, whether a~patient with LGG should receive resection, radiation therapy or chemotherapy, are not fully standardized.\nClinicians usually base their decisions about extent of surgical resection, timing of radiotherapy and long-term benefits and risks of chemotherapy on a large number of factors including age, performance status, location of tumour and patient preference. \n\nCurrent standard of care is first to perform maximal resection with minimum impact on the brain's functional areas. Support for such an approach comes from studies which demonstrate that the extent of resection is a~prognostic factor for LGG patients \\cite{Keles,Smith}. Unfortunately, because of the LGGs infiltrative characteristics, surgery alone fails to cure these tumours in most cases as only the tumour bulk can be resected. There is a~trend to use more active treatments and various alternative approaches have been considered. Post-operative radiotherapy could be a~therapeutic option for low-grade gliomas, but it causes long-term neurocognitive toxicity and fails to show a significant improvement in patient survival~\\cite{Karim,Karim1,Shaw}. Therefore radiotherapy is usually deferred and performed routinely only in patients with tumours facing a~high risk of malignant transformation \\cite{Pouratian}. \n\nIn this context, there is an increasing interest in the use of chemotherapeutic agents which could influence tumour evolution and at the same time allow the delay of more aggressive treatments. Currently there are two chemotherapeutic drugs effective for the treatment of glioma patients: temozolomide (TMZ), an oral alkylating agent and procarbazine, lomustine and vincristine (PCV), a~combination of alkylating agents and cell-cycle specific microtubule inhibitor. About 25-50\\% of LGGs show chemotherapeutic responses to treatment with either TMZ~or PCV. PCV has been used from the 1970's and has been suggested as a~clinically relevant option especially for some subtypes of anaplastic gliomas with the 1p\/19q codeletion \\cite{Cairncross, Bent2}. Unfortunately it causes significant myelosuppression, nausea and peripheral neuropathy \\cite{Mason,Buckner}. \n\nTMZ~is a~cell-cycle non-specific prodrug, absorbed with almost 100\\% bioavailability \\cite{Newlands}.~This chemotherapeutic agent can cross the blood-brain barrier and is spontaneously converted in tumour cells to its active metabolite \\cite{Marchesi}. The main cytotoxic action of TMZ occurs at the O$^6-$ position of guanine. During subsequent cycles of replication, the futile mismatch repair system is initiated and O$^6$MeG is incorrectly paired with thymine instead of cytosine, eventually inducing cell death, see \\emph{e.g.} \\cite{Jiricny,Barciszewska}. A~phase III~trial showed the efficacy of TMZ for high-grade gliomas~\\cite{Stupp} and since then it has been used routinely for patients with newly diagnosed glioblastoma (the most malignant type of glioma)~\\cite{Bent3}. Phase II~trials have demonstrated its effectivity against both previously irradiated and unirradiated LGGs \\cite{Kesari,Kaloshi,Pouratian1}. In addition, there are reports of cases where neoadjuvant chemotherapy given to surgically unresectable tumours has allowed subsequent gross total resection \\cite{Blonski,Jo}, which is of great importance especially when the tumour is highly infiltrative or located in~eloquent areas. Large prospective studies with long-term follow-up are under development to verify statistically significant improvement in resections following TMZ~treatment. It has also been reported that TMZ~treatment may lead to an important reduction in seizure frequency in LGG patients \\cite{Koekkoek}. Thus prolonged TMZ~treatment until evidence of resistance is a~clinically interesting option for selected patients as an up-front or adjuvant treatment. Clinical trials are on the way to study the effect of this treatment on survival. \n\nTemozolomide has a~better toxicity profile than PCV, being well tolerated by patients \\cite{Liu,Pace,Pouratian1}. This is due to the fact that, in contrast to procarbazine and lomustine, TMZ does not cause a chemical cross-linking of the DNA strands and it is less toxic to the haematopoietic progenitor cells in the bone marrow \\cite{Agarwala}. Patients treated with TMZ are also seeing a reduced risk of cumulative haematologic toxicity thanks to the rapid elimination of this drug \\cite{Agarwala,Portnow,Newlands}. \n\nThe response of glioma cells to chemotherapy is a~subject of many clinical and biological studies. Recently, it has been shown that in some LGG patients a~metabolic response to TMZ~therapy is observed even 3 months after the end of~the treatment \\cite{Wyss,Guillevin}. Moreover, dynamic volumetric studies have shown that a~treatment-related volume decrease can be observed for many months after the chemotherapy is discontinued \\cite{Peyre,Ricard}. The time to maximum tumour response was reported to be in some cases larger than 2 years \\cite{Chamberlain}.~Other researchers investigated relations between some molecular characteristics of LGGs and response to TMZ~\\cite{Hoang,Kaloshi,Ricard, Ohba}. However, the question of the correct timing of chemotherapy remains unanswered, namely whether it should be given first or when progression has been observed. Another issue to be addressed is~the optimal fractioning of TMZ.\n\nThe typical plan of TMZ~treatment is to give doses of $150$--$200$~mg per m$^2$~of patient body surface once per day for 5 days every 28 consecutive days. The number of such cycles in~clinical practise is usually between 12 and 30 \\cite{Ricard, Ribba,Chamberlain} and it depends on patient-related characteristics and the haematological toxicity observed. There have been many clinical studies on alternative treatment regimens for gliomas. Among others in \\cite{Kesari} patients were treated in cycles with doses of 75 mg\/m$^2$ given daily for 7~weeks followed by four-week breaks. Some trials on dose escalation \\cite{Wick,Taal,Viaccoz} intended to overcome the DNA-repair activity of the enzyme MGMT (O$^6$MeG methyltransferase), which reverses alkylation at the O$^6$ position of guanine \\cite{Everhard,Hegi}. \nHowever, these TMZ~regimes were either not effective or had to be rejected because of a~high toxicity \\cite{Hammond}. Recent studies show that O$^6$MeG is able to trigger not only apoptosis (programmed cell death), but also autophagy (mechanism allowing the degradation and recycling of cellular components) and senescence (state of permanent cell-cycle arrest in~which cells are resistant to apoptotic death) \\cite{Knizhnik}, which may provide an explanation why previous studies failed to improve treatment efficacy. Many clinicians conclude that the chemotherapy fractionation scheme providing the best tumour response and acceptable haematologic toxicity are still to be determined.\n \nMathematical modelling has great potential to help in finding appropriate therapeutic timings and\/or fractionations and in individual patient treatment decision. Even models simple from mathematical point of view can be useful for clinical practice. \n\nIn this paper we describe a~simple mathematical model for LGG growth and response to chemotherapy that fits very well with longitudinal volumetric data of patients diagnosed with LGGs. Interestingly, the model suggests that the response of the tumour to chemotherapy may be related to~its aggressiveness. We also provide an approximate explicit formula for the time of tumour response to chemotherapy. This equation may be helpful to clinicians in selecting patients who will benefit most from early treatment and finding the best personalised therapy.\n\nOur plan in this paper is as follows: in Section 2 we develop the model and describe its parameters. In Section 3 we present the results of model fitting to patient data and suggest the relation between time of response to chemotherapy and patient prognosis. The analytical estimates are shown in Section 4. We discuss the results of our model and therapeutic implications in Section 5.\n\n\\section{Mathematical model} \\label{sec:model}\n\\subsection{The dynamics of tumour cells}\nComputational modelling of glioma growth started 20 years ago \\cite{Murray} and has received strong attention in the last few years (see \\emph{e.g.} \\cite{Unkelbach,Konukoglu,Barazzoul, Bastogne,Kirkby}). \nFor solid tumours mathematical models have considered different chemothe\\-rapy-related factors such as drug diffusion, uptake\/binding, clearance and their effect on cell cycle progression (see \\emph{e.g.} \\cite{Au,Jackson,Tzafriri,Swierniak}). Many mechanistic mathematical models models have been developed to improve the design of chemotherapy regimes (see \\emph{e.g.} \\cite{Gardner} for a~summary). However, very few models have considered chemotherapy of LGGs \\cite{Ribba}.\n\nIn order to keep our description as simple as possible we will build a~continuous macroscopic model assuming that the tumour grows due to net cell division. The simplest choice for the proliferative term is to assume that a~tumour cell number $P(t)$ is governed by a~logistic growth with coefficient $\\rho,$~its inverse giving an estimate of the typical cell doubling times. \n \nTMZ is a small molecule and is easily absorbed in the digestive tract with 100\\% of bioavailability and a half-life of absorption of 7 minutes \\cite{Ostermann}. It should be noted that the intact TMZ molecule easily crosses the blood brain barrier due to its lipophilicity, and is then activated in the brain compartment. It hydrolyses to MTIC, which subsequently undergoes spontaneous breakdown to an inactive metabolite AIC and an active methyldiazonium cation. It is the methyldiazonium ion that causes the damage in patients' DNA, namely it transfers methyl groups to DNA. The most common sites of methylation are the N$^7$ position of guanine (N$^7$-MeG; 60-80\\%) followed by the N$^3$ position of adenine (N$^3$-MeA; 10-20\\%) and the O$^6$ position of guanine (O6-MeG; 5-10\\%) \\cite{Fu}, the last one being responsible for major cytotoxicity.\n\nAs to the TMZ effect on tumour cells, we choose here to formulate a macroscopic model. It is known that TMZ-induced damage leads to cell death long after the end of therapy ~\\cite{Peyre,Bent1,Ricard,Chamberlain}. It has been verified \\emph{in vitro} that the glioma cells death after administration of TMZ is induced most typically in one of the post-treatment cell cycles \\cite{Roos} due to futile mismatch repair cycles (see \\emph{e.g.} \\cite{Marchesi} for a detailed description of this mechanism). This delayed cell death is a key feature of glioma response to chemotherapy. Thus, in line with this biological evidence we will assume that tumour cells treated with chemotherapy die after a~time $k\/\\rho$ of the order of mean $k$~effective tumour doubling times. This type of model has been used successfully to describe the effect of radiotherapy on LGGs \\cite{Victor,Victor2}.\n\nThen, we will complement the equation for the number of functionally alive tumour cells $P(t)$ with an equation for the evolution of the number of cells irreversibly damaged by chemotherapy $D(t)$. The number of cells damaged by the~drug in a~time unit is considered to be proportional to the concentration of the drug $C(t)$ multiplied by the number of proliferating tumour cells with the rate~$\\alpha,$~measuring the influence of TMZ~on cells.\nWe assume that irreversibly damaged tumour cells try to enter mitosis with the same probability as those active, but die after a mean value of $k$ such attempts, which results in the growth rate $\\rho\\left(1-\\frac{P+D}{K}\\right)$ for proliferative cells and the death rate $-\\frac{\\rho}{k}\\left(1-\\frac{P+D}{K}\\right)$ for damaged cells, $K$~being the carrying capacity for both populations $P+D$, which leads to the following set of equations\n\\begin{subequations}\n \\label{ode}\n\t\\begin{eqnarray} \n\t\t\\frac{\\mathrm{d} P}{\\mathrm{d} t} & = & \\rho P\\left(1-\\frac{P+D}{K}\\right) - \\alpha PC, \\label{ode1} \\\\ \n\t\t\\frac{\\mathrm{d} D}{\\mathrm{d} t} & = & -\\frac{\\rho}{k}D\\left(1-\\frac{P+D}{K}\\right) + \\alpha PC, \\label{ode2} \n\t\\end{eqnarray}\nwhere $C(t)$ accounts for the brain chemotherapy concentration to be described in more details later. Taking an average cell volume, we can easily treat the tumour mass $P+D$ as the total tumour volume, which is easier to compare with results obtained from magnetic resonance imaging (MRI), usually used in brain tumour diagnosis and follow-up observation.\n\n\\subsection{Kinetics of chemotherapy drug} \\label{subs:CT}\nThe systemic pharmacokinetic and pharmacodynamic properties of TMZ has been studied in detail in several studies \\cite{Baker,Ostermann}. Baker \\textit{et~al.}\\xspace \\cite{Baker} described the concentrations of TMZ, MTIC and AIC in plasma in detail. Ostermann \\textit{et~al.}\\xspace~\\cite{Ostermann} collected data on TMZ concentration from blood and cerebrospinal fluid (CSF) obtained via lumbar puncture in patients with malignant gliomas. \n\nHowever due to the physiological separation of brain and tumour from both blood and CSF (through blood-brain, blood-tumour, blood-CSF and CSF-tumour barriers) the amount of drug reaching the tumour differs from the amount of drug circulating in blood and CSF \\cite{Shannon,Blakeley}. \nTherefore instead of describing a~complicated mechanism with many unknown parameters based on data collected from blood or CSF, we chose a simpler dynamics based directly on the brain tissue data. Thus we will base our model on data from the study by Portnow \\textit{et~al.}\\xspace~\\cite{Portnow} who examined TMZ concentration in intracerebral microdialysis samples from peritumoural brain interstitium obtained from patients with central nervous system tumours.\n\nWith regard to chemotherapy pharmacokinetics, we will assume, as usual \\cite{Jacqmin,Ribba}, that the concentration of TMZ, $C(t)$, measured in units of days decays exponentially due to the drug clearance with a~constant rate $\\lambda$. It is consistent with the fact that TMZ has linear pharmacokinetics \\cite{Newlands}. Thus, we have to complement Eqs.~\\eqref{ode} with \n\\begin{equation} \\label{ode3}\n\t\t\\frac{\\mathrm{d} C}{\\mathrm{d} t} = -\\lambda C. \n\t\\end{equation}\n\\end{subequations}\nTMZ~reaches a~maximal drug concentration in the brain about two hours after administration~\\cite{Hammond,Portnow,Baker}, what is very short in comparison to the time scale of tumour evolution in the model (of the order of years). Thus, we may treat the whole time of oral drug administration, absorption and transport to the brain as a~discontinuous change in the drug concentration that occurs at given administration times. Such a formulation of the problem enables also the following mathematical analysis and estimations.\n\nChemotherapy will consist of a~sequence of doses $d_1,d_2,\\ldots,d_n$ given at times \n$t_1 < t_2 <\\ldots < t_n$. \nWe assume that initially (at time $t=t_0$, taken to be the start of the tumour observation) the tumour has a~certain mass $P_0=P(t_0)$ and there are neither damaged cells, nor chemotherapy drugs within the tumour, thus \\mbox{$D(t_0) = 0, \\ C(t_0) = 0.$}\nThus for $t\\in (0, t_1)$ solving Eqs.~\\eqref{ode} with initial conditions given above leads to \n\\begin{equation} \\label{solP}\nP(t) = \\frac{ KP_0 \\e^{\\rho t}}{K - P_0 \\left(1-\\e^{\\rho t} \\right)}, \\quad D(t) = 0, \\quad C(t)=0.\n\\end{equation}\nFor $t\\in (t_j, t_{j+1})$, $j=1,2,3,\\ldots,n$ with $n$ being the total number of doses, the values of $P$, $D$ and $C$ change according to Eqs.~\\eqref{ode}, while at $t=t_j$ subsequent doses of the drug are administrated which we model as impulses, so that \n\\begin{equation} \\label{impulses}\n P(t_j)=P(t_j^-), \\quad \n\t D(t_j)=D(t_j^-), \\quad \n\t C(t_j)=C(t_j^-)+C_j, \n\\end{equation}\nwhere $f(t_j^-) = \\lim_{t\\to t_j^-} f(t)$ and $C_{j}$ is the fraction of the dose $d_{j}$ which reaches the tumour tissue, accounting for drug loss during transport to the brain.\n\nThe interval between doses (typically 1 day) will be chosen to be larger than the time of whole dose elimination (reported to be around 7h \\cite{Agarwala}) and the typical damage repair times (of the order of a~few hours \\cite{VanderKogel}), so that one dose will not alter the effect of the next one. \n\nThe asymptotic behaviour of model~\\eqref{ode} under the effect of a~finite number of doses is easy to obtain. When no drug is given for time $t \\ge t_n$ (with $n$ being the index of the last dosis), then \\mbox{$C(t)=C(t_n) \\exp(-\\lambda (t-t_n))\\to 0$} as $t\\to+\\infty$. As $C \\to \\infty$ and $P$ is bounded, then $\\alpha P C \\to 0$ for $t \\to +\\infty.$ As a consequence $D'(t)<0$ for sufficiently large time. \nThen it is easily seen that $D(t)\\to 0$ and $P(t)\\to K$ as $t\\to+\\infty$. This behaviour is not surprising and can be interpreted as patient death due to drug clearance and tumour regrowth. Patient death usually occurs when the tumour reaches a~critical size called the fatal tumour burden considered to be in high-grade glioma models to be around 6 cm in diameter \\cite{Swanson2008,Woodward}.\n\nIt is important to emphasise that model \\eqref{ode} intends to describe the effect of first-line chemotherapy, since after the treatment resistant phenotypes arise leading to the acquisition of drug resistance. Thus a~detailed analysis of second-line chemotherapy would require the introduction of more phenotypes in the model and is beyond the scope of this research.\n\n\\subsection{Parameter estimation} \\label{sec:parameters}\nTo work with Eqs. \\eqref{ode} we need to provide realistic values for the model parameters. \n\nThe saturation coefficient $K$ for LGG growth will be set to the volume of a~sphere of diameter $10$~cm reported to be the maximal mean tumour diameter observed in LGG patients \\cite{Ricard}. In fact, most patients die when the tumour diameter is about 6 cm in size as discussed before \\cite{Swanson2008,Woodward}.\n\nWe can estimate the rate of drug decay $\\lambda$ using values of TMZ~half-life clearance time $t_{1\/2}$. From the definition of $t_{1\/2}$ and assuming exponential decay as in Eqs.\\eqref{ode} we have\n$$\\frac{1}{2}=\\e^{\\textstyle -\\lambda t_{1\/2}}.$$\n\nTo account also for the drug loss during transport to the brain we calculate value $C_{j}$ of the maximal dose $d_{j}$ reaching the tumour as \n\\begin{equation} \\label{dose}\nC_j = \\beta \\cdot d_{j} \\cdot b,\n\\end{equation}\nwhere $\\beta$ is the fraction of TMZ~getting to 1ml of brain interstitial fluid (from a~unit dose) and $b$~is a~surface of a~patient body with $j \\in \\{1,\\ldots, n\\}$ and $n$~being the total number of doses $d$ administered. Then $C_{j}$ can be interpreted as an effective dose per fraction. \n\nThe most typical chemotherapy schedule consists of cycles of 28 days with five TMZ~oral doses on days 1 to 5 followed by a~break of 23~days. Standard dose per day is 150 mg per m$^2$ of patient body surface, which is usually around 1.6~m$^2$ for women and 1.9 m$^2$ for men \\cite{Mosteller} with an average of 1.7~m$^2$~\\cite{Sparreboom}. \nThen in the case of the standard chemotherapy scheme we will fix $d_{j}=d=150$mg\/m$^2$ and $C_{j}=C_0=\\beta \\cdot d \\cdot b.$\n\nThe parameter $\\beta$ can be calculated using the value of maximal TMZ~concentration $C_{max}$ for a~dose of 150 mg\/m$^2$ taken from the literature \\cite{Hammond,Portnow}. Assuming that time to reach peak drug concentration in the brain is negligible (equals $0.85-2$h) in comparison to the time scale of the model, we set the initial drug concentration $C_{0}$ in the moment of its administration to the value $C_{max}.$ \n\nA~summary of the biological parameter values is presented in Table~\\ref{table-parameters}. \n\\begin{table}[ht] \\small \n\t\\begin{center}\n\t\t\\caption{Biological parameters describing TMZ concentration in brain.}\n\t\t\\begin{tabular}{l l l c}\n\t\t\t\\hline\n\t\t\tParameter & Description & Value, references & \\\\ \n\t\t\t\\hline\n\t\t\t$t_{1\/2}$ & TMZ~half-life clearance time & $\\simeq 2$h & \\cite{Hammond} \\\\\n\t\t\t$C_{max}$ & mean peak TMZ~concentration & $0.6 \\mu$g\/ml & \\cite{Portnow} \\\\\n\t\t\t& in brain interstitium & &\\\\\n\t\t\t$\\lambda$ & rate of decay of TMZ~& 0.3466\/h & Calculated \\\\\n\t\t\t& & & from \\cite{Hammond,Portnow} \\\\\n\t\t\t$\\beta$ & fraction of TMZ~getting & $2.1\\cdot 10^{-6}$\/ml (m)& Estimated\\\\ \n\t\t\t& to brain interstitium & $2.5 \\cdot 10^{-6}$\/ml (w)& from \\cite{Hammond,Portnow} \\\\ \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\label{table-parameters}\n\t\\end{center}\n\\end{table} \n\n\\section{Numerical results}\n\\subsection{Model fitting to patient data} \\label{sec:model fitting}\nTo test if our simple model given by Eqs.~\\eqref{ode} is able to reflect the dynamics of LGG response to chemotherapy, we have used the model to describe volumetric longitudinal data of patients followed at the Bern University Hospital between 1990 and 2013. In this study we selected data on 18 patients who had been treated with TMZ~out of a total number of 82 LGGs patients, see Table~\\ref{tab:patients data}. Radiological glioma growth was quantified by manual measurements of tumour diameters on successive MRIs. The three largest tumour diameters ($D_1,\\ D_2, \\ D_3$) according to three reference orthogonal planes (axial, coronal and sagittal planes) have been measured and tumour volumes have been estimated using the ellipsoidal approximation: $V=(D_1 \\cdot D_2 \\cdot D_3)\/2$, following the standard clinical practice \\cite{Pallud,Mandonnet}. \n\nThe inclusion criteria for patients for the purpose of model fitting in this study included: (i) biopsy\/surgery confirmed LGG (astrocytoma, oligoastrocytoma or oligodendroglioma), (ii) availability of at least 2 MRIs before the onset of TMZ~treatment, (iii) no other treatment given in the period of study and (iv) availability of at least 4 MRIs after TMZ~onset with at least one after the end of the chemotherapy. 7 patients satisfied these criteria. All patients in this group received more than 4~TMZ~cycles and the mean duration of TMZ~treatment was 6.26 months. \n\n\\begin{table}[ht] \\small \n\\begin{center}\n\\caption{Characteristics of patients treated with TMZ}\n\\begin{tabular}{l c c}\n\\hline\nAge at diagnosis, mean (st.~deviation), yr & 47.19 (7.54) \\\\\nSex, M\/F & 14\/4 \\\\\n\\emph{Histology at diagnosis} & \\\\\n\\hspace{0.5cm} Oligodendroglioma & 7 \\\\\n\\hspace{0.5cm} Oligoastrocytoma & 9 \\\\\n\\hspace{0.5cm} Astrocytoma & 1 \\\\\n\\hspace{0.5cm} Unknown & 1 \\\\\n\\emph{Type of surgery} & \\\\\n\\hspace{0.5cm} Biopsy & 10 \\\\\n\\hspace{0.5cm} Resection & 9 \\\\\n\\emph{Radiotherapy} & 8 \\\\\n\\emph{Chemotheraphy (CT)} & 18 \\\\\n\\hspace{0.5cm} Age at CT onset, mean (st.~deviation), yr & 51.8 (8.35) \\\\\n\\hspace{0.5cm} Time from surgery to CT, mean (st.~deviation), yr & 3.7 (4) \\\\ \n\\hspace{0.5cm} Second-line CT & 8 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:patients data}\n\\end{center}\n\\end{table}\n\nThe rate of tumour cell proliferation $\\rho,$~the coefficient $k$ describing the delay in damaged cell death and the parameter of TMZ-cell kill strength $\\alpha$ were considered to be tumour-specific and fitted for each patient. Thus only three parameters are unknown and the others are taken as in Table \\ref{table-parameters}.\n \nTo estimate the parameter $\\rho$ we used patient data before the start of TMZ~administration since it is the only relevant parameter during that time, see Eq.~\\eqref{solP}. The initial tumour volume $P_0$in this equation was fixed to be the volume from first MRI~done after surgery. Then, having obtained the value of $\\rho$, MRI~data after the onset of chemotherapy was used to estimate parameters $\\alpha$~and $k$ in Eqs.~\\eqref{ode}. Model fitting was done using a~weighted least squares method. To simulate Eqs.~\\eqref{ode}, we have used the standard Matlab ODE solver based on the Runge-Kutta 4th-order method.\n\nFigs.~\\ref{fig:proliferacja}, \\ref{fig:tTTP} show both the real tumour volume data obtained from the MRIs (circles) together with the best fit obtained with Eqs.~\\eqref{ode} (solid line). The model dynamics fit the real volumetric tumour evolution well, showing an impressive agreement with a~minimal number of parameters for patients with delayed response to chemotherapy. The minimal value of the fitted proliferation rate for some patients is one order of magnitude smaller than values $(1-5) \\cdot 10^{-3}$ day$^{-1}$ observed in other studies \\cite{Gerin,Victor,Victor3} as in these studies the model for tumour growth also considered a diffusive term. Some of the tumours were relatively large, however no formation of neoangiogenesis or necrotic core was observed. See also Supplementary material for all patients data and estimated parameters.\n\n\\begin{figure}[h!p] \n\\begin{center}\n\\includegraphics[width=0.525\\textwidth]{108} \n\\includegraphics[width=0.525\\textwidth]{159} \n\\includegraphics[width=0.525\\textwidth]{25} \n\\caption{Tumour volume evolution for three selected patients treated with TMZ. The beginning and the end of TMZ~treatment are marked with vertical dashed lines. There are shown the volumes calculated from MRIs (circles) and from the fitted mathematical model (solid lines). The number of TMZ~cycles and the values of parameters were different for each patient.\n(top) Patient treated with 5 TMZ~cycles, $\\alpha=0.971918$ml\/$\\mu$g\/day, $\\rho = 0.001761$\/day, $ \\ k = 0.555867.$\n(center) Patient treated with 11 TMZ~cycles, \\mbox{$\\alpha = 0.279911$ml\/$\\mu$g\/day,} $\\rho = 0.000136$\/day, $ \\ k = 0.025617.$ \n(bottom) Patient treated with 4 TMZ~cycles, $\\alpha=1.387798$ml\/$\\mu$g\/day, \\mbox{$\\rho = 0.002416$\/day,} $ \\ k = 0.272291.$}\n\\label{fig:proliferacja}\n\\end{center}\n\\end{figure}\n\n\\subsection{Tumours with faster response have worse prognosis} \\label{sec:relations}\n\\begin{figure}[h!t]\n\\begin{center}\n\\includegraphics[trim={0.5cm 0cm 1.8cm 0.9cm},clip=true,width=0.45\\textwidth]{fig_rho}\n\\includegraphics[trim={0.5cm 0cm 1.8cm 0.9cm},clip=true,width=0.45\\textwidth]{fig_alpha}\n\\caption{Tumour volume evolution for different values of parameters. Virtual patients were treated with 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.\\ref{sec:parameters}) with $k=0.5.$ (left)~Parameter $\\alpha$ was fixed to value 0.7ml\/$\\mu$g\/day, $\\rho_1=0.004\/$day$, \\ \\rho_2=0.002\/$day. (right) Parameter $\\rho$ was fixed to value 0.0025\/day, \n$\\alpha_1=0.4$ml\/$\\mu$g\/day, $\\alpha_2=0.8$ml\/$\\mu$g\/day. The horizontal dotted lines correspond to tumour sizes equal to the fatal tumour burden.}\n\\label{fig_rho,alpha}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!t]\n\\centering\n\\includegraphics[trim={0.865cm 0cm 1.73cm 0.9cm},clip=true,width=0.49\\textwidth]{alpha,times} \n\\includegraphics[trim={0.865cm 0cm 1.73cm 0.9cm},clip=true,width=0.49\\textwidth]{rho,times}\n\\caption{Characteristic times of tumour response for different proliferation rates $\\rho$ and different levels of TMZ~cell kill strength $\\alpha$.~We considered 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.~\\ref{sec:parameters}) for virtual patients with LGG of initial volume 40 cm$^3$. (left) Results for $\\rho = 0.0008$\/day, $k=0.3$ and $\\alpha \\in [0.1,1]$ml\/$\\mu$g\/day.~(right) Results for $\\alpha=0.8$ml\/$\\mu$g\/day, $k=0.3$ and $\\rho \\in [0.7,8] \\times 10^{-3}$\/day.}\n\\label{fig:relacje}\n\\end{figure}\n\n\\begin{figure}[h!t]\n\\centering\n\\includegraphics[trim={1cm 0.6cm 1.03cm 1.4cm},clip=true,width=0.47\\textwidth]{surfOS} \n\\includegraphics[trim={0.63cm 0.6cm 0.95cm 1.4cm},clip=true,width=0.47\\textwidth]{surfTRP,GD}\n\\caption{Characteristic times of tumour response for different proliferation rates $\\rho$ and different levels of TMZ~cell kill strength $\\alpha$.~We considered 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.~\\ref{sec:parameters}) with $k=0.3.$\nValues of time to radiological progression (left), growth delay and overall survival (right) are shown for virtual patients with LGG of an initial volume of 40~cm$^3$.\n}\n\\label{fig:surf}\n\\end{figure}\n\n\\begin{figure}[h!t]\n\\centering \n\\includegraphics[trim={0.83cm 0cm 1.8cm 1.02cm},clip=true,width=0.5\\textwidth]{TRP,OS}\n\\caption{Time to radiological response and overall survival of LGGs patients treated with TMZ. Patients were divided into two groups: in the first, denoted by circles, patients were treated with TMZ~for less than 10 months (average: 5.72, st.~deviation: 3.8), in the second group, denoted by triangles, patients were treated with TMZ~for more than 10 months (average: 13.65, st.~deviation: 3.14).}\n\\label{fig:patients-PFS,OS}\n\\end{figure}\nWe have also studied how the tumour response depends on parameters fitted, see Figs. \\ref{fig:relacje}, \\ref{fig:surf}. \nWe will denote by ``time to radiological progression'' (TRP) the time when the tumour attains its minimum volume after the chemotherapy onset and starts regrowing. We will refer to ``growth delay'' as the time for which the tumour volume equals the initial one when regrowing after the therapy, see~\\cite{Victor}. We will refer to ``early response'' when TRP is attained shortly after the end of chemotherapy and ``no response'' when there was no decrease in tumour volume detectable by radiologist. In case of frequent MRIs these times can be easily obtained from model simulations and compared with the values obtained from patient's MRI~volumetry. It can be estimated with an error being the time between two subsequent MRIs. We have also computed the overall survival (OS) as the time until reaching the fatal tumour burden defined in Sec.\\ref{subs:CT}. We have considered only the cases of virtual tumours whose volumes decreased below the volume at TMZ onset. Note that for the purpose of analysis of response to TMZ, OS is computed for virtual tumours responding to TMZ and without any other treatment in the following course of disease, thus in general could be overestimated and thus should not be compared to the values of real-patients overall survival. \n\nAfter performing many simulations for different initial values and chemotherapy schemes, we conclude that both larger proliferation rates $\\rho$ and smaller TMZ~cell kill strengths $\\alpha$, lead to an earlier response to TMZ~treatment. A~more systematic study is shown in Fig.~\\ref{fig_rho,alpha} for two specific parameter sets, providing representative examples. Virtual patients who responded earlier to TMZ~(had smaller TRP) had a~faster regrowth and reached the fatal tumour burden earlier. Thus, a shorter TRP is an indicator of worse prognosis. \n\nAre these model features also present in the patient's data? Fig.~\\ref{fig:patients-PFS,OS} shows how the overall survival rate correlates with the time to radiological progression. For the purpose of this analysis we also included patients with only one MRI before treatment with TMZ. We excluded (i) two patients in which only two MRIs were available after the end of chemotherapy showing no tumour regrowth and (ii) one patient treated with TMZ~for only 1.5 month showing no response. For each patient TRP was estimated as the time to the MRI in which tumour volume was the smallest after the onset of treatment. Due to the differing duration of TMZ treatments we divided patients into a group receiving less than 10 TMZ cycles and those receiving 10 or more cycles.\n\nThe Spearman rank correlation coefficient between TRP and OS equals 1 for data of patients treated with TMZ~for less than 10 months and 0.9047619 for those treated with TMZ~for longer time. The exact Spearman coefficient test significance levels equal 0.008333 and 0.002282 for right-tailed tests for group of patients treated with less and more than 10 cycles of TMZ, respectively. This result indicates a~positive correlation between TRP and OS. The significance levels were calculated using R. \nData on overall survival is right-censored, however the results suggest that the early regrowth of the tumour after chemotherapy is related to its aggressiveness. Despite therapies used after progression, those tumours that responded faster to TMZ~treatment progressed faster, suggesting either larger proliferation potential and\/or smaller TMZ~cell kill strength. \n\n\\section{Results (II): Analytical estimates of tumour response} \n\n\\subsection{Survival fraction}\nUp to now our analysis has been based on numerical simulations of Eqs. \\eqref{ode}. We can calculate the fraction of tumour cells eliminated by a~single dose of chemotherapy, which in the context of radiotherapy is usually referred to as \\emph{survival fraction}. To do so, we will assume that the time of drug absorption, distribution and elimination from the human body is much shorter than the doubling time of the tumour cell population, what is true for LGGs. Therefore, focusing on short-term effects of the drug we may neglect the term describing tumour proliferation. Consequently the size of damaged cells population remains zero and we consider instead the~simplified model\n\\begin{subequations}\n\\begin{eqnarray}\n\t\t\\frac{d P}{d t} & = - \\alpha P C, \\\\ \n\t\t\\frac{d C}{d t} & = -\\lambda C.\n\\end{eqnarray}\n\t\\end{subequations}\nFor time before the second drug administration $t_1\\le t0$, thus showing a radiologically visible decrease in total volume. Thus, in our approach, in which only first-line chemotherapy is described and resistant cells do not arise, tumour progression will occur after the end of chemotherapy ($s_{\\textrm{TRP}} \\gg s_{n}$) provided tumour growth is slow as happens in the case of LGGs. Thus we will try to obtain explicit formulae approximating $x$ and $y$~for $s \\gg s_{n}.$\n\nWe will consider tumours with initial sizes significantly smaller than the carrying capacity, then the Eq. \\eqref{ode_scaled} takes the simpler form\n\\begin{subequations}\\label{ode_final}\n\\begin{eqnarray} \n\t\t\\frac{d x}{d s} & = & x - xz, \\\\ \n\t\t\\frac{d y}{d s} & = & -\\frac{1}{k}y + xz, \\\\ \n\t\t\\frac{d z}{d s} & = & -\\mu z,\n\t\\end{eqnarray}\n\\end{subequations}\nwith initial conditions: $ x(0)=x_0, \\ y(0)=0, \\ z(0)=z_0.$ Then for rescaled time TRP we have $x'(s_{\\textrm{TRP}}) + y'(s_{\\textrm{TRP}})=0,$ therefore \n\\begin{equation} \\label{formula s_ttp}\nx(s_{\\textrm{TRP}})=\\frac{1}{k}y(s_{\\textrm{TRP}}).\n\\end{equation}\nEqs.~\\eqref{ode_final} are a~set of ordinary differential equations with impulses, the functions $x$ and $y$ being continuous, and $z$~being discontinuous at times $s_{j}$ for $j \\in \\{2,\\ldots,n\\}$ as \n\\begin{equation}\n z(s_j)=z(s_j^-)+z_0.\n \\end{equation}\nSince dose clearance time is about two hours we may assume that each dose is cleared in one day, then, in the rescaled units\n\\mbox{$z((s_{j}+\\rho)^{-}) \\approx 0$} for \\mbox{$j \\in \\{1,\\ldots,n\\}.$} Therefore, we can approximate\n\\begin{equation} \\label{z approximation}\nz(s) \\approx \n\\left\\{\n\t\\begin{aligned}\n\t\t& z_0 \\e^{\\textstyle -\\mu(s-s_{j})} && s \\in (s_{j}, s_{j}+\\rho), \\\\ \n\t\t& 0 && \\mbox{for other $s$},\n\t\\end{aligned}\n\t\\right.\n\\end{equation} \nwhere $j=\\displaystyle \\operatorname*{arg\\,max}_{i \\in \\{1,\\ldots, n\\}} \\left\\{ s_{i} \\leq s \\right\\}.$\nLet us define\n\\begin{subequations}\n\\begin{align}\n& w(s) =\\int_{0}^{s} z(t) \\mathrm{d} t, \\\\\n& w_0 =w(\\rho)=\\int_{0}^{\\rho} z(t) \\mathrm{d} t= \\frac{z_0}{\\mu}\\left(1- \\e^{\\textstyle -\\mu \\rho} \\right).\n\\end{align}\n\\end{subequations}\nWe should emphasise that for $s> s_2,$ $w(s)\\neq z_0\\left(1- \\e^{\\textstyle -\\mu s} \\right)\/\\mu$ due to the administration of the next drug dose.\nFurthermore, from the definition of function $z$ we have \n\\begin{equation} \\label{periodicity}\n\\begin{aligned}\nw(s) & = \\int_{0}^{s_j} z(t) \\mathrm{d} t + \\int_{s_{j}}^{s} z(t) \\mathrm{d} t = w(s_{j}) + \\int_{0}^{s-s_{j}} z(t) \\mathrm{d} t \n\\approx (j-1)w_{0}+ w(s-s_{j}) \\\\\n& \\approx \\left\\{ \\begin{array}{ll}\n(j-1)w_{0}+ \\frac{z_0}{\\mu}\\left(1- \\e^{\\textstyle -\\mu (s-s_{j})} \\right) & s-s_{j} \\leq \\rho,\\\\\njw_0 & \\textrm{otherwise,}\n\\end{array} \\right.\n\\end{aligned}\n\\end{equation}\nwhere $j=\\displaystyle \\operatorname*{arg\\,max}_{i \\in \\{1,\\ldots, n\\}} \\left\\{ s \\geq s_{i} \\right\\}.$\nHence for $s>s_{n}+\\rho$ the formulae for rescaled proliferating and the damaged part of tumour take the~form\n\\begin{subequations}\\label{xy}\n\\begin{eqnarray} \\label{x}\nx(s) &= & x_0 \\e^{\\textstyle s-w(s)}=x_0 \\e ^{\\textstyle s - n w_0 }, \\\\\n\\label{y}\ny(s) & = & \\int_{0}^{s} \\e^{\\textstyle -\\frac{s-t}{k}} x(t) z(t) \\mathrm{d} t. \n\\end{eqnarray}\n\\end{subequations}\nWe look for $s_{\\textrm{TRP}}$ which fulfils condition \\eqref{formula s_ttp}. Using Eqs. \\eqref{xy} we have \n\\begin{equation}\nk\\e^{ \\textstyle \\tilde{k}s_{\\textrm{TRP}} -n w_0} = \\int_{0}^{s_{\\textrm{TRP}}} \\e^{\\textstyle \\tilde{k}t - w(t) } z(t) \\mathrm{d} t,\n\\end{equation}\n\\begin{equation} \\label{sTTP1}\ns_{\\textrm{TRP}}=\\frac{1}{\\tilde{k}}\\Bigg[ nw_0 + \\ln \\left( \\frac{1}{k} \\int_{0}^{s_{\\textrm{TRP}}} \\e^{\\textstyle \\tilde{k}t - w(t) } z(t) \\mathrm{d} t \\right) \\Bigg],\n\\end{equation}\nwhere $\\tilde{k}=1+1\/k$. \nAs a~result of approximations \\eqref{z approximation} and \\eqref{periodicity} we conclude that for \\mbox{$s\\geq s_{n}+ \\rho$} \n\\begin{align} \\label{rhs}\n& \\displaystyle \\int_{0}^{s} \\e^{\\textstyle \\tilde{k}t - w(t) } z(t) \\mathrm{d} t =\n\\displaystyle \\sum_{j=1}^{n} z_0 \\displaystyle \\int_{s_{j}}^{s_{j}+\\rho} \\e^{\\textstyle \\tilde{k}t - w(t)} \\e^{\\textstyle -\\mu(t-s_{j})} \\mathrm{d} t \n\\nonumber \\\\\n & = \\displaystyle z_0 \\sum_{j=1}^{n} \\e^{\\textstyle -(j-1)w_{0} +\\tilde{k}s_{j}} \\displaystyle \\int_{s_{j}}^{s_{j}+\\rho} \\e^{\\textstyle \n(\\tilde{k} -\\mu)(t-s_{j}) + \\frac{\\textstyle z_0}{\\textstyle \\mu}\\left(\\e^{\\textstyle -\\mu (t-s_{j})} -1 \\right)} \\mathrm{d} t \n\\nonumber \\\\\n& = z_0 \\left( \\sum_{j=1}^{n} \\e^{\\textstyle -(j-1)w_{0} +\\tilde{k}s_{j}} \\right) \\int_{0}^{\\rho}\n\\e^{\\textstyle \n(\\tilde{k} -\\mu)t + \\frac{\\textstyle z_0}{\\textstyle \\mu}\\left( \\e^{\\textstyle -\\mu t}-1\\right)} \\mathrm{d} t. \n\\end{align} \nUsing Taylor expansion of an exponential function for $t<1\/\\mu$ we approximate \n\\[\n\t\\e^{-\\mu t}-1 \\approx \\begin{cases}\n\t\t\t-\\mu t & 0\\le t<1\/\\mu, \\\\\n\t\t\t-1 & t\\ge 1\/\\mu,\n\t\t\\end{cases}\n\\]\nand obtain\n\\begin{flalign} \\label{integral}\n& \\displaystyle \\int_{0}^{\\rho}\n\\e^{\\textstyle (\\tilde{k} -\\mu)t + \\frac{\\textstyle z_0}{\\textstyle \\mu}\\left( \\e^{\\textstyle -\\mu t}-1\\right)} \\mathrm{d} t \\approx\n\\displaystyle \\int_{0}^{\\frac{1}{\\mu}} \\e^{\\textstyle (\\tilde{k} -\\mu)t - z_0 t} \\mathrm{d} t \n+ \\int_{\\frac{1}{\\mu}}^{\\rho} \\e^{\\textstyle (\\tilde{k} -\\mu)t - \\frac{z_0}{\\mu}} \\mathrm{d} t \\nonumber \\\\\n&= \\frac{1}{\\tilde{k}-\\mu -z_{0}}\n\\left( \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -1 \\right) +\n\\frac{1}{\\tilde{k}-\\mu} \\left( \\e^{\\textstyle (\\tilde{k}-\\mu)\\rho -\\frac{z_0}{\\mu}}-\\e^{\\textstyle\\frac{\\tilde{k}-\\mu-z_0}{\\mu}} \\right) \\nonumber \\\\\n&= \\frac{1}{\\left(\\tilde{k}-\\mu -z_{0}\\right)\\left(\\tilde{k}-\\mu\\right)} \\Bigg[ z_0 \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -\\tilde{k} +\\mu + \n\\left(\\tilde{k} -\\mu -z_0\\right) \\e^{\\textstyle \\left(\\tilde{k}-\\mu\\right)\\rho - \\frac{z_0}{\\mu}} \n\\Bigg].\n\\end{flalign}\nTo compute the sum term in Eq.~\\eqref{rhs} we need the relation between dose indexes $j$~and the times of their administration $s_{j}.$ Taking into consideration assumptions from Sec. \\ref{sec:estimates} and Eqs.~(\\ref{dosingtimes}-\\ref{dosingindexes}) we get\n$ s_{j}= \\big[(i-1)r+ mT\\big]\\rho, $\nwhere $m\\in \\{0, n\/p -1\\} $ is the~number of completed chemotherapy cycles before dose $s_{j},$ $j \\in \\{1,\\ldots, n\\}$ and $i \\in \\{1, \\ldots, p\\}$ is an index of dose within the~TMZ~cycle.\nAs a~result we obtain\n\\begin{multline} \n\\sum_{j=1}^{n} \\exp \\left(-(j-1)w_{0} +\\tilde{k}s_{j}\\right) \\\\\n= \\sum_{m=0}^{ n\/p-1} \\sum_{i=1}^{p}\n\\exp \\left( m \\left(-pw_{0} +\\tilde{k}\\rho T \\right)+ (i-1)(-w_{0} +\\tilde{k}\\rho r) \\right)\n \\\\\n = \\frac{1-\\e^{\\textstyle \\left(-p w_{0} +\\tilde{k}\\rho T \\right) \\frac{n}{p} }}{1-\\e^{\\textstyle -pw_{0} + \\tilde{k}\\rho T}} \n\\cdot \\frac{1-\\e^{\\textstyle \\left(-w_{0} +\\tilde{k}\\rho r \\right)p}}{1-\\e^{\\textstyle -w_{0} +\\tilde{k}\\rho r}},\n \\end{multline}\nwhere $n$ is the total number of doses and $n\/p$ is the total number of chemotherapy cycles as previously stated. \nThen using Eqs. \\eqref{sTTP1}, \\eqref{rhs} and \\eqref{integral} we get \n\\begin{align}\n& s_{\\textrm{TRP}} = \n\\frac{n w_0 }{\\tilde{k}}\n+ \\frac{1}{\\tilde{k}}\\ln \n\\left\\{ z_0 \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -\\tilde{k} +\\mu + \n\\left(\\tilde{k} -\\mu -z_0\\right) \\e^{\\textstyle \\left(\\tilde{k}-\\mu\\right)\\rho - \\frac{ z_0}{\\mu}} \n\\right\\} \\nonumber \\\\\n& + \\frac{1}{\\tilde{k}}\\ln \\left\\{\n\\frac{\\textstyle z_0\\left( 1-\\e^{\\textstyle \\left(-pw_{0} +\\tilde{k}\\rho T \\right) \\frac{n}{p} }\\right) \n\\left(1-\\e^{\\textstyle \\left(-w_{0} +\\tilde{k}\\rho r\\right)p}\\right)}\n{\\textstyle k\\left( 1-\\e^{\\textstyle -pw_{0} + \\tilde{k}\\rho T} \\right) \n\\left(1-\\e^{\\textstyle -w_{0} +\\tilde{k}\\rho r}\\right) \n\\left(\\tilde{k}-\\mu -z_{0} \\right)\\left(\\tilde{k}-\\mu\\right)} \\right\\} .\n\\end{align}\nIn terms of the initial time scale, the time to tumour progression can be estimated as \n$t_{\\textrm{TRP}}=s_{\\textrm{TRP}}\/ \\rho,$ giving the final result\n\\begin{align} \\label{formula}\n& t_{\\textrm{TRP}} = \n\\frac{n w_0 }{\\tilde{k}\\rho}\n+ \\frac{1}{\\tilde{k}\\rho}\\ln \n\\left\\{ z_0 \\e^{\\textstyle \\frac{\\tilde{k}-\\mu-z_0}{\\mu}} -\\tilde{k} +\\mu + \n\\left(\\tilde{k} -\\mu -z_0\\right) \\e^{\\textstyle \\left(\\tilde{k}-\\mu\\right)\\rho - \\frac{ z_0}{\\mu}} \n\\right\\} \\nonumber \\\\\n& + \\frac{1}{\\tilde{k}\\rho}\\ln \\left\\{\n\\frac{\\textstyle z_0\\left( 1-\\e^{\\textstyle \\left(-pw_{0} +\\tilde{k}\\rho T \\right) \\frac{n}{p} }\\right) \n\\left(1-\\e^{\\textstyle \\left(-w_{0} +\\tilde{k}\\rho r\\right)p}\\right)}\n{\\textstyle k\\left( 1-\\e^{\\textstyle -pw_{0} + \\tilde{k}\\rho T} \\right) \n\\left(1-\\e^{\\textstyle -w_{0} +\\tilde{k}\\rho r}\\right) \n\\left(\\tilde{k}-\\mu -z_{0} \\right)\\left(\\tilde{k}-\\mu\\right)} \\right\\}.\n\\end{align}\nEq. \\eqref{formula} gives the time to radiological progression as a~function of parameters with relevant biological and\/or therapeutical meaning.\nSince TRP is a~metric of practical relevance, it is very interesting that it is possible to estimate its value analytically. \n\n\\subsection{Validation}\nEq. \\eqref{formula} has been obtained via a~number of approximations and thus it is relevant to compare its predictions with the results of the original equation \\eqref{ode} and real patient data.\n\nTo do the latter we first fix the treatment parameters to match those routinely used for TMZ therapeutic schedules. Since TMZ~is given on 5 consecutive days in cycles consisting of 28 days, we get $p=5$, $T=28$, $r=1$. Taking dose per fraction to be 150~mg\/m$^2$ and the other parameters as in Sec.~\\ref{sec:estimates} we can then estimate the time of progression for the individual patients studied previously (accounting for the number of cycles received by each patient). \n\nFig.~\\ref{fig:tTTP}~shows how well formula \\eqref{formula} estimates the response to chemotherapy for three patients chosen from our database. The task of comparing simulation results with real patients MRI data requires a lot of caution. In particular, we need to take into account the limitations of calculations of tumour volume using the method of three largest diameters. The method is only an approximation of real tumour volume and its accuracy is limited by slice thickness, changes in head position \\cite{Schmitt} or even by perception of medical doctor who calculate these diameters. However in this case one can conclude that we have obtained a satisfactory result in fit. In the future we hope that MRI data will be analysed through automatic segmentation, \\textit{e.g.}~with algorithm suggested by Porz \\textit{et~al.}\\xspace \\cite{Porz} and the real tumour volume will be calculated more accurately. \n\nFig.~\\ref{fig:diff} presents relative differences between TRP from the estimated formula and simulations for different sets of parameters, suggesting a very good approximation.\n\n\\begin{figure}[h!p]\n\\centering\n\\includegraphics[width=0.525\\textwidth]{10_TRP}\n\\includegraphics[width=0.525\\textwidth]{57_TRP}\n\\includegraphics[width=0.525\\textwidth]{151_TRP}\n\\caption{Tumour volume evolution for three patients treated with TMZ. \nVertical dashed lines mark the start and the end of TMZ~treatment. \nCircles denote the volumes obtained from MRIs and solid lines the results of the best fit using Eqs.~\\eqref{ode}. \n(top) Woman treated with 6~TMZ cycles, $\\alpha=0.199094$ml\/$\\mu$g\/day, $\\rho = 0.00022\/$day, $k = 0.075644.$\n(center) Man treated with 11~TMZ~cycles, $\\alpha = 0.17367$ml\/$\\mu$g\/day, $\\rho = 0.000338\/$day, $k = 0.019279.$ \n(bottom) Man treated with 4~TMZ~cycles, $\\alpha=0.236439$ml\/$\\mu$g\/day, $\\rho = 0.000701\/$day, $k = 0.257806.$\nThe times to radiological progression computed using Eq.~\\eqref{formula} are marked with vertical dashed-dotted lines, showing a very good agreement with the data and the simulations of Eqs.~\\eqref{ode}. The relative differences between $t_{\\textrm{TRP}}$ calculated from Eq.~\\eqref{formula} and from Eqs.~\\eqref{ode} were respectively $0.042043, \\ 0.041671$ and $0.037481$ years, for the selected patients. \n}\n\\label{fig:tTTP}\n\\end{figure}\n\n\\begin{figure}[h!p]\n\t\\begin{center}\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth]{diff_k_rho0004alpha4}\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth]{diff_k_rho0008alpha8}\n\t\t\\ \\\\ \\ \\\\\t\t\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_rho_k3alpha8}\t\t\n\t\t\\includegraphics[trim={0.6cm 0.1cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_rho_k6alpha4}\n\t\t\\ \\\\ \\ \\\\\t\n\t\t\\includegraphics[trim={0.6cm 0cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_alpha_k6rho0008}\n\t\t\\includegraphics[trim={0.6cm 0cm 1.4cm 0.9cm},clip=true,width=0.49\\textwidth, height=0.2\\textheight]{diff_alpha_k3rho0004}\n\t\t\\caption{Relative percentage difference between time to radiological progression estimated from simulations of model~\\eqref{ode} and formula \\eqref{formula}. We considered 12 cycles of TMZ~as in the standard fractionation scheme (see Sec.~\\ref{sec:parameters}) for virtual patients with LGG having an initial volume of 40 cm$^3$. Results for $\\rho = 0.0004$\/day, $\\alpha=0.4$ml\/$\\mu$g\/day, $k \\in [0.02, 1]$ (left) and $\\rho = 0.0008$\/day, $\\alpha=0.8$ml\/$\\mu$g\/day, $k \\in [0.02, 1].$ (center) Results for $\\alpha=0.8$ml\/$\\mu$g\/day, $k=0.3$ for $\\rho \\in [0.2,8] \\times 10^{-3}$\/day (left) and $\\alpha=0.8$ml\/$\\mu$g\/day, $k=0.3$ for $\\rho \\in [0.2,3] \\times 10^{-3}$\/day. (bottom) Results for $\\rho = 0.0008$\/day, $k=0.6$ and $\\alpha \\in [0.3,1.5]$ml\/$\\mu$g\/day (left) and $\\rho = 0.0004$\/day, $k=0.3$ and $\\alpha \\in [0.15,1.5]$ml\/$\\mu$g\/day (right).\n\t\t}\n\t\t\\label{fig:diff}\n\t\\end{center}\n\\end{figure} \n\n\\subsection{The study of tumour response for other chemotherapy protocols}\nWe have also verified that Eq.~\\eqref{formula} provides a good approximation of TRP for model \\eqref{ode} for other fractionation schemes. Fig.~\\ref{fig:tTTP1} shows some examples. \n\n\\begin{figure}[h!t]\n\\begin{center}\n\\includegraphics[trim={0.8cm 0cm 1.8cm 0.9cm},clip=true,width=0.46\\textwidth]{TRP2}\n\\includegraphics[trim={0.8cm 0cm 1.8cm 0.9cm},clip=true,width=0.46\\textwidth]{TRP5}\n\\caption{Tumour volume evolution for virtual patients simulated from Eqs.~\\eqref{ode}. Times to radiological progression estimated from Eq.~\\eqref{formula} are marked with vertical dashed-dotted lines. Values of parameters were $k =0.5$, $\\alpha = 0.4$ml\/$\\mu$g\/day and $\\rho = 0.005\/$day. The start and the end of TMZ~treatment are marked with vertical dashed lines. (left) 9 TMZ~cycles of 34 days with doses given every 2 days for a~total of 10 doses per cycle. The dose per fraction was $d$=100~mg\/m$^2.$ (right) 18~TMZ~cycles of 77 days with doses given every 7 days for a~total of 10 doses per cycle. The dose per fraction was $d$=50 mg\/m$^2.$ Relative differences between times free of progression calculated from Eq.~\\eqref{formula} and estimated from simulations were 0.026756 and 0.025303 years, respectively.}\n\\label{fig:tTTP1}\n\\end{center}\n\\end{figure} \n\n\\section{Discussion and therapeutic implications}\n\nDue to the observed clinical significance of TMZ there is an increasing interest in studying its characteristics. Up to now there have been a great deal of relevant research into the pharmacokinetic\/pharmacodynamic properties of TMZ \\cite{Baker,Hammond,Ostermann,Portnow}, its specific mechanism of action \\cite{Marchesi,Roos,Barciszewska,Agarwala} and modelling its concentration dynamics \\emph{in vitro} and \\emph{in vivo} \\cite{Zhou,Rosso,Ballesta}. However there are fewer mathematical studies of patient response to TMZ in LGGs. \nHere we intended to construct a mathematical model which would enable understanding of delayed response to chemotherapy observed in LGGs without using an excessive number of unknown parameters. Cells were assumed to grow logistically, chemotherapy drug kinetics and its effect on glioma cells was based on TMZ concentration in brain tissue \\cite{Portnow} and clinical observations \\cite{Bent1,Ricard,Chamberlain}. Note that even the authors of the very complicated model \\cite{Ballesta}, constructed for the purpose of describing pharmacokinetics and pharmacodynamis of TMZ, validated their model for human cerebral tumours on the basis of data of Portnow \\textit{et~al.}\\xspace \\cite{Portnow}.\nIt is remarkable that a~simple model such as the one presented here with essentially only three unknown parameters ($\\alpha, \\rho, k$) is able to describe the response of real patients to a~variable number of cycles of TMZ.\n\nThe model also shows a~correlation between a~short time to radiological progression and a~poor virtual patient outcome. We may conclude that time to radiological progression can be useful as a~measure of tumour aggressiveness due to its dependence on tumour-specific parameters: proliferation rate $\\rho$ and TMZ~cell kill strength $\\alpha$ (see Figs.~\\ref{fig:relacje},\\ref{fig:surf}). \nOur data on patients treated with first-line TMZ~suggests likewise that despite other therapies used in the follow-up, patients who \nhad shorter estimated TRP had worse prognosis. Such observation has been made for radiotherapy \\cite{Pallud1, Ducray}, but so far no similar analysis of response to TMZ~has been done. The velocity of tumour decrease after radiotherapy (or equivalently time of progression-free survival) is strongly associated with the risk of rapid progression and poor overall survival. Here we suggest a similar result for the response to chemotherapy, namely that short time to radiological progression results in shorter overall survival. \n\nThis outcome makes us think of the possibility of using chemo\\-therapy to probe tumours, hence providing estimates of tumour-specific parameters $\\rho$ and $\\alpha.$ We could apply a small number of cycles of TMZ~causing minimal toxicity and monitor with MRI~the tumour response to chemotherapy. In order to assure the reasonable measurement error, we would need at least two measurements before and three after TMZ onset. We predict that the time horizon would be of around 2 years from the time of the first MRI. We believe it could be feasible as even up to now there were cases when MRI was performed three times a year. Based on our database there will be no progression at this time horizon. Such a procedure can be used as a~novel way to assess tumour aggressiveness. Our mathematical model suggests that tumour which attains its minimal volume early after a short course of TMZ~treatment (has shorter TRP)~may be more aggressive, therefore in such a~case the remaining TMZ~dose has to be finished as soon as possible and other therapeutic options (further surgery if feasible or radiotherapy) should be considered. Such a concept can be supported also by the\\emph{in vitro} results of Roos \\textit{et~al.}\\xspace \\cite{Roos}, who shown that higher proliferation rates accelerate apoptosis after TMZ treatment, thus in terms of our idea, shorter TRP.\n\nThis idea resembles that described in \\cite{Victor}, but with chemotherapy instead of radiotherapy.~Although the modelling principles are similar, from the clinical point of view the use of TMZ~is a much more interesting as a way to probe a tumour than the use of radiotherapy as its side-effects are long-term and non-reversible. On the other hand, TMZ~has significantly lower and largely reversible side-effects. Moreover, radiotherapy is well known to induce changes in the MRI~images due to inflammation which may distort the analysis of the tumour response. Finally, TMZ~is easily managed because it is administered orally.\n\n\\section{Conclusions}\nWe have build a~model which is simple from the mathematical point of view, but which incorporates the basic biological features of LGG growth and the response to chemotherapy. \n\nThe model is able to describe response to TMZ with a minimal number of parameters and suggests that tumours having a shorter time to radiological response after TMZ~treatment~may be more aggressive in terms of proliferation potential.~We plan to reassess this observation using a~larger data set, if possible.~In this case we would like also to verify whether the survival curves obtained can be fitted for a~cohort of virtual patients, as done by Kirkby \\textit{et~al.}\\xspace \\cite{Kirkby2007}. \n\nMoreover, we propose a~paradigm for probing tumour with TMZ~which could be used in clinical practice. We have also found an equation giving the time free of progression as a~function of the relevant biological and therapeutic parameters. In future studies it may be helpful in designing treatment schedules giving the longest TRP possible with the additional condition for the toxicity level.\n\nIn order to address other clinically relevant issues and the biological perspective several improvements to the present model will be implemented in the near future. First it may be appropriate to include more biological details such as the potential existence of the so-called cancer initiating cells or cancer stem cells. Also incorporating different phenotypes may be relevant to describe the process of acquiring resistances to TMZ.\n\nIt would also be interesting to find the minimal doses and minimal frequency of therapy such that the solution (namely tumour mass) stays below a~given threshold. In principle, survival could be improved and chemoresistance deferred using metronomic fractionations, \\emph{i.e.} schedules consisting of many, equally spaced and generally low doses of chemotherapeutic drugs without extended rest periods (see \\textit{e.g.}~\\cite{Benzekry,Andre}).\n\nWe hope that optimized cancer treatment protocols on the basis of models such as the one presented in this paper may become in the future a~standard element of personalised medicine.\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{}{}{0pt}{\\Large\\scshape\\bfseries\\filcenter}\n\\tableofcontents\n\\titleformat\\section{}{}{0pt}{\\Large\\scshape\\bfseries\\filcenter\\thesection{} - }\n\n\\section{Introduction}\\label{intro}\n\nThe aim of this paper is to derive an error estimate for approximate solutions of the compressible barotropic Navier-Stokes equations obtained by a discretisation scheme.\nThese equations are posed on the time-space domain $Q_T=(0,T)\\times\\Omega$, where $\\Omega$ is a bounded polyhedral domain of $\\R^d$, $d=2,3$ and $T>0$, and read:\n\\begin{subequations}\n\t \\begin{align}\n\t\t& \\partial_t \\vr + \\dv ( \\vr \\bu) = 0, \\label{cont2} \\\\\n\t\t& \\partial_t (\\vr\\bu) +{\\rm div}(\\vr\\bu\\otimes\\bu) - \\mu \\Delta \\bu -(\\mu+\\lambda) \\nabla \\dv \\bu+ \\nabla_x p(\\vr) = \\bm{0}, \\label{mov2}\n\t\\end{align}\n\t\\label{pbcont}\n\\end{subequations}\nsupplemented with the initial conditions\n\\begin{equation}\\label{ci1}\n\t \\vr(0,x) = \\vr_0 (x),~\\vr\\bu(0,x) =\\vr_0\\bu_0,\n\\end{equation}\nwhere $\\varrho_0$ and $\\bu_0$ are given functions from $\\Omega$ to $\\R_+$ and $\\R^d$ respectively, and boundary conditions\n\\begin{equation}\\label{ci2}\n\t\\bu_{|(0,T)\\times\\partial \\Omega} =0.\n\\end{equation}\nIn the above equations, the unknown functions are the scalar density field $\\vr(t,x)\\ge 0$ and vector velocity field $\\bu=(u_1,\\ldots,u_d)(t,x)$, where $t \\in (0,T)$ denotes the time and $x\\in \\Omega$ is the space variable.\nThe viscosity coefficients $\\mu$ and $\\lambda$ are such that\n\\begin{equation}\\label{visc}\n\t\\mu > 0,~ \\lambda + \\frac{2}{d}\\mu \\ge 0.\n\\end{equation}\nThe pressure $p$ is a given by an equation of state, that is a function of density which satisfies\n\\begin{equation}\\label{hypp}\n\tp \\in C([0,\\infty)) \\cap C^1(0,\\infty),\\;p(0)=0,\\; p'(\\vr)>0.\n\\end{equation}\nIn addition to (\\ref{hypp}), in the error analysis, we shall need to prescribe the asymptotic behavior of the pressure at large densities\n\\begin{equation}\\label{pressure1}\n\t\\lim_{\\vr\\to \\infty}\\frac {p'(\\vr)}{\\vr^{\\gamma-1}}=p_\\infty>0\\quad\\mbox{with some } \\gamma\\ge 1;\n\\end{equation}\nfurthermore, if $\\gamma <2$ in \\eqref{pressure1}, we need the additional condition (for small densities):\n\\begin{equation}\\label{pressure2}\n\\liminf_{\\vr\\to 0}\\frac {p'(\\vr)}{\\vr^{\\alpha+1}}=p_0>0\\quad\\mbox{with some }\\alpha\\le 0.\n\\end{equation}\n\nThe main underlying idea of this paper is to derive the error estimates for approximate solutions of problem \\eqref{pbcont}{--(\\ref{ci2})} obtained by time and space discretization by using the discrete version of the {\\em relative energy method} { introduced on the continuous level in \\cite{FeJiNo,FENOSU, FENO7}. In spite of the fact that the relative energy method looks at the first glance pretty much similar to the widely used {\\em relative entropy method} (and both approaches translate the same thermodynamic stability conditions), they are very different in appearance and formulation and may provide different results.}\nThe notions of relative entropy and relative entropy inequality were first introduced by Dafermos \\cite{Daf4} in the context of systems of conservation laws and in particular for the compressible Euler equations.\nThe relative energy functional was suggested and successfully used for the investigation of the stability of weak solutions to the equations of viscous compressible and heat conducting fluids in \\cite{FENO7}.\nIn contrast with the relative entropy of Dafermos, { for the viscous and heat conducting fluids}, the relative energy approach is able to provide the structural stability of weak solutions, while the relative entropy approach fails in this case.\n\\\\ \\\\\nBoth functionals coincide { (modulo a change of variables)} in the case of (viscous) compressible flows in the barotropic regime. The relative energy functional and the intrinsic version of the relative energy inequality have been recently employed to obtain several stability results for the weak solutions to these equations, including the weak strong uniqueness principle, see \\cite{FeJiNo,FENOSU}.\nNote that particular versions of the relative entropy inequality with particular specific test functions had been previously derived in the context of low Mach number limits, see e.g. \\cite{MASS,WanJia}.\n\\\\ \\\\\nThe { discrete} version of the Dafermos relative entropy was employed in the non viscous case to derive an error estimate for the numerical approximation to a hyperbolic system of conservation laws and, in particular, to the compressible Euler equations \\cite{CancesMathisSeguin2014Relative}.\nIn this latter paper, the authors assume an $L^\\infty$ bound for the discrete solution, which is uniform with respect to the size of the space and time disretization (usually called stability hypothesis), that is not provided by the discrete equations. { The same method with the same severe hypotheses have been used in\n\\cite{YOVAN} to treat the compressible Navier-Stokes equations}.\nThe error analysis in the present paper relies on the theoretical background introduced in \\cite{FeJiNo} and yields an {\\it unconditional result}; in particular, {\\it we do not need any assumed bound} on the solution to get the error estimate.\n\\\\ \\\\\nThe mathematical analysis of numerical schemes for the discretization of the steady and\/or non steady compressible Navier-Stokes and\/or compressible Stokes equations has been the object of some recent works.\nThe convergence of the discrete solutions to the weak solutions of the compressible stationary Stokes was shown for a finite volume-- non conforming P1 finite element \\cite{GHL2009iso,EGHL2010isen,eym-10-convII} and for the wellknown MAC scheme which was introduced in \\cite{har-65-num} and is widely used in computational fluid dynamics {\\ (see e.g. \\cite{Sun-2014})}.\nThe unsteady Stokes problem was also discretized by some other discretization schemes on a reformulation of the problem, which were proven to be convergent \\cite{KK2010Stokes,KK2011Stokes,KK2012Stokes}.\n{ The unsteady barotropic Navier-Stokes equations was recently { investigated} in \\cite{KARPER} in the case $\\gamma>3$ (there is a real difficulty in the realistic case $\\gamma \\le 3$ arising from the treatment of the non linear convective term).}\nHowever, in these works, the rate of convergence is not provided; in fact, to the best of our knowledge, no error analysis has yet been performed for any of the numerical schemes that have been designed for the compressible Navier-Stokes equations, in spite of its great importance for the numerical analysis of the equations and for the mathematical simulations of compressible fluid flows.\nWe present here a general technique to obtain an error analysis and apply it to one of the available numerical schemes.\nTo the best of our knowledge, this is the first result of this type in the mathematical literature on the subject.\n\\\\ \\\\\nTo achieve the goal, we systematically use the relative energy method on the discrete level.\nFrom this point of view, this paper is as valuable for the introduced methodology as for the result itself.\nHere, we apply the method to the scheme of \\cite{KARPER}.\n{ In spite of the fact that this latter scheme is not used in practice (see e.g. \\cite{KHL2013baro} for a related schemes used in industrial codes), we begin the error analysis with the scheme \\cite{KARPER}} because of its readily available convergence proof.\nIn fact, we aim to use this approach to investigate the numerical errors of less academic numerical schemes, such as the finite volume -- non conforming P1 finite element \\cite{GHL2010drift,KHL2013baro,GGHL2008baro,GHKLL2011allspeed} or the MAC scheme \\cite{bab-11-dis,her-14-inc}.\n\\\\ \\\\\nThe paper is organized as follows.\nAfter recalling the fundamental setting of the problem and the relative energy inequality in the continuous case in Section \\ref{1}, we proceed in Section \\ref{2} to the discretization: we introduce the discrete functional spaces and the definition of the numerical scheme, { and state the main result of the paper, that is the error estimate formulated} in Theorem \\ref{Main}.\nThe remaining sections are devoted to the proof of Theorem \\ref{Main}:\n\\begin{list}{$\\bullet$}{}\n\\item\n In Section \\ref{4} we recall the existence theorem for the numerical scheme (Lemma \\ref{Theorem1}) and derive estimates provided by the scheme. \n\\item In Section \\ref{5}, we derive the discrete intrinsic version of the relative energy inequality for the solutions of the numerical scheme (see Theorem \\ref{Theorem4}).\n\\item The relative energy inequality is transformed to a more convenient form in Section \\ref{6}, see Lemma \\ref{refrelenergy}.\n\\item Finally, in Section \\ref{7}, we investigate the form of the discrete relative energy inequality with the test function { being a strong} solution to the original problem. This investigation is formulated in Lemma \\ref{strongentropy} and finally leads to a Gronwall type estimate formulated in Lemma \\ref{Gronwall}. The latter yields the error estimates and finishes the proof of the main result.\n \\end{list}\n Fundamental properties of the discrete functional spaces\nneeded throughout the paper are reported in Appendix (Section \\ref{3}). Some of them (especially those referring to the $L^p$ setting, $p\\neq 2$ that are not currently available in the mathematical literature) are proved. Section \\ref{3} is therefore of the independent interest.\n\n\\section{The continuous problem}\\label{1}\nThe aim of this section is to recall some fundamental notions and results.\nWe begin by the definition of weak solutions to problem \\eqref{pbcont}-- \\eqref{ci2}.\n\n\\begin{df}[Weak solutions]\\label{ws}\n{ Let $\\varrho_0 : \\Omega \\to [0, +\\infty) $ and $\\bu_0 :\\Omega \\to \\R^d$ with finite energy $E_0=\\int_\\Omega (\\frac{1}{2} \\varrho_0 |\\bu_0|^2 + {{H}}(\\vr_0)) \\dx$ and finite mass $0d\/(d-1)$, see Lions \\cite{LI4} for \"large\" values of $\\gamma$, Feireisl and coauthors \\cite{FNP} for $\\gamma>d\/(d-1)$.\n\n\\medskip\n\nLet us now introduce the notion of relative energy.\nWe first introduce the function\n\\begin{equation}\\label{E}\n\t\\begin{array}{lll}\n\t\t&E: &[0,\\infty)\\times(0,\\infty)\\to \\R,\\\\\n\t\t& & (\\vr,r) \\mapsto E(\\vr|r)=H(\\vr)-H'(r)(\\vr-r) -H(r),\n\t\\end{array}\n\\end{equation}\nwhere $H$ is defined by \\eqref{H}.\nDue to the monotonicity hypothesis in (\\ref{hypp}), $H$ is strictly convex on $[0,\\infty)$, and therefore\n\\[\n\tE(\\vr|r)\\ge 0\\quad\\mbox{and}\\quad E(\\vr|r)=0\\;\\Leftrightarrow\\;\\vr=r.\n\\]\nIn order to measure a ``distance'' between a weak solution $(\\vr,\\bu)$ of the compressible Navier-Stokes system and any other state $(r,\\bU)$ of the fluid , we introduce the relative energy functional, defined by\n\\begin{equation}\\label{ent}\n\t{\\cal{E}}(\\vr,\\bu\\Big|r,\\bU) = \\int_\\Omega \\Big(\\frac{1}{2} \\vr| \\bu -\\bU|^2 + E(\\vr\\,|\\,r)\\Big) \\dx.\n\\end{equation}\nIt was proved recently in \\cite{FeJiNo} that, provided assumption \\eqref{hypp} holds, any weak solution satisfies the following so-called relative energy inequality\n\\begin{equation}\\label{p5}\n\\begin{aligned}\n\t&{\\mathcal E} \\left( \\vr, \\bu \\Big| r, \\bU \\right) (\\tau) -{\\mathcal E} \\left( \\vr, \\bu \\Big| r, \\bU \\right)(0)+ \\int_0^\\tau \\intO{ \\Big( \\mu|\\nabla(\\bu-\\bU)|^2 +(\\mu+\\lambda)|\\dv(\\bu-\\bU)|^2 \\Big) } \\dt \\\\\n\t&\\phantom{{\\mathcal E} \\left( \\vr, \\bu \\right)} \\le \\int_0^\\tau \\intO{ \\Big( \\mu \\nabla\\bU:\\nabla(\\bU-\\bu) +(\\mu+\\lambda)\\dv \\bU \\dv(\\bU-\\bu) \\Big) } \\dt \\\\\n\t & \\phantom{{\\mathcal E} \\left( \\vr, \\bu \\right) ) \\le } + \\int_0^\\tau \\intO{ \\vr \\partial_t \\bU\\cdot (\\bU - \\bu )}\\dt + \\int_0^\\tau\\intO{\\vr\\bu {\\cdot} \\nabla \\bU \\cdot(\\bU - \\bu )}\\dt \\\\\n\t & \\phantom{{\\mathcal E} \\left( \\vr, \\bu \\right) \\le } -\\int_0^\\tau \\intO{p(\\vr)\\dv \\bU} \\dt + \\int_0^\\tau \\intO{ (r - \\vr) \\partial_t H'(r)}\\dt -\\int_0^\\tau \\intO{\\vr \\nabla H'(r) \\cdot\\bu }\\dt\n\\end{aligned}\n\\end{equation}\nfor a.a. $\\tau\\in (0,T)$, and for any pair of test functions\n\\[\nr \\in C^1 ([0,T] \\times \\Ov{\\Omega}),\\ r > 0,\\ \\bU \\in C^1([0,T] \\times \\Ov{\\Omega};\\R^3 ),\\ \\bU|_{\\partial \\Omega} = 0.\n\\]\n\n\nThe stability of strong solutions in the class of weak solutions is stated in the following proposition.\n\\begin{Proposition}[Estimate on the relative energy]\\label{proposition}\n Let $\\Omega$ be a Lipschitz domain. Assume that the viscosity coefficients satisfy assumptions (\\ref{visc}), that the pressure $p$ is a twice continuously differentiable function on $(0,\\infty)$ satisfying (\\ref{hypp}) and (\\ref{pressure1}), and that $(\\vr,\\bu)$ is a weak solution to problem \\eqref{pbcont}--\\eqref{ci2} emanating from initial data $(\\vr_0\\ge 0,\\bu_0)$, with finite energy $E_0$ and finite mass $M_0{\\ =\\int_\\Omega\\vr_0{\\rm d} x}>0$.\nLet $(r,\\bU)$ in the class\n\\begin{equation} \\label{r,U}\n\t\\left\\{\\begin{array}{l}\n\t\t \\displaystyle r\\in C^1([0,T]\\times\\overline\\Omega),\\; 0<\\underline r =\\min_{(t,x)\\in\\overline Q_T}r(t,x)\\le r(t,x)\\le \\overline r= \\max_{(t,x)\\in\\overline Q_T}r(t,x), \\\\ ~ \\\\\n \t\t \\bU\\in C^1([0,T]\\times \\overline\\Omega;\\R^3),\\; \\bU|_{(0,T)\\times\\partial\\Omega}=0\n\t\\end{array}\\right.\n \\end{equation}\nbe a (strong) solution of problem \\eqref{pbcont} emanating from the initial data $(r_0,\\bU_0)$.\nThen there exists\n\\[\nc=c(T,\\Omega, M_0, E_0, \\underline r, \\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|(\\nabla r, \\partial_t r, \\bU, \\nabla\\bU, \\partial_t \\bU) \\|_{L^\\infty(Q_T;\\Rm^{19})} )>0\n\\]\nsuch that for almost all $t\\in (0,T)$,\n \\begin{equation}\\label{cerrorestimate}\n {\\cal E}(\\vr,\\bu\\Big|r,\\bU)(t)\\le c{\\cal E}(\\vr_0,\\bu_0\\Big|r_0,\\bU_0).\n\\end{equation}\n\\eP\nThis estimate (implying among others the weak-strong uniqueness) was proved in \\cite{FeJiNo} (see also \\cite{FENOSU}) for pressure laws (\\ref{pressure1}) with $\\gamma>d\/(d-1)$.\nIt remains valid under weaker hypothesis on the pressure, such as (\\ref{pressure1}) with $\\gamma\\ge 1$; this can be proved using ideas introduced in \\cite{BEFENO} and \\cite{MANO}.\n\n\n\n\n\n\n\\section{The numerical scheme}\\label{2}\n\\subsection{Partition of the domain}\\label{3.1}\nWe suppose that $\\Omega$ is a bounded domain of $\\R^d$, polygonal if $d=2$ and polyhedral if $d=3$.\nLet ${\\cal{T}}$ be a decomposition of the domain $\\Omega$ in { tetrahedra}, which we call hereafter a triangulation of $\\Omega$, regardless of the space dimension. By ${\\cal{E}}(K)$, we denote the set of the edges ($d=2$) or faces ($d=3$) $\\sigma$ of the element $K \\in {\\cal{T}}$ called hereafter faces, regardless of the dimension.\nThe set of all faces of the mesh is denoted by ${\\cal{E}}$; the set of faces included in the boundary $\\partial\\Omega$ of $\\Omega$ is denoted by ${\\cal{E}}_{\\extt}$ and the set of internal faces (i.e ${\\cal{E}} \\setminus {\\cal{E}}_{\\extt} $) is denoted by ${\\cal{E}}_{\\intt}$.\nThe triangulation ${\\cal{T}}$ is assumed to be regular in the usual sense of the finite element literature (see e.g. \\cite{cia-91-bas}), and in particular, ${\\cal{T}}$ satisfies the following properties:\n\\begin{itemize}\n\t\\item[$\\bullet$] $ \\overline{\\Omega} = \\cup_{K \\in {\\cal{T}}} \\overline{K} $;\n\t\\item[$\\bullet$] if $(K,L) \\in {\\cal T}^2$, then $ \\overline{K} \\cap \\overline{L} = \\emptyset $ or $\\overline{K} \\cap \\overline{L} $ is a vertex or $\\overline{K} \\cap \\overline{L} $ is a common face of $K$ and $L$; in the latter case it is denoted by $K|L$.\n\\end{itemize}\nFor each internal face of the mesh $\\sigma=K|L$, $\\bn_{\\sigma, K}$ stands for the normal vector of $\\sigma$, oriented from $K$ to $L$ (so that $\\bn_{\\sigma, K}=-\\bn_{\\sigma,L}$).\nWe denote by $ |K|$ and $ |\\sigma|$ the ($d$ and $d-1$ dimensional) Lebesgue measure of the { tetrahedron} $K$ and of the face $\\sigma$ respectively, and by $h_K$ and $h_\\sigma$ the diameter of $K$ and $\\sigma$ respectively.\nWe measure the regularity of the mesh thanks to the parameter $\\theta$ defined by\n{\n\\begin{equation}\\label{reg}\n\t{\\ \\theta = \\inf \\{ \\frac{\\xi_K}{h_K}, K \\in {\\cal{T}} \\} }\n\\end{equation}\n}\nwhere $\\xi_K$ stands for the diameter of the largest ball included in $K$.\nLast but not least we denote by $h$ the maximal size of the mesh,\n\\begin{equation}\\label{maxh}\n {\\\th=\\max_{K \\in {\\cal{T}}} h_K.}\n \\end{equation}\nThe triangulation $\\mathcal T$ is said to be regular if it satisfies\n\\begin{equation}\\label{reg1-}\n \t{\\ \\theta\\ge\\theta_0>0.}\n\\end{equation}\n\n\\subsection{Discrete {\\ function} spaces}\nLet $\\mathcal T$ be a mesh of $\\Omega$.\nWe denote by $L_h({ \\Omega})$ the space of piecewise constant functions on the cells of the mesh;\nthe space $L_h{ (\\Omega)}$ is the approximation space for the pressure and density.\nFor $1\\le p<\\infty$, the mapping\n\\[\n\tq \\mapsto \\|q\\|_{L_h^p(\\Omega)}= \\|q\\|_{L^p(\\Omega)}=\\Big(\\sum_{K\\in{\\cal T}}|K| |q_K|^p\\Big)^{1\/p}\n\\]\nis a norm on $L_h(\\Omega)$.\nWe also introduce spaces of non-negative and positive functions:\n\\[\n L_h^+{ (\\Omega)} = \\{q \\in L_h{ (\\Omega)}, ~q_K \\ge 0, ~\\forall K \\in \\T \\},\\quad L_h^{++}{ (\\Omega)} = \\{q \\in L_h{ (\\Omega)}, ~q_K > 0, ~\\forall K \\in \\T \\}.\n\\]\nThe approximation space for the velocity field is the space $\\bW_h(\\Omega)=V_h(\\Omega;{\\R^d})$, where $V_h(\\Omega)$ is the non conforming piecewise linear finite element space \\cite{cro-73-con,ern-04-the} defined by:.\n\\begin{multline}\n\t V_h(\\Omega) = \\{ v \\in L^2(\\Omega), ~\\forall K \\in {\\cal{T}}, ~v_{|K} \\in \\mathbb{P}_1(K),\\\\\n\t \\forall \\sigma \\in {\\cal{E}}_{\\intt} ,\\; \\sigma=K|L,\\; \\int_{\\sigma}v_{|K} \\dS=\\int_{\\sigma}v_{|L}\\dS,\\quad\n\t\\forall \\sigma \\in {\\cal{E}}_{\\extt},\\; \\int_\\sigma v \\dS=0 \\},\n \\end{multline}\nwhere $\\mathbb{P}_1(K)$ denotes the space of affine functions on $K$ and $\\dS$ the integration with respect to the $(d-1)$-dimensional Lebesgue measure { on the face $\\sigma$}.\nEach element $v\\in V_h(\\Omega)$ can be written in the form\n\\begin{equation}\\label{CRP}\n\tv{ (x)}=\\sum_{\\sigma\\in{\\cal E}_{\\rm int}}v_\\sigma\\varphi_\\sigma{ (x)},\\quad x\\in \\Omega\n\\end{equation}\nwhere the set $\\{\\varphi_\\sigma\\}_ {\\sigma\\in{\\cal E}_{\\rm int}}\\subset V_h(\\Omega)$ is the classical basis determined by\n\\begin{equation}\\label{CRB}\n\\forall (\\sigma,\\sigma')\\in {\\cal E}^2_{\\rm int},\\;\\int_{\\sigma'}\\varphi_\\sigma \\dS=\\delta_{\\sigma,\\sigma'},\\quad\\forall\\sigma'\\in {\\cal E}_{\\rm ext},\\; \\int_{\\sigma'}\\varphi_\\sigma \\dS=0\n\\end{equation}\n{ and $\\{v_\\sigma\\}_{\\sigma\\in {\\cal E}_{\\rm int}}\\subset R$ is the set of degrees of freedom relative to $v$. }\nNotice that $V_h(\\Omega)$ approximates the functions with zero traces in the sense that for all elements in $V_h(\\Omega)$, $v_\\sigma=0$ provided $\\sigma\\in {\\cal E}_{\\rm ext}$.\nSince only the continuity of the integral over each face of the mesh is imposed, the functions in $V_h(\\Omega)$ may be discontinuous through each face; the discretization is thus nonconforming in $ W^{1,p}(\\Omega;\\R^d)$, $1\\le p\\le \\infty$.\nFinally, we notice that for any $1\\le p<\\infty$ the expression\n\\[\n\t|v|_{V_h^p(\\Omega)}=\\Big(\\sum_{K\\in {\\cal T}}\\|\\nabla v\\|_{L^p(K;\\Rm^d)}^p\\Big)^{1\/p}\n\\]\nis a norm on $V_h(\\Omega)$ and we denote by $V^p_h{ (\\Omega)}$ the space $V_h{ (\\Omega)}$ endowed with this norm.\n\n\n\nWe finish this section by introducing some notations.\nFor a function $v$ in $L^1(\\Omega)$, we set\n\\begin{equation}\n \tv_K = \\frac 1{|K|}\\int_K v \\dx \\textrm{ for } K \\in {\\cal T} \\textrm{ and } \\hat v{ (x)}= \\sum_{K\\in {\\cal T}} v_K 1_K{ (x)},\\; { x\\in \\Omega}\n\t\\label{vhat}\n\\end{equation}\nso that $\\hat v \\in L_h(\\Omega)$. { Here and in what follows, $1_K$ is the characteristic function of $K$}.\n\n{ If} $v \\in W^{1,p}(\\Omega)$, we set\n\\begin{equation}\n\tv_\\sigma=\\frac 1{|\\sigma|}\\int_\\sigma v{\\rm d} S\\textrm{ for } \\sigma\\in {\\cal E}.\n\\label{vtilde}\n\\end{equation}\nFinally, if $v \\in W^{1,p}_0(\\Omega)$, we set\n\\begin{equation}\n\n v_h{ (x)}=\\sum_{\\sigma\\in {\\cal E_{\\rm int}}} v_\\sigma \\varphi_\\sigma{ (x)},\\; { x\\in \\Omega.}\n\t\\label{vh}\n\\end{equation}\nso that $v_h \\in V_h(\\Omega)$.\n{ In accordance with the above notation, for $v\\in W^{1,p}_0(\\Omega)$, the symbol $\\hat v_{h}$ means $\\widehat{v_h}(x)=\\sum_{\\sigma\\in {\\cal E}_{\\rm int}}v_\\sigma\\hat\\phi_\\sigma(x)$, and the symbol $v_{h,K}=\\frac 1{|K|}\\int_K v_h(x){\\rm d}x$ and the symbol $\\hat v_{h,\\sigma}^{\\rm up}= [\\widehat{(v_h)}]_\\sigma^{\\rm up}$.}\n\n\n\n\n\n\\subsection{Discrete equations}\nLet us consider a partition $0=t_0 0\\\\ q_L\\; \\mbox{if } \\bu_\\sigma\\cdot\\bn_{\\sigma, K}{ \\le} 0,\n \\end{cases}\n\\end{equation}\nso that\n\\[\n\\sum_{\\sigma\\in{\\cal E}(K)} q_\\sigma^{\\rm up}{\\bu}_\\sigma\\cdot{\\vc n}_{\\sigma,K}=\\stik \\Big(q_K [{\\bu}_\\sigma\\cdot{\\vc n}_{\\sigma,K}]^+ - q_L [{\\bu_\\sigma}\\cdot{\\vc n}_{\\sigma,K}]^-\\Big),\n\\]\nwhere $a^+ = \\max(a,0)$, $a^- =- \\min(a,0)$.\n\n\n\nLet us then consider the following numerical scheme \\cite{KARPER}:\n\n\\vspace{2mm}\n\n{\\it Given $(\\vr^0,\\bu^0) \\in L_h^+(\\Omega) \\times \\bW_h(\\Omega) $\n{find}} $ (\\vr^n)_{1\\le n \\le N}\\subset (L_h(\\Omega))^N, (\\bu^n)_{1\\le n \\le N} \\subset (\\bW_h(\\Omega))^N $ {\\it such that for all} $ n=1,...,N $\n\\vspace{2mm}\n\\begin{subequations}\\label{scheme}\n\\begin{align}\n \\label{dcont}\n\t&|K| \\frac{ \\vr_{K}^n - \\vr_{K}^{n-1}}{\\deltat} + \\sum_{\\sigma \\in {\\cal E}(K)} |\\sigma| \\vr_\\sigma^{n,{\\rm up}}[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]=0,\\quad\\forall K \\in {\\cal{T}}, \\\\\n\t& \\sum_{K\\in{\\cal T}}\\frac{|K|}{\\deltat} \\Big({\\vr^n_K{{\\bu}}^n_{K}- \\vr^{n-1}_K{{\\bu}}^{n-1}_{K}} \\Big)\\cdot \\bv_K+ \\sum_{K\\in {\\cal T}}\\sum_{\\sigma \\in {\\cal E}(K)} |\\sigma|\\vr^{n,{\\rm up}}_\\sigma {{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}[\\bu^n_\\sigma\\cdot \\bn_{\\sigma,K}]\\cdot \\bv_K \\nonumber \\\\\n\t&\\vspace{-.3cm}\\qquad \\qquad - \\sum_{K\\in{\\cal T}}p(\\vr^n_K)\\sum_{\\sigma \\in {\\cal E}(K)} |\\sigma|\\bv_\\sigma\\cdot {\\vc n}_{\\sigma,K}+\\mu\\sum_{K\\in{\\cal T}}\\int_K \\nabla\\bu^n : \\nabla\\bv \\ \\dx \\label{dmom}\\\\ \\vspace{-.3cm}\n\t&\\qquad \\qquad \\qquad \\qquad\\qquad + ( \\mu +\\lambda)\\sum_{K\\in{\\cal T}}\\int_K{\\rm div}\\bu^n{\\rm div}\\bv\\, \\dx =0,\n\t\\; \\forall \\bv \\in {\\bW_h(\\Omega) }. \\nonumber\n\\end{align}\n\\end{subequations}\nNote that the boundary condition $\\bu^n_{\\sigma}=0\\quad\\mbox{if $\\sigma\\in {\\cal E}_{\\rm ext}$}$ is ensured by the definition of the space $V_h(\\Omega)$.\nNote also that if $\\sigma \\in {\\cal E}_{\\rm int}$, $\\sigma = K|L$, one has, following \\eqref{vhat} and \\eqref{upwind1},\n\\[\n{{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}=u^n_K=\\frac 1 {|K]} \\int_K \\bu^n(x) \\dx \\textrm{ if } \\bu^n_\\sigma\\cdot \\bn_{\\sigma,K}>0 \\textrm{ and }\n{{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}=u^n_L=\\frac 1 {|L]} \\int_L \\bu^n(x) \\dx \\textrm{ if } \\bu^n_\\sigma\\cdot { \\bn_{\\sigma,K}< 0}.\n\\]\n\nIt is well known that any solution $(\\vr^n)_{1\\le n \\le N}\\subset (L_h(\\Omega))^N$ satisfies $\\vr^n >0$ thanks to the upwind choice in \\eqref{dcont} { (see e.g. \\cite{GGHL2008baro,KARPER})}.\nFurthermore, summing \\eqref{dcont} over $K \\in \\mesh$ immediately yields the total conservation of mass, which reads:\n\\begin{equation}\\label{masscons}\n\t\\forall n=1,...N,\\quad \\int_\\Omega \\vr^n \\dx = \\int_\\Omega \\vr^0 \\dx.\n\\end{equation}\n\nWe finally state in this section the existence result, which can be proved by a topological degree argument,\n\\cite{GGHL2008baro,KARPER}.\n\\begin{Proposition}[Existence]\\label{Theorem1}\n\tLet $(\\vr^0,\\bu^0){ \\in L_h^{++}}(\\Omega) \\times \\bW_h(\\Omega)$.\n\tUnder assumptions \\eqref{visc} and \\eqref{hypp}, Problem \\eqref{scheme} admits at least one solution\n\t\\[\n\t\t(\\vr^n)_{1\\le n \\le N}\\in [{ L_h^{++}}(\\Omega)]^N, (\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N.\n\t\\]\n\\end{Proposition}\n\n\n\n\\subsection{Main result: error estimate}\n\nLet $(r,\\bU) : [0,T]\\times\\overline\\Omega\\mapsto (0,\\infty)\\times \\R^3$ be $C^2$ functions such that $\\bU=\\bzero$ on $\\partial \\Omega$.\nLet $(\\vr,\\bu)$ be a solution of the discrete problem \\eqref{scheme}.\nInspired by (\\ref{ent}), we introduce the discrete relative energy functional\n\\begin{align}\\label{dent}\n\t{\\cal E}(\\vr^n,\\bu^n\\Big|r^n,\\bU^n) &=\\int_{\\Omega}\\Big(\\frac 12\\vr^n|\\hat\\bu^n-\\hat{\\bU}_h^n|^2\n\n\t+ E(\\vr^n|\\hat r^n)\\Big)\\dx \\\\\n\t&=\\sum_{K\\in{\\cal T}}|K|\\Big(\\frac 12\\vr_K|\\bu_K^n-\\bU^n_{h,K}|^2+ E(\\vr_K^n|r_K^n)\\Big), \\nonumber\n\\end{align}\nwhere\n\\begin{equation}\\label{notation2-}\nr^n(x)=r(t_n,x),\\;\\bU^n(x)=\\bU(t_n,x),\\;{ n=0,\\ldots,N,}\n\\end{equation}\n $(\\vr^n,\\bu^n)$ is defined in (\\ref{notation0}), and $E$ is defined by \\eqref{E}.\nLet us finally introduce the notations\n\\[\nM_0=\\sum_{K\\in{\\cal K}}|K|\\vr_K^0,\\mbox{ and } E_0=\\sum_{K\\in{\\cal K}}|K|\\Big(\\frac 12\\vr_K^0|\\bu_K^0|^2 + H(\\vr_K^0)\\Big).\n\\]\n\nNow, we are ready to state the main result of this paper.\nFor the sake of clarity, we shall state the theorem and perform the proofs only in the most interesting three dimensional case.\nThe modifications to be done for the two dimensional case, which is in fact more simple, are mostly due to the different Sobolev embedings, and are left to the interested reader.\n\n\n\\begin{Theorem}[Error estimate]\\label{Main}\nLet $\\theta_0 > 0$ and ${\\cal{T}} $ be a { regular} triangulation of a bounded polyhedral domain $ \\Omega\\subset\\R^3 $ { introduced in Section \\ref{3.1}} such that $ \\theta \\ge \\theta_0 $, where $ \\theta $ is defined in (\\ref{reg}).\nLet $p$ be a twice continuously differentiable function satisfying assumptions (\\ref{hypp}), (\\ref{pressure1}) with $\\gamma\\ge 3\/2$, and the additional assumption (\\ref{pressure2}) in the case $\\gamma<2$.\nLet the viscosity coefficients satisfy assumptions (\\ref{visc}).\nSuppose that $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and that $(\\vr^n)_{1\\le n \\le N}\\subset [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\subset [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme}.\n Let $(r,\\bU)$ in the class\n\\begin{subequations}\\label{dr,U}\n\\begin{align}\n& r\\in C^2([0,T]\\times\\overline\\Omega),\\quad 0<\\underline r:=\\min_{(t,x)\\in \\overline Q_T}\\le r(t,x)\\le\\overline r:= \\max_{(t,x)\\in \\overline Q_T}r(t,x), \\label{dr}\\\\\n & \\bU\\in C^2([0,T]\\times\\overline\\Omega;\\R^3), \\;\\bU|_{\\partial\\Omega}=0 \\label{U}\n\\end{align}\n \\end{subequations}\n be a (strong) solution of problem \\eqref{pbcont}.\n Then there exists\n\\begin{multline*}\n c=c\\Big(T,|\\Omega|, {\\rm diam}(\\Omega),\\theta_0,\\gamma, M_0, E_0, \\underline r,\\overline r,\\\\ |p'|_{C^1([\\underline r,\\overline r])},\n \\|(\\nabla r, \\partial_t r, \\partial_t\\nabla r, \\partial^2_t r, \\bU, \\nabla\\bU, { \\nabla^2\\bU}, \\partial_t\\bU, { \\partial_t^2\\bU,} \\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;{ \\Rm^{68}})}\\Big) \\in (0,+\\infty)\n\\end{multline*}\n (independent of $h$, $\\deltat$) such that for any $m=1,\\ldots,N,$\n \\begin{equation}\\label{errorestimate}\n {\\cal E}(\\vr^{ m},\\bu^{ m}\\Big|r^{ m},\\bU^{ m}){ +\\deltat\\sum_{n=1}^m\\sum_{{K}\\in {\\cal T}}\\int_K|\\Grad(u^n-\\vc U^n_h)|^2{\\rm d x}}\n \\le c\\Big({\\cal E}(\\vr^0,\\bu^0\\Big|r^0,\\bU^0)+ h^{A}\n +\\sqrt{\\deltat}\\Big),\n \\end{equation}\nwhere\n\\begin{equation}\\label{defAmain}\n A = \\begin{cases}\n\t\t\\frac {2\\gamma-3}\\gamma\\quad\\mbox{if }\\gamma\\in (3\/2,2],\\\\\n\t\t1\/2\\quad\\mbox{if }\\gamma> 2.\n \\end{cases}\n\\end{equation}\n\\end{Theorem}\n\n{ Starting from this point, unlike in Section \\ref{intro}, here and hereafter, the symbol ${\\cal E}$ refers always to the {\\it discrete} relative energy functional defined in (\\ref{dent}).}\n\n\n\\begin{Remark}\\label{rem1}{\\rm ~}\\\\\n{\nAssumptions (\\ref{dr,U}) on the regularity of the strong solution $(r,\\vc U)$ in Theorem \\ref{Main} may be slightly relaxed: It is enough to suppose\n$$\n(r,\\vc U)\\in C^1([0,T]\\times\\overline\\Omega;\\R^4),\\;\\nabla^2 \\vc U\\in C([0,T]\\times\\overline\\Omega;\\R^3),\\;\n0<\\inf_{(t,x)\\in \\overline Q_T} r(t,x),\n$$\n$$\n\\partial_t^2r\\in L^1(0,T;L^{\\gamma'}(\\Omega)),\\;\\partial_t\\nabla r\\in L^2(0,T; L^{6\\gamma\/(5\\gamma-6)}(\\Omega;\\R^3)),\\;\n(\\partial_t^2\\vc U,\\partial_t\\nabla\\vc U)\\in L^2(0,T; L^{6\/5}(\\Omega;\\R^{12})).\n$$\nThe constant in the error estimate depends on $\\underline r$ and the norms of $r$ and $\\vc U$ in these spaces.\nThis improvement is at the price of more technicalities in estimates of several residual terms, namely in estimates (\\ref{R2.1}--\\ref{R2.2}), (\\ref{R5.2}), (\\ref{R6.4}), (\\ref{cR2.1}), (\\ref{cR2.3}--\\ref{cR2.4})\nand (\\ref{cP1}).}\n\\end{Remark}\n\\begin{Remark}\\label{rmq-thmp}{\\rm ~ }\\\\\n\\vspace{-3mm}\n\\begin{enumerate}\n\\item\nTheorem \\ref{Main} holds also for two dimensional bounded { polygonal} domains under the assumption that $\\gamma\\ge1$.\nAssumption (\\ref{pressure2}) on the asymptotic behavior of pressure near $0$ is no more necessary in this case.\nThe value of $A$ in the error estimate (\\ref{errorestimate}) is\n\\[\nA =\\left\\{\n\\begin{array}{c}\n\\frac {2\\gamma-2}\\gamma\\quad\\mbox{if $\\gamma\\in (1,2]$},\\\\\n1\\quad\\mbox{if $\\gamma> 2$}.\n\\end{array}\n\\right.\n\\]\n{\n\\item Suppose that the discrete initial data $(\\vr^0,\\vc u^0)$ coincide with the projection $(\\hat r^0,\\hat {\\vc U}_{h}^0)$ of the initial data determining the strong solution. Then formula (\\ref{errorestimate}) provides in terms of classical Lebesgue spaces the following bounds:\n $$\n \\|\\vr^m-r^m\\|^2_{L^2(\\Omega\\cap\\{\\underline r\/2\\le\\vr^m\\le 2\\overline r\\})}+\n \\|\\hat{\\vc u}^m- {\\vc U}^m\\|^2_{L^2(\\Omega\\cap\\{\\underline r\/2\\le\\vr^m\\le 2\\overline r\\})}\\le\n c\\Big(h^{A}\n +\\sqrt{\\deltat}\\Big)\n $$\n for the \"essential part\" of the solution (where the numerical density remains bounded from above and from below outside zero), and\n $$\n |\\{\\vr^m\\le\\underline r\/2\\}|+ |\\{\\vr^m\\ge 2\\overline r\\}| +\\|\\vr^m\\|^\\gamma_{L^\\gamma(\\Omega\\cap \\{\\vr^m\\ge 2\\overline r\\})}+\\|\\vr^m|\\hat{\\vc u}^m-{\\vc U}^m|^2\\|_{L^1(\\Omega\\cap\\{\\vr^m\\ge 2\\overline r\\})}\\le c\\Big(h^{A}\n +\\sqrt{\\deltat}\\Big)\n $$\n for the \"residual part\" of the solution, where the numerical density can be \"close\" to zero or infinity.\n (In the above formula, for $B\\subset\\Omega$, $|B|$ denotes the Lebesgue measure of $B$.)\n\n Moreover, in the particular case of $p(\\vr)=\\vr^2$ (that however represents a non physical situation)\n $E(\\vr| r)=(\\vr- r)^2$ and the error estimate (\\ref{errorestimate}) reads\n $$\n \\|\\vr^m- r^m\\|^2_{L^2(\\Omega)}+\n \\|\\vr^m|\\hat{\\vc u}^m- {\\vc U}^m|^2\\|_{L^1(\\Omega)}\\le\n c\\Big(\\sqrt h\n +\\sqrt{\\deltat}\\Big)\n $$\n}\n\n\\item Theorem \\ref{Main} can be viewed as a discrete version of Proposition \\ref{proposition}.\nIt is to be noticed that the assumptions on the constitutive law for pressure guaranteeing the error estimates for the scheme \\eqref{scheme} are somewhat stronger ($\\gamma\\ge 3\/2$) than the assumptions needed for the stability in the continuous case ($\\gamma\\ge 1$).\nThe threshold value $\\gamma=3\/2$ is however in accordance with the existence theory of weak solutions.\nThe assumptions on the regularity of the strong solution to be compared with the discrete solution in the scheme are slightly stronger than those needed to establish the stability estimates in the continuous case.\n\n\\item If $d=3$, we notice that the assumptions on the pressure (as function of the density) in Theorem \\ref{Main} are compatible with the isentropic case $p(\\vr)=\\vr^\\gamma$ for all values $\\gamma\\ge 3\/2$.\n\n \\item { The scheme \\cite{KARPER} contains in addition artificial stabilizing terms both in the continuity and momentum equations. These terms are necessary for the convergence proof in \\cite{KARPER} even for the large values of $\\gamma$. It is to be noticed that the error estimate in Theorem \\ref{Main} is formulated for the numerical scheme without these stabilizing terms. Of course similar error estimate is a fortiori valid also for the scheme with the stabilizing terms, however, this issue is not discussed in the present paper.}\n\n\\end{enumerate}\n\\end{Remark}\n\n{ The rest of the paper is devoted to the proof of Theorem \\ref{Main}. For the sake of simplicity, and in order to simplify notation, we present the proof for the uniformly regular mesh meaning that there exist positive numbers $c_i=c_i(\\theta_0)$ such that\n\\begin{equation}\\label{reg1}\nc_1h_K\\le h\\le c_2 h_\\sigma\\le c_3 h_K,\\quad c_1|K|\\le |\\sigma| h\\le c_2|\\sigma|h_K\\le c_3|\\sigma|h_\\sigma\\le c_4 |K|\n\\end{equation}\nfor any $K\\in {\\cal T}$ and any $\\sigma\\in {\\cal E}$.\nThe necessary (small) modifications needed to accommodate the regular mesh satisfying only (\\ref{reg1-}) are straightforward. Even with this simplification the proof is quite involved, and some details have to be necessarily omitted to keep its length within reasonable bounds. The reader can eventually find them in the extended version of this paper available on ArXiv \\cite{GHMNArxive}.\n}\n\n\\section{Mesh independent estimates}\\label{4}\n\nWe start by a remark on the notation.\nFrom now on, the letter $c$ denotes positive numbers that may tacitly depend on $T$, $|\\Omega|$, ${\\rm diam}(\\Omega)$, $\\gamma$, $\\alpha$, $\\theta_0$, $\\lambda$ and $\\mu$, and on other parameters;\nthe dependency on these other parameters (if any) is always explicitly indicated in the arguments of these numbers.\nThese numbers can take different values even in the same formula.\nThey are always independent of the size of the discretisation $\\deltat$ and $h$.\n\n\n\\subsection{Energy { Identity}}\nOur analysis starts with an energy inequality, which is crucial both in the convergence analysis and in the error analysis.\nWe recall this energy estimate which is already given in \\cite{KARPER}, along with its proof for the sake of completeness.\n\n\\begin{lm}\\label{Theorem3}\n Let $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and suppose that $(\\vr^n)_{1\\le n \\le N}\\in [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme} with the pressure $p$ satisfying condition (\\ref{hypp}).\nThen there exist\n\\begin{align*}\n&\\overline \\vr^n_{\\sigma}\\in [\\min(\\vr^n_K,\\vr^n_L), \\max(\\vr^n_K,\\vr^n_L)],\\;\\sigma=K|L\\in {\\cal E}_{\\rm int},\\; n=1,\\ldots,N \\\\\n&\\overline\\vr_K^{n-1,n}\\in [\\min(\\vr^{n-1}_K,\\vr^n_K), \\max(\\vr^{n-1}_K,\\vr^n_K)],\\; K\\in {\\cal T},\\; n=1,\\ldots,N\n\\end{align*}\nsuch that\n\\begin{multline}\n\\sum_{K\\in {\\cal T}}{|K|}\\Big(\\frac 12\\vr^m_K|{\\bu}^m_K|^2\n +H(\\vr_K^m)\\Big)\n-\\sum_{K\\in {\\cal T}}|K|\\Big(\\frac 12\\vr^{0}_K|{\\bu}^{0}_K|^2\n+H(\\vr_K^{0})\\Big)\n\\\\\n+\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad\\bu^n|^2 \\dx+( \\mu +\\lambda)\\int_K|{\\rm div}\\bu^n|^2 \\dx\\Big)\n\\\\\n+ [D^{m,|\\Delta\\bu|}_{\\rm time}]+ [D^{m,|\\Delta\\vr|}_{\\rm time}]+ [D^{m,|\\Delta\\bu|}_{\\rm space}] + [D^{m, |\\Delta\\vr|}_{\\rm space}]= 0,\n\\label{denergyinequality}\n\\end{multline}\nfor all $m=1,\\ldots,N$,\nwhere\n\\begin{subequations}\\label{upwinddissipation}\n\t\\begin{align}\n\t\t& [D^{m,|\\Delta\\bu|}_{\\rm time}]=\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{|K|}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2\n\t\t\\label{upwinddissipation_1}\\\\\n\t\t&[D^{m,|\\Delta\\vc \\vr|}_{\\rm time}]=\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}|K|H''(\\overline\\vr_K^{n-1,n})\\frac {|{\\vr}_K^n-{\\vr }_K^{n-1}|^2} 2,\n\t\t\\label{upwinddissipation_2}\\\\\n\t\t&[D^{m,|\\Delta\\bu|}_{\\rm space}]=\\deltat\\sum_{n=1}^m\\sum_{\\sigma=K|L\\in{\\cal E}_{\\rm int}}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\frac{({\\bu}^n_K-{{\\bu}}^n_L)^2} 2\\;|{\\bu}^n_\\sigma\\cdot\\bn_{\\sigma,K}|,\n\t\t\\label{upwinddissipation_3}\\\\\n\t\t&[D^{m, |\\Delta\\vr|}_{\\rm space}]=\\deltat\\sum_{n=1}^m\\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|H''(\\overline \\vr^n_{\\sigma})\\frac{(\\vr^n_K-\\vr^n_L)^2}2 \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|.\n\t\t\\label{upwinddissipation_4}\n\\end{align}\n \\end{subequations}\n\\end{lm}\n\n\\begin{proof}\nMimicking the formal derivation of the total energy conservation in the continuous case we take as test function $\\bv = \\bu^n$ in the discrete momentum equation (\\ref{dmom})$^n$ and obtain\n\\begin{equation}\\label{denergy1}\n\t I_1+ I_2+I_3+I_4 = 0,\n\\end{equation}\nwhere\n\\begin{align*}\n& { I_1=\\sum_{K\\in {\\cal T}}\\frac {|K|}\\deltat(\\vr_k^n\\bu_K^n-\\vr_K^{n-1}\\bu_K^{n-1})\\cdot\\bu_K^n},\n \t&&{ I_2}= \\sum_{K\\in {\\cal T}}\\stik |\\sigma|\\vr^{n,{\\rm up}}_\\sigma {{ \\hat\\bu}}_{\\sigma}^{n,{\\rm up}}\\cdot\\bu^n_K\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n \\\\\n &{ I_3} = -\\sum_{K\\in {\\cal T}}\\stik |\\sigma|p(\\vr^n_K)[\\bu^n_{\\sigma}\\cdot{\\vc n}_{\\sigma,K}], &&\n\t{ I_4= \\sum_{K\\in{\\cal T}}\\int_K\\Big(\\mu\\nabla\\bu^n:\\nabla\\bu^n +( \\mu +\\lambda){\\rm div}{\\bu^n}{\\rm div}{{\\bu}^n}\\Big) \\dx.}\n\\end{align*}\nNext, we multiply the continuity equation (\\ref{dcont})$^n_K$ by $\\frac 12|\\bu^n_K|^2$ and sum over all\n$K\\in {\\cal T}$. We get\n\\begin{equation}\\label{denergy2}\nI_5+I_6 = 0\n\\end{equation}\n\\[\n\\mbox{with }I_5 = -\\sum_{K\\in {\\cal T}}\\frac 12\\frac{|K|}{\\deltat} ({ \\vr_{K}^n - \\vr_{K}^{n-1}})|\\bu^n_K|^2 \\mbox{ and } I_6=-\\sum_{K\\in {\\cal T}}\\stik\\frac 12 |\\sigma| \\vr_\\sigma^{n,\\rm up}[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]|\\bu^n_K|^2.\n\\]\nFinally, we multiply the continuity equation (\\ref{dcont})$^n_K$ by $H'(\\vr_K^n)$ and sum over all $K\\in {\\cal T}$.\nWe obtain\n\\begin{equation}\\label{denergy3}\nI_7+I_8 =0,\n\\end{equation}\n\\[\n\\mbox{with }I_7=\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat} ({ \\vr_{K}^n - \\vr_{K}^{n-1}}) H'(\\vr^n_K)\n \\mbox{ and } I_8=\\sum_{K\\in {\\cal T}}\\stik |\\sigma| \\vr_\\sigma^{n,\\rm up}[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}] H'(\\vr^n_K).\n\\]\n\nWe now sum formulas (\\ref{denergy1})--(\\ref{denergy3}) in several steps.\n\n\\vspace{2mm}\n\n{\\bf Step 1:} {\\it Term $I_1+ I_7$.}\nWe verify by a direct calculation that\n\\[\nI_1= \\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(\\frac 12 \\vr^n_K|\\bu^n_K|^2-\\frac 12 \\vr^{n-1}_K|\\bu^{n-1}_K|^2\\Big)\n+\\sum_{{ K\\in{\\cal T}}}\\frac{|K|}{\\deltat}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2.\n\\]\nIn order to transform the term $I_7$, we employ the Taylor formula\n\\[\nH'(\\vr_K^n)\\Big(\\vr^n_K-\\vr^{n-1}_K\\Big)= H(\\vr_K^n)-H(\\vr_K^{n-1}) +\\frac 12 H''(\\overline\\vr_K^{n-1,n})(\\vr_K^n-\\vr_K^{n-1})^2,\n\\]\nwhere $\\overline\\vr_K^{n-1,n}\\in [\\min(\\vr_K^{n-1},\\vr_K^{n}), \\max(\\vr_K^{n-1},\\vr_K^{n})]$.\nConsequently,\n\\begin{multline}\\label{denergy4}\nI_1+I_7=\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(\\frac 12 \\vr^n_K|\\bu^n_K|^2-\\frac 12 \\vr^{n-1}_K|\\bu^{n-1}_K|^2\\Big)\n+ \\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(H(\\vr_K^n)-H(\\vr_K^{n-1})\\Big)\n\\\\+\\sum_{K\\in{\\cal T}}\\frac{|K|}{\\deltat}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2\n+\\sum_{K\\in{\\cal T}}\\frac{|K|}{\\deltat}H''(\\overline\\vr_K^{n-1,n})\\frac {|{\\vr}_K^n-{\\vr }_K^{n-1}|^2} 2.\n\\end{multline}\n{\\bf Step 2:} {\\it Term $I_2+I_6$}.\nThe contribution of the face $\\sigma=K|L$ to the sum $I_2+I_6$ reads, by virtue of (\\ref{upwind1}),\n\\begin{multline*}\n|\\sigma|\\, [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]^+\\,\\vr_K \\Big(|\\bu_K^n|^2 -\\bu^n_K\\cdot\\bu_L^n -\n\\frac 12|\\bu^n_K|^2 + \\frac 12|\\bu^n_L|^2\\Big) \\\\\n+ |\\sigma|\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,L}]^+\\,\\vr_L \\Big(|\\bu_L^n|^2 -\\bu^n_K\\cdot\\bu_L^n -\n\\frac 12|\\bu^n_L|^2 + \\frac 12|\\bu^n_K|^2\\Big).\n \\end{multline*}\nConsequently,\n\\begin{equation}\\label{denergy5}\nI_2+I_6=\\sum_{\\sigma=K|L\\in{\\cal E}_{\\rm int}} |\\sigma| |\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}|\\vr_{\\sigma}^{n,{\\rm up}}\n\\frac{(\\bu^n_K-\\bu^n_L)^2} 2.\n\\end{equation}\n{\\bf Step 3:} {\\it Term $I_3+I_8$.}\nWe have\n\\begin{multline*}\nI_8=\\sum_{K\\in {\\cal T}}\\stik|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]\\,\\Big( H'(\\vr^n_K)( \\vr_\\sigma^{n,\\rm up}-\\vr^n_K)+ H(\\vr^n_K)\\Big)\n\\\\+ \\sum_{K\\in {\\cal T}}\\stik|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]\\,\\Big( \\vr^n_K H'(\\vr^n_K) -H(\\vr^n_K)\\Big).\n \\end{multline*}\nRecalling (\\ref{upwind1}), we may write the contribution of the face $\\sigma=K|L$ to the first sum in $I_8$; it reads\n\\begin{multline*}\n|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}]^+\\,\\Big(H(\\vr^n_K)- H'(\\vr^n_L)(\\vr^n_K-\\vr^n_L)-H(\\vr^n_L)\\Big)\n\\\\+|\\sigma|\\,[{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,L}]^+\\,\\Big(H(\\vr^n_L)- H'(\\vr^n_K)(\\vr^n_L-\\vr^n_K)-H(\\vr^n_K)\\Big).\n \\end{multline*}\nRecalling that $rH'(r)-H(r)=p(r)$, we get, employing the Taylor formula\n\\[\nI_3+I_8=\\sum_{\\sigma=K|L\\in {\\cal E}_{\\rm int}}\\,|{\\bu_\\sigma^n}\\cdot\\bn_{\\sigma,K}| H''(\\overline \\vr^n_{\\sigma})\\frac{(\\vr^n_K-\\vr^n_L\\Big)^2} 2\n\\]\nwith some $\\overline \\vr^n_{\\sigma}\\in [\\min(\\vr^n_K,\\vr^n_L), \\max(\\vr^n_K,\\vr^n_L)]$.\n\n\\vspace{2mm}\n\n{\\bf Step 4:} {\\it Conclusion}\n\nCollecting the results of Steps 1-3 we arrive at\n\\begin{multline}\\label{denergyinequality1}\n\\sum_{K\\in {\\cal T}}\\frac 12\\frac{|K|}{\\deltat}\\Big(\\vr^n_K|{\\bu}^n_K|^2-\\vr^{n-1}_K|{\\bu}^{n-1}_K|^2\\Big)\n+ \\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\Big(H(\\vr_K^n)-H(\\vr_K^{n-1})\\Big) +\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad\\bu^n|^2 \\dx\n\\\\+( \\mu+\\lambda)\\int_K|{\\rm div}\\bu^n|^2 \\dx\\Big)\n+\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}\\vr_K^{n-1}\\frac {|{\\bu}_K^n-{\\bu}_K^{n-1}|^2} 2\n+\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat}H''(\\overline\\vr_K^{n-1,n})\\frac {|{\\vr}_K^n-{\\vr }_K^{n-1}|^2} 2\n\\\\+ \\sum_{\\substack{\\sigma\\in{\\cal E}_{\\rm int} \\\\ \\sigma=K|L}}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\frac{({\\bu}^n_K-{{\\bu}}^n_L)^2} 2\\;|{\\bu}^n_\\sigma\\cdot\\bn_{\\sigma,K}|\n+\\sum_{\\substack{\\sigma\\in{\\cal E}_{\\rm int} \\\\ \\sigma=K|L}}|\\sigma|H''(\\overline \\vr^n_{\\sigma})\\frac{(\\vr^n_K-\\vr^n_L\\Big)^2} 2 \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|= 0.\n\\end{multline}\nAt this stage, we get the statement of Lemma \\ref{Theorem3} by multiplying (\\ref{denergyinequality1})$^{n}$ by $\\deltat$ and summing from $n=1$ to $n=m$. Lemma \\ref{Theorem3} is proved.\n\\end{proof}\n\n\\subsection{Estimates}\n\n\n\nWe have the following corollary of Lemma \\ref{Theorem3}.\n\\begin{cor}\\label{Corollary1}\n\\begin{description}\n\\item{(1)}\nUnder assumptions of Lemma \\ref{Theorem3}, there exists $c=c(M_0,E_0)>0$ (independent of $h$ and $\\deltat$) such that\n\\begin{equation}\\label{est0}\n|\\bu|_{L^2(0,T;V^2_h(\\Omega;\\Rm^3)}\\le c\n\\end{equation}\n\\begin{equation}\\label{est1}\n\\|\\bu\\|_{L^2(0,T;L^6(\\Omega;\\Rm^3))}\\le c\n\\end{equation}\n\\begin{equation}\\label{est2}\n\\|\\vr\\hat{\\bu}^2\\|_{L^\\infty(0,T;L^1(\\Omega))}\\le c.\n\\end{equation}\n \\item{(2)} If in addition the pressure satisfies assumption (\\ref{pressure1}) then\n\\begin{equation}\\label{est3}\n\\|\\vr\\|_{L^\\infty(0,T;L^\\gamma(\\Omega))}\\le c\n\\end{equation}\n\\item{(3)} If the pair $(r,\\bU)$ belongs to the class (\\ref{dr,U}) there exists $c=c(M_0,E_0,\\underline r,\\overline r, \\|\\bU, \\nabla \\bU\\|_{L^\\infty(Q_T;\\Rm^{12})})>0$\nsuch that for all $n=1,\\ldots,N$,\n\\begin{equation}\\label{est4}\n{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n)\\le c,\n\\end{equation}\nwhere the discrete relative energy ${\\cal E}$ is defined in (\\ref{dent}).\n\\end{description}\n\\end{cor}\n\\begin{proof}\nRecall that\n\\[|\\bu|^2_{L^2(0,T;V^2_h(\\Omega;\\Rm^3)} = \\deltat \\sum_{n=1}^N\\sum_{K\\in {\\cal T}}\\int_K|\\Grad\\bu^n|^2 \\dx;\\]\nthe estimate (\\ref{est0}) follows from (\\ref{denergyinequality}).\nThe estimate (\\ref{est1}) holds due to imbedding (\\ref{sob1}) in Lemma \\ref{Lemma2+} and bound (\\ref{est0}).\nThe estimate (\\ref{est2}) is just a short transcription of the bound for the kinetic energy in (\\ref{denergyinequality}).\n\n{ We prove estimate (\\ref{est3}). First, we deduce from (\\ref{hypp}) and the definition (\\ref{H}) of $H$ that $0\\le -H(\\vr)\\le c_1$ with some $c_1>0$, provided $0<\\vr \\le 1$ and $H(\\vr)> 0$ if $\\vr>1$. This fact in combination with the bound for $\\int_\\Omega H(\\vr){\\rm d}x$ derived in (\\ref{denergyinequality}) yields\n \\begin{equation}\\label{H+}\n \\int_\\Omega |H(\\vr)|{\\rm d}x\\le c<\\infty.\n\\end{equation}\nSecond, relations (\\ref{hypp}--\\ref{pressure2}) imply that there are $\\overline\\vr>1$ and $0<\\underline p<\\overline p<\\infty$ such that\n$$\n\\left\\{\n\\begin{array}{c}\n\\vr^{\\alpha}p_0\/2\\le\\frac {p(\\vr)}{\\vr^2}\\mbox{if $0<\\vr<1\/\\overline\\vr$},\\\\\n\\underline p\\le \\frac {p(\\vr)}{\\vr^2}\\le\\overline p\\;\\mbox{if $1\/\\overline\\vr\\le\\vr\\le \\overline\\vr$},\\\\\n\\vr^{\\gamma-2} {p_\\infty}\/2\\le \\frac {p(\\vr)}{\\vr^2}\\;\\mbox{if $\\vr>2\\overline\\vr$}\n\\end{array}\n\\right\\}.\n$$\nUsing these bounds and the definition (\\ref{H}) of $H$ we verify that\n$$\n\\vr^\\gamma\\le c (|H(\\vr)|+ \\vr +1)\n$$\nwith a convenient positive constant $c$. Now, bound (\\ref{est3}) follows readily\nfrom the boundedness of\n$\\int_\\Omega\\vr^m{\\rm d}x\\equiv\\sum_{K\\in {\\cal T}}|K|\\vr_K^m$ and $\\int_\\Omega H(\\vr^m){\\rm d}x\\equiv\\sum_{K\\in {\\cal T}}{|K|} H(\\vr_K^m)$ established in (\\ref{masscons}) and\n(\\ref{denergyinequality}).\n}\n\n Finally, to get (\\ref{est4}), we have employed (\\ref{E}), (\\ref{dent}), (\\ref{masscons}), (\\ref{H+})\nto estimate $\\int_\\Omega E(\\vr^n|\\hat r^n)\\dx$ and (\\ref{est2}), (\\ref{L2-3}), (\\ref{L1-3}) to evaluate $\\sum_{K\\in {\\cal T}}\\int_K\\vr_K^n|\\bU_{h,K}^n-\\bu_K^n|^2\\dx$.\n\\end{proof}\n\nThe following estimates are obtained thanks to the numerical diffusion due to the upwinding, as is classical in the framework of hyperbolic conservation laws, see e.g. \\cite{egh-book}.\n\n\\begin{lm}[Dissipation estimates on the density]\\label{Lemma5}\nLet $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$.\nSuppose that $(\\vr^n)_{1\\le n \\le N}\\subset [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\subset [\\bW_h]^N(\\Omega)$ is a solution of problem \\eqref{scheme}.\nFinally assume that the pressure satisfies hypotheses (\\ref{hypp}) and\n(\\ref{pressure1}).\nThen we have:\n\\begin{description}\n\\item {(1)} If $\\gamma\\ge 2$ then there exists $c=c(\\gamma,\\theta_0, E_0)>0$ such that\n\\begin{equation}\\label{dissipative2}\n\\deltat \\sum_{n=1}^N\\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|\\frac{(\\vr^n_K-\\vr^n_L)^2}{{\\rm max}(\\vr^n_K,\\vr^n_L)} \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|\\le c.\n\\end{equation}\n\\item{(2)} If $\\gamma\\in [1, 2)$ and the pressure satisfies additionally assumption (\\ref{pressure2})\nthen there exists $c=c(M_0, E_0)>0$ such that\n\\begin{multline}\\label{dissipative1}\n\\deltat\\sum_{n=1}^N \\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|\\frac{(\\vr^n_K-\\vr^n_L)^2}{[{\\rm max}(\\vr^n_K,\\vr^n_L)]^{2-\\gamma}} 1_{\\{\\overline\\vr^n_\\sigma\\ge 1\\}} \\;|{\\bu}^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|\n\\\\+\n\\deltat \\sum_{n=1}^N\\sum_{\\sigma=K|L \\in{\\cal E}_{\\rm int}}|\\sigma|(\\vr^n_K-\\vr^n_L)^2 1_{\\{\\overline\\vr^n_\\sigma <1\\}} \\;|\\bu^n_\\sigma\\cdot{\\vc n}_{\\sigma,K}|\\le c,\n\\end{multline}\nwhere the numbers $\\overline\\vr^n_\\sigma$ are defined in Lemma \\ref{Theorem3}.\n\\end{description}\n\\end{lm}\n\\begin{proof} We start by proving the simpler statement \\textit{ (2)}.\nTaking into account the continuity of the pressure, we deduce from assumptions (\\ref{pressure1}) and (\\ref{pressure2}) that there exist numbers $\\overline p_0>0$, $\\overline p_\\infty>0$ such that\n\\[\nH''(s)\\ge \\begin{cases}\n \\frac{\\overline p_\\infty}{s^{2-\\gamma}},\\quad\\mbox{if } s\\ge 1,\n\\\\ \\overline p_0 s^\\alpha\\ge\n\\overline p_0, \\quad\\mbox{if }s< 1,.\n \\end{cases}\n\\]\nwhence, splitting the sum in the definition of the term $[D^{N,\\Delta\\vr}_{\\rm space}]$ (see (\\ref{upwinddissipation_4})) into two sums, where $(\\sigma, n)$ satisfies $\\overline\\vr_\\sigma^n\\ge 1$ for the first one and $\\overline\\vr_\\sigma^n< 1$ for the second, we obtain the desired result.\n\nLet us now turn to the proof of statement \\textit{(1)}.\nMultiplying the discrete continuity equation (\\ref{dcont})$_K^n$ by $\\ln \\vr_K^n$ and summing over $K\\in {\\cal T}$, we get\n\\begin{equation*}\n\\sth |K| \\frac{ \\vr_{K}^n - \\vr_{K}^{n-1}}{\\deltat}\\ln\\vr_K^n + \\sth \\sum_{\\sigma\\in {\\cal E}(K),\\sigma=K|L}(\\ln\\vr_K^n)\\vr_\\sigma^{n,{\\rm up}} \\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}=0.\n\\end{equation*}\nBy virtue of the convexity of the function $\\vr\\mapsto \\vr\\ln\\vr-\\vr$ on the positive real line, and due to the Taylor formula, we have\n\\[\n\\vr_K^n\\ln \\vr_K^n-\\vr_K^{n-1}\\ln \\vr_K^{n-1} -( \\vr_K^n-\\vr_K^{n-1})\\le \\ln \\vr_K^n (\\vr_K^n-\\vr_K^{n-1});\n\\]\nwhence, thanks to the mass conservation \\eqref{masscons} and the definition of $\\vr_\\sigma^{{\\rm up}}$, we arrive at\n\\begin{multline*}\n\\sth |K| \\frac{ \\vr_{K}^n\\ln\\vr_K^n- \\vr_{K}^{n-1}\\ln\\vr_K^{n-1}}{\\deltat}\n+ \\stkl |\\sigma| \\vr_K^n [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ \\Big( \\ln\\vr_K^n-\\ln\\vr_L^n\\Big)\n\\\\+ \\stkl |\\sigma| \\vr_L^n [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+ \\Big( \\ln\\vr_L^n-\\ln\\vr_K^n\\Big)\\le 0,\n\\end{multline*}\nor equivalently\n\\begin{multline}\\label{est5}\n\\deltat\\stkl |\\sigma| [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ \\Big( \\vr_K^n(\\ln\\vr_K^n-\\ln\\vr_L^n) -(\\vr_K^n-\\vr_L^n)\\Big)\n+\\deltat\\stkl |\\sigma| [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+ \\Big( \\vr_L^n ( \\ln\\vr_L^n-\\ln\\vr_K^n)-(\\vr_L^n-\\vr_K^n)\\Big)\\le\n\\\\-\\sth |K| \\Big({ \\vr_{K}^n\\ln\\vr_K^n- \\vr_{K}^{n-1}\\ln\\vr_K^{n-1}}\\Big)\n +\\deltat\\stkl |\\sigma| \\Big([\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ (\\vr_L^n-\\vr_K^n))+ [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+ (\\vr_K^n-\\vr_L^n))\\Big).\n\\end{multline}\nFrom \\cite[Lemma C.5]{FettahGallouet2013Stokes}, we know that if $\\varphi$ and $\\psi$ are functions in $C^1((0,\\infty);\\R)$ such that $s\\psi'(s)=\\varphi'(s)$ for all $ s \\in (0,\\infty)$, then for any $(a,b)\\in (0,\\infty)^2$ there exits $c \\in [a,b]$ such that\n\\[\n\t\t(\\psi(b)-\\psi(a))b -(\\varphi(b)-\\varphi(a)) = \\frac{1}{2}(b-a)^2 \\psi'(c).\n\\]\nApplying this result with $\\psi(s)=\\ln s$, $\\varphi(s)=s$ we obtain that the left hand side of \\eqref{est5} is greater or equal to\n\\[\n\\deltat\\stkl |\\sigma| \\Big([\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}]^+ + [\\bu_\\sigma^n \\cdot \\bn_{\\sigma,L}]^+\\Big)\n\\frac{(\\vr^n_K-\\vr^n_L)^2}{\\max(\\vr^n_K,\\vr^n_L)}.\n\\]\nOn the other hand, the first term at the right hand side is bounded from above by $\\|\\vr^n\\|_{L^\\gamma(\\Omega)}^\\gamma$.\n Finally the second term at the right hand side is equal to\n\\[\n-\\deltat\\sum_{K\\in {\\cal T}}\\int_K\\vr_K^n{\\rm div}\\bu^n {\\le \\deltat\\sum_{K\\in {\\cal T}}\\|\\vr_K\\|_{L^2(K)}\n\\|{\\rm div}\\vc u^n\\|_{L^2(K)},}\n\\]\nwhence bounded from above by $\\deltat\\|\\bu^n\\|_{V_h^2(\\Omega;\\Rm^3)}\\|\\vr^n\\|_{L^2(\\Omega)}$, { where we have used the H\\\"older inequality and the definition of the $V_h^2(\\Omega)$-norm}. The statement \\textit{(1)} of Lemma \\ref{Lemma5} now follows from the estimates of Corollary \\ref{Corollary1}.\n\n\\end{proof}\n\n\n\\section{Exact relative energy inequality for the discrete problem}\\label{5}\n\nThe goal of this section is to prove the discrete version of the relative energy inequality.\n\n\\begin{Theorem}\\label{Theorem4}\n\nSuppose that $\\Omega\\subset \\R^3$ is a polyhedral domain and ${\\cal T}$ its regular triangulation { introduced in Section \\ref{3.1}}. Let $p$ satisfy hypotheses (\\ref{hypp}) and the viscosity coefficient $\\mu$, $\\lambda$ obey (\\ref{visc}).\nLet $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and suppose that $(\\vr^n)_{1\\le n \\le N}\\in [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme}.\nThen there holds for all $m=1,\\ldots,N$,\n\\begin{equation}\\label{drelativeenergy}\n\\begin{aligned}\n&\\sum_{K\\in {\\cal T}}\\frac 12{|K|}\\Big(\\vr^{ m}_K|{\\bu}^m_K-{\\bU}^m_{h,K}|^2-\\vr^{0}_K|{\\bu}^{0}_K-{\\bU}^{0}_{h,K}|^2\\Big)\n+ \\sum_{K\\in {\\cal T}}{|K|}\\Big(E(\\vr_K^m| r_K^m)-E(\\vr_K^{0}| r_K^{0})\\Big)\n\\\\ &\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad(\\bu^n-\\bU^n_h)|^2 \\dx+( \\mu +\\lambda)\\int_K|{\\rm div}(\\bu^n-\\bU^n_h)|^2 \\dx\\Big)\n\\\\&\\phantom{\\sum_{K\\in {\\cal T}}} \\le\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K\\Grad\\bU^n_h:\\Grad(\\bU^n_h-\\bu^n) \\dx+(\\mu +\\lambda)\\int_K{\\rm div}\\bU^n_h{\\rm div}(\\bU^n_h-\\bu^n) \\dx\\Big)\n\\\\&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}{|K|}\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big(\\frac{{\\bU}_{h,K}^{n-1} + {\\bU}_{h,K}^{n} }2 -\n\\bu_K^{n-1}\\Big)\n\\\\\n&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad-\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\stik|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\frac{\\bU^n_{h,K}+\\bU^n_{h,L}}2-\n\\hat{\\bu}_{\\sigma}^{n,{\\rm up}}\\Big)\\cdot{\\bU}^n_{h,K} [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]\n\\\\\n&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad-\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\stik |\\sigma|p(\\vr^n_K)[{\\bU}_{h,\\sigma}^{n}\\cdot\\bn_{\\sigma,K}]\n\\\\&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\n\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\frac{|K|}{\\deltat} ( r^n_K-\\vr^n_K)\\Big(H'( r^n_K)-H'( r^{n-1}_K)\\Big)\n\\\\\n&\\phantom{\\sum_{K\\in {\\cal T}}}\\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\stik |\\sigma|\\vr_\\sigma^{n,{\\rm up}}H'( r_K^{ n-1})[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n\\end{aligned}\n\\end{equation}\nfor any $ 00\n\\]\n(where $\\overline r=\\max_{(t,x)\\in \\overline {Q_T}} r(t,x)$, $\\underline r=\\min_{(t,x)\\in \\overline {Q_T}} r(t,x)$),\nsuch that for all $m=1,\\ldots,N$, we have:\n\\begin{equation}\\label{relativeenergy-}\n\\begin{aligned}\n\t&{\\cal E}(\\vr^m,\\bu^m\\Big| r^m, \\bU^m)- {\\cal E}(\\vr^0,\\bu^0\\Big|r(0),\\bU(0))\n\t\\\\& \\qquad \\qquad \\qquad +\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K|\\Grad(\\bu^n-\\bU^n_h)|^2 \\dx+( \\mu +\\lambda)\\int_K|{\\rm div}(\\bu^n-\\bU^n_h)|^2 \\dx\\Big)\n\t\\\\& \\qquad \\le \\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K\\Grad\\bU^n_h:\\Grad(\\bU^n_h-\\bu^n) \\dx+(\\mu +\\lambda)\\int_K{\\rm div}\\bU^n_h{\\rm div}(\\bU^n_h-\\bu^n) \\dx\\Big)\n\t\\\\& \\qquad+\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n} -\\bu_K^{n}\\Big)\n\t\\\\ &\\qquad { + \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\hat\\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot\\bn_{\\sigma,K}}\n\t\\\\& \\qquad-\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} \\int_K p(\\vr_K^n)\\dv\\bU^n\\dx\n\t+\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n} [\\partial_t r]^n\\dx\n\t\\\\ &\\qquad -\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\int_K\\frac{\\vr^n_K}{r^n_K} p'(r^n_K) { \\bu^n}\\cdot\\nabla r^n\\dx\t\n\\, { + R^m_{h,\\deltat}} { + G^m}\n\\end{aligned}\n\\end{equation}\nfor any pair $(r,\\bU)$ belonging to the class (\\ref{dr,U}), where\n\\begin{equation}\\label{A1}\n{ |G^m|\\le c\\,\\deltat\\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n\\Big| r^n, U^n),}\\;\\;\n{ |R^m_{h,\\deltat}|\\le c (\\sqrt{\\deltat} + h^A)},\\quad \\mbox{ and } A=\\left\\{\\begin{array}{c}\n{ \\frac{2\\gamma-3}{\\gamma}}\\;\\mbox{if $\\gamma\\in [3\/2,2)$}\n\\\\\n1\/2 \\;\\mbox{ if { $\\gamma\\ge 2$}},\n\\end{array}\n\\right.\n\\end{equation}\n { and where we have used notation (\\ref{notation2-}) for $r^n$, $\\bU^n$ and (\\ref{vhat}--\\ref{vtilde}) for $\\vc U^n_h$, $\\vc U^n_{h,K}$, $r^n_K$, $\\vc u^n_{\\sigma}$.}\n\\end{lm}\n\n\n\n\\begin{proof}\n\\label{6.0}\nThe right hand side of the relative energy inequality (\\ref{drelativeenergy}) is a sum $\\sum_{i=1}^6T_i$, where\n\\begin{align*}\n& T_1=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\Big(\\mu\\int_K\\Grad\\bU^n_h:\\Grad(\\bU^n_h-\\bu^n) \\dx+(\\mu +\\lambda)\\int_K{\\rm div}\\bU^n_h{\\rm div}(\\bU^n_h-\\bu^n) \\dx\\Big),\n\\\\ & T_2=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}{|K|}\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big(\\frac{{\\bU}_{h,K}^{n-1} + {\\bU}_{h,K}^{n} }2 -\n\\bu_K^{n-1}\\Big),\n\\\\ & T_3=-\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\frac{\\bU^n_{h,K}+\\bU^n_{h,L}}2-\n\\hat {\\bu}_{\\sigma}^{n,{\\rm up}}\\Big)\\cdot{\\bU}^n_{h,K} [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n\\\\ & T_4 = -\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|p(\\vr_K)[{\\bU}_{h,\\sigma}^{n}\\cdot\\bn_{\\sigma,K}],\n\\\\ & T_5=\n\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}|K| ( r^n_K-\\vr^n_K)\\frac{H'( r_K^{n})-H'(r^{n-1}_K)}{\\deltat},\n\\\\& T_6=\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}H'( r_K^{ n-1})[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}].\n \\end{align*}\n\nThe term $T_1$ will be kept as it is; all the other terms $T_i$ will be transformed to a more convenient form, as described in the following steps.\n\n\\vspace{2mm}\\noindent\n{\\bf Step 1:} {\\it Term $T_2$.}\n\\label{6.1}\nWe have\n\\[\nT_2=T_{2,1}+ R_{2,1},\\mbox{ with }T_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K\n}^{n-1} -\n\\bu_K^{n-1}\\Big),\n\\mbox{ and }R_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}R^{n,K}_{2,1},\n\\]\nwhere\n$$\n R_{2,1}^{n,K}=\n\\frac {|K|}2\n\\vr_K^{n-1}\\frac{({\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1})^2}{\\deltat}={ \\frac {|K|}2\n\\vr_K^{n-1}\\frac{([{\\bU}^{n}-{\\bU}^{n-1}]_{h,K})^2}{\\deltat}}.\n$$\nWe may write by virtue of the first order Taylor formula applied to function $t\\mapsto \\vc U(t,x)$,\n$$\n\\Big|\\frac{[{\\bU}^{n}-{\\bU}^{n-1}]_{h,K}}{\\deltat}\\Big|=\n\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\Big[\\int_{t_{n-1}}^{t_n} \\partial_t\\vc U(z,x) {\\rm d} z\\Big]_h\\Big]{\\rm d}x\\Big|\n$$\n$$\n=\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\int_{t_{n-1}}^{t_n} [\\partial_t\\vc U(z) \\Big]_h(x){\\rm d} z\\Big]{\\rm d}x\\Big|\\le \\|[\\partial_t\\vc U ]_h\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))}\\le \\|\\partial_t\\vc U \\Big\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))},\n$$\nwhere we have used the property (\\ref{ddd}) of the projection onto the space $V_h{ (\\Omega)}$.\nTherefore, thanks to the mass conservation \\eqref{masscons}, we finally get\n\\begin{equation}\\label{R2.1}\n|R^{n,K}_{2,1}|\\le\\frac {M_0} 2|K|\\deltat\\|\\partial_t\\bU\\|^2_{L^\\infty(0,T; L^{\\infty}(\\Omega;\\Rm^3))}.\n\\end{equation}\n\nLet us now decompose the term $T_{2,1}$ as\n\\begin{equation}\\label{T2}\nT_{2,1}=T_{2,2}+R_{2,2},\\mbox{ with }T_{2,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K\n}^{n} - \\bu_K^{n}\\Big),\n \\mbox{ and } R_{2,2}=\\deltat\\sum_{n=1}^m R_{2,2}^n,\n\\end{equation}\nwhere $\\displaystyle\n\tR^n_{2,2}=\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n-1} - \\bU_{h,K}^{n}\\Big)\n \t- \\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bu}_{K}^{n-1} \t\t -\\bu_{K}^{n}\\Big).$\n\nBy the same token as above, we may estimate the residual term as follows\n\\[\n|R^n_{2,2}|\\le \\deltat\\; c M_0 \\|\\partial_t\\bU\\|^2_{L^\\infty(0,T;W^{1,\\infty}(\\Omega;\\Rm^3)}+\nc M_0^{1\/2}\\Big(\\sum_{K\\in{\\cal T}}|K|\\vr_K^{n-1}|{\\bu}_{K\n}^{n-1} -\n\\bu_{K}^{n}|^2\\Big)^{1\/2} \\|\\partial_t\\bU\\|_{L^\\infty(0,T;L^{\\infty}(\\Omega;\\Rm^3))},\n\\]\nwhere we have used the H\\\"older inequality to treat the second term;\nwhence, by virtue of estimate (\\ref{upwinddissipation_1}),\n\\begin{equation}\\label{R2.2}\n|R_{2,2}|\\le \\sqrt{\\deltat} \\,c(M_0, E_0, \\|(\\partial_t\\bU, \\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})}).\n\\end{equation}\n\n\n\n\n\\vspace{2mm}\\noindent\n{\\bf Step 2:} {\\it Term $T_3$.}\n\\label{6.2}\nEmploying the definition (\\ref{upwind1}) of upwind quantities, we easily establish that\n\\begin{align*}\n& T_3= T_{3,1} + R_{3,1}, \\\\\n& \\mbox{with }T_{3,1}= \\deltat\\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E(K)}}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\hat {\\bu}_{\\sigma}^{n,{\\rm up}}-\n\\hat {\\bU}^{n, {\\rm up}}_{h,\\sigma}\\Big)\\cdot{\\bU}^n_{h,K} \\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}, \\quad\nR_{3,1}=\n\\deltat\\sum_{n=1}^m { \\sum_{\\sigma \\in {\\cal E}_{\\rm int}}R_{3,1}^{n,\\sigma}}, \\\\\n&\\mbox{and }{ R_{3,1}^{n,\\sigma}}= |\\sigma|\\vr_K^n \\frac{|\\bU_{h,K}^n-\\bU_{h,L}^n|^2}2\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]^+\n+ |\\sigma|\\vr_L^n \\frac{|\\bU_{h,L}^n-\\bU_{h,K}^n|^2}2\\,[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,L}]^+, \\; \\forall \\sigma=K|L \\in {\\cal E}_{\\rm int}.\n\\end{align*}\n{ Writing\n$$\n\\bU^n_{h,K}-\\bU^n_{h,L}= \\bU^n_{h,K}-\\bU^n_{h,\\sigma}+\\bU^n_{h,\\sigma}-\\bU^n_{h,L},\\; \\sigma=K|L\\in {\\cal E}_{\\rm int},\n$$\n}\nemploying estimates (\\ref{L2-1}) and (\\ref{L1-3})$_{s=1}$ and the continuity of the mean value $\\bU^n_\\sigma{ =\\bU_{h,\\sigma}^n}$ of $\\bU^n_h$ over faces $\\sigma$, we infer by using the Taylor formula applied to function\n$x\\mapsto U^n(x)$,\n\\[\n\t{ |R_{3,1}^{n,\\sigma}|}\\le h^2\\;c \\|\\nabla\\bU\\|^2_{L^\\infty (Q_T;\\Rm^9)} |\\sigma|(\\vr^n_K+\\vr^n_L) |\\bu^n_\\sigma|, \\; \\forall \\sigma=K|L \\in {\\cal E}_{\\rm int},\n\\]\nwhence\n\\begin{equation}\\label{R3.1}\n\\begin{aligned}\n |R_{3,1}| & \\le h\\;c \\|\\nabla\\bU\\|^2_{L^\\infty (Q_T;\\Rm^9)}\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)} h|\\sigma|(\\vr^n_K+\\vr^n_L)^{6\/5}\\Big)^{5\/6}\n\\Big[\\deltat \\sum_{n=1}^m\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma||\\bu^n_\\sigma|^6\\Big)^{1\/3}\\Big]^{1\/2}\n\\\\ &\\le h \\; c(M_0,E_0,\\|\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^{9})}),\n\\end{aligned}\n\\end{equation}\nprovided $\\gamma\\ge 6\/5$,\nthanks to the discrete H\\\"older inequality, the equivalence relation (\\ref{reg1}), the equivalence of norms (\\ref{norms1}) and energy bounds listed in Corollary \\ref{Corollary1}.\n\nEvidently, for each face $\\sigma=K|L\\in {\\cal E}_{\\rm int}$,\n$\n\\bu_{\\sigma}^n\\cdot\\bn_{\\sigma,K}+\\bu_\\sigma^n\\cdot{\\vc n}_{\\sigma,L}=0;$ whence, finally\n\\begin{equation}\\label{T3.1}\nT_{3,1}= \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\\Big(\\hat\n{\\bu}_{\\sigma}^{n,{\\rm up}}-\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}\\Big)\\cdot\\Big({\\bU}^n_{h,K}-\\bU^n_\\sigma\\Big) \\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}\n\\end{equation}\n{ Let us now decompose the term $ T_{3,1}$ as\n\\begin{equation*}\n \\begin{aligned}\n&T_{3,1}= T_{3,2}+ R_{3,2}, \\mbox{ with } { R_{3,2}=\\deltat\\sum_{n=1}^mR^{n}_{3,2}}, \\\\\n&T_{3,2}= \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\hat\\bu_\\sigma^{n,{\\rm up}}\\cdot\\bn_{\\sigma,K}, \\mbox{ and }\\\\\n&{ R^{n}_{3,2}}=\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\Big(\\bu_{\\sigma}^n-\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big)\\cdot\\bn_{\\sigma,K}.\n \\end{aligned}\n\\end{equation*}\nBy virtue of { discrete} H\\\"older's inequality and the { first order Taylor formula applied to function $x\\mapsto \\vc U^n(x)$ in order to evaluate the difference $\\bU^n_\\sigma-{\\bU}^n_{h,K}$}, we get\n\\begin{equation*}\n \\begin{aligned}\n\t{ |R^{n}_{3,2}|} & \\le\n\tc \\|\\nabla\\vc U\\|_{L^\\infty(Q_T;\\Rm^9)}\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n\\vr_\\sigma^{n,{\\rm up}}\\Big|\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}}\\Big|^2\\Big)^{1\/2}\\\\\n& \\times\n\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n|\\vr_\\sigma^{n,{\\rm up}}|^{\\gamma_0}\\Big)^{1\/(2{\\gamma_0})}\\Big(\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n\\Big|\\bu_\\sigma^{n}-\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big|^q\\Big)^{1\/q},\n \\end{aligned}\n\\end{equation*}\nwhere $\\frac 12+\\frac 1{2\\gamma_0}+\\frac 1q=1$, $\\gamma_0={\\rm min}\\{\\gamma, 2\\}$ and $\\gamma\\ge 3\/2$. For the sum in the last term of the above product, we have\n$$\n\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n\\Big|\\bu_\\sigma^{n}-\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big|^q\\le c \\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}h|\\sigma|\n|\\bu_\\sigma^{n}-\\bu_K^{n}|^q\n$$\n$$\n\\le c \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\n\\Big(\\|\\bu_\\sigma^{n}-\\bu^n\\|_{L^q(K;\\Rm^3)}^q+\n\\sum_{K\\in {\\cal T}}\n\\|\\bu^n-\\bu_K^{n}\\|_{L^q(K;\\Rm^3)}^q\\Big)\n\\le c h^{\\frac {2\\gamma_0-3}{2\\gamma_0}q}|\\bu^n|_{V^2_h(\\Omega;\\Rm^3)}^q,\n$$\nwhere we have used the definition (\\ref{upwind1}), the Minkowski inequality and the interpolation inequalities\n(\\ref{L2+-1}--\\ref{L2+-2}). Now we can go back to the estimate of $R_{3,2}^n$ taking into account the upper bounds\n(\\ref{est0}), (\\ref{est3}--\\ref{est4}), in order to get\n \\begin{equation}\\label{R3.2}\n |R_{3,2}|\\le h^A\\;c (M_0,E_0,\\|\\nabla\\vc U\\|_{L^\\infty(Q_T;\\Rm^9)})\n \\end{equation}\n provided $\\gamma\\ge 3\/2$, where $A$ is given in (\\ref{A1}).\n\n Finally, we rewrite term $T_{3,2}$ as\n \\begin{equation}\\label{T3}\n \\begin{aligned}\n&T_{3,2}= T_{3,3}+ R_{3,3}, \\mbox{ with } { R_{3,3}=\\deltat\\sum_{n=1}^mR^{n}_{3,3}}, \\\\\n&T_{3,3}= \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\hat\\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot\\bn_{\\sigma,K}, \\mbox{ and }\\\\\n&{ R^{n}_{3,3}}=\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr_\\sigma^{n,{\\rm up}}\n\\Big(\\hat{\\bU}^{n,{\\rm up}}_{h,\\sigma}-\\hat{\\bu}^{n,{\\rm up}}_{\\sigma}\\Big)\\cdot\\Big(\\bU^n_\\sigma-{\\bU}^n_{h,K}\\Big) \\Big(\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\Big)\\cdot\\bn_{\\sigma,K};\n \\end{aligned}\n\\end{equation}\nwhence\n\\begin{equation}\\label{R3.3}\n|R_{3,3}|\\le c(\\|\\nabla\\bU\\|_{L^\\infty(Q_T,\\Rm^9)})\\; \\deltat\\sum_{n=1}^m{\\cal E}(\\vr^n,\\vc u^n\\,|\\, r^n,\\bU^n).\n\\end{equation}\n\n }\n\n\n\n\\vspace{2mm}\\noindent\n{\\bf Step 3:} {\\it Term $T_4$.}\n\\label{6.3}\nUsing the Stokes formula and the property (\\ref{L1-1}) in Lemma \\ref{Lemma1}, we easily see that\n \\begin{equation}\\label{T4}\n T_4=-\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{{\\rm \\int_K}} p(\\vr_K^n)\\dv\\bU^n\\dx.\n \\end{equation}\n\n\\vspace{2mm}\\noindent\n{\\bf Step 4:} {\\it Term $T_5$.}\n \\label{6.4}\nUsing the Taylor formula, we get\n\\[\nH'(r_K^n)-H'(r_K^{n-1})=H''(r_K^{n})(r_K^n-r_K^{n-1}) -\\frac 12H'''(\\overline r_K^n)(r_K^n-r_K^{n-1})^2,\n\\]\nwhere $\\overline r_K^n\\in[\\min(r_K^{n-1},r_K^n), \\max(r_K^{n-1},r_K^n)]$;\nwe infer\n\\begin{equation*}\n\\begin{aligned}\n\t& T_5= T_{5,1}+ R_{5,1},\\mbox{ with } T_{5,1}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}|K| ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n} \\frac{r_K^n-r_K^{n-1}}{\\deltat}, \\, R_{5,1}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} R_{5,1}^{n,K}, \\mbox{ and } \\\\\n\t& R_{5,1}^{n,K}= \\frac 12|K|H'''(\\overline r_K^n)\\frac{(r_K^n-r_K^{n-1})^2}{\\deltat}(\\vr_K^n-r_K^n).\n \\end{aligned}\n\\end{equation*}\nConsequently, by the { first order Taylor formula applied to function $t\\mapsto r(t,x)$ on the interval $(t_{n-1}, t_n)$} and thanks to the mass conservation \\eqref{masscons}\n\\begin{equation}\\label{R5.1}\n\t|R_{5,1}|\\le \\deltat \\;c(M_0,\\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r]},\\|\\partial_t r\\|_{L^\\infty(Q_T)}),\n\\end{equation}\nwhere $\\underline r$, $\\overline r$ are defined in (\\ref{dr,U}).\n\n\\vspace{2mm}\nLet us now decompose $T_{5,1}$ as follows:\n\\begin{equation}\\label{T5}\n\\begin{aligned}\n\t& T_{5,1}=T_{5,2}+ R_{5,2}, \\mbox{ with }T_{5,2}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{ \\int_K} ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n} [\\partial_t r]^n { {\\rm d}x}, \\, R_{5,2}=\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} R_{5,2}^{n,K}, \\mbox{ and} \\\\\n\t& R_{5,2}^{n,K}= {\\int_K} ( r^n_K-\\vr^n_K)\\frac{p'(r_K^n)}{r_K^n}\\Big(\\frac{r_K^n-r_K^{n-1}}{\\deltat} -[\\partial_t r]^n\\Big){ {\\rm d} x}. \\end{aligned}\n\\end{equation}\nIn accordance with (\\ref{notation2-}), here and in the sequel, $[\\partial_t r]^n(x)=\\partial_t r(t_n,x)$.\n{ We write using twice the Taylor formula in the integral form and the Fubini theorem,\n$$\n|R_{5,2}^{n,K}|= \\frac 1\\deltat\\Big|{p'(r^n_K)}{r^n_K} (\\vr^n_K-r^n_K)\\int_K\\int^{t_n}_{t_{n-1}}\\int_s^{t_n}\\partial_t ^2 r(z){\\rm d}z{\\rm d}s {\\rm d}x\\Big|\n$$\n$$\n\\le \\frac {p'(r^n_K)}{r^n_K}\\int^{t_n}_{t_{n-1}}\\int_K|\\vr^n_K-r^n_K|\\Big|\\partial_t ^2 r(z)\\Big|{\\rm d}x{\\rm d}z{\\rm d}s\n$$\n$$\n\\le \\frac{p'(r^n_K)}{r^n_K}\n\\|\\vr^n-\\hat r^n\\|_{L^{\\gamma}(K)}\\int^{t_n}_{t_{n-1}}\\|\\partial_t^2 r(z)\\|_{L^{\\gamma'}(K)}{\\rm d z}{\\rm d s}.\n$$\nTherefore, by virtue of Corollary \\ref{Corollary1}, we have estimate\n\\begin{equation}\\label{R5.2}\n\t|R_{5,2}|\\le \\deltat\\; c(M_0, E_0,\\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r]},\\|\\partial^2_t r\\|_{L^1(0,T; L^{\\gamma'}(\\Omega)}).\n\\end{equation}\n}\n\n\n\n\\vspace{2mm}\\noindent\n{\\bf Step 5:} {\\it Term $T_6$.}\nUsing the same argumentation as in formula (\\ref{T3.1}), we may write\n\\begin{equation}\n \\begin{aligned}\n\t &T_6=T_{6,1} + R_{6,1},\\quad R_{6,1}=\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}R_{6,1}^{n,\\sigma,K}, \\mbox{ with} \\\\\n\t&T_{6,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\vr_K^{n}\\Big( H'( r_K^{ {n-1}})-H'(r_\\sigma^{ {n-1}})\\Big)\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}, \\mbox{ and} \\\\\n\t& R_{6,1}^{n,\\sigma,K}=|\\sigma|\\Big(\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n}\\Big)\\Big( H'( r_K^{ {n-1}})-H'(r_\\sigma^{ {n-1}})\\Big)\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}, { \\mbox{ for } \\sigma=K|L.}\n \\end{aligned}\n\\end{equation}\nWe estimate this term separately for $\\gamma\\le 2$ and $\\gamma>2$.\nIf $\\gamma\\le 2$, motivated by Lemma \\ref{Lemma5}, we may write\n\\begin{multline}\\label{R6.1a}\n\t |R_{6,1} ^{n,\\sigma,K}| \\le \\sqrt h\\, \\|\\nabla H'(r)\\|_{L^\\infty(Q_T;\\Rm^3)} |\\sigma| \\\\\n\t\\times \\Big( \\frac { |\\vr_\\sigma^{n,{\\rm up}} - \\vr_K^{n} |} {\\max(\\vr_K,\\vr_L)^{(2-\\gamma)\/2}} \\,\\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}}| 1_{\\overline\\vr_\\sigma^n\\ge1} \\sqrt h(\\vr^n_K+\\vr^n_L)^{(2-\\gamma)\/2} \\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}}|\n\t\\\\ + |\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n}|\\,\\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|} 1_{\\overline\\vr_\\sigma^n<1}\n\\sqrt h \\sqrt{|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|} \\Big),\n\\end{multline}\nwhere we again use the { first order Taylor formula applied to function $H'$ between endpoints $r_K^{n-1}$,\n$r_\\sigma^{n-1}$,} and where the numbers $\\overline\\vr_\\sigma^n$ are defined in Lemma \\ref{Theorem3}.\nConsequently, an application of the H\\\"older and Young inequalities yields\n\\begin{equation}\\label{R6.1b}\n \\begin{aligned}\n |R_{6,1}|\n\t& \\le \\sqrt h\\, { c} \\|\\nabla H'(r)\\|_{L^\\infty(Q_T;\\Rm^3)}\n\t\t\\deltat \\sum_{n=1}^m\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma| \\frac {(\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2}{\\max(\\vr_K,\\vr_L)^{(2-\\gamma)}}\\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}| 1_{\\overline\\vr_\\sigma^n\\ge1}\\Big)^{1\/2}\n\t\t\\\\ & \\hspace{6cm} \\times \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\\vr_K^{2-\\gamma} \\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|\\Big)^{1\/2}\n\t\t\\\\ & \\quad + \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|h (\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2\\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}| 1_{\\overline\\vr_\\sigma^n<1}\\Big)^{1\/2}\n \t\t \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h \\,|\\bu_\\sigma^n\\cdot\\bn_{\\sigma,K}|\\Big)^{1\/2}\\Big]\n\t\\\\ & \\le \\sqrt h\\, { c} \\|\\nabla H'(r)\\|_{L^\\infty(Q_T;\\Rm^3)} \\deltat \\sum_{n=1}^m\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in \t{\\cal E}(K)}|\\sigma| \\frac {(\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2} {\\max(\\vr_K,\\vr_L)^{(2-\\gamma)}} \\, |\\bu_\\sigma^n \\cdot \\bn_{\\sigma,K}| 1_{\\overline\\vr_\\sigma^n\\ge1}\n\t\t\\\\ & \\qquad \\qquad + \\Big(\\sum_{K\\in {\\cal T}} |K|\\vr_K^{6(2-\\gamma)\/5}\\Big)^{5\/6} \\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\n\t\t\\\\ & \\qquad \\qquad + \\sum_{K\\in {\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|h (\\vr_\\sigma^{n,{\\rm up}}-\\vr_K^{n})^2\\,|\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}| 1_{\\overline\\vr_K^n<1}+ |\\Omega|^{5\/6} \\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\\Big]\n\\end{aligned}\n\\end{equation}\nWe deduce employing the discrete H\\\"older inequality\n{\n$$\n\\deltat \\sum_{n=1}^m\\Big(\\sum_{K\\in {\\cal T}} |K|\\vr_K^{6(2-\\gamma)\/5}\\Big)^{5\/6} \\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\n$$\n$$\n\\le \\Big[\\deltat \\sum_{n=1}^m \\Big(\\sum_{K\\in {\\cal T}} |K|\\vr_K^{6(2-\\gamma)\/5}\\Big)^{5\/3} \\Big]^{1\/2}\\Big[\\deltat \\sum_{n=1}^m\\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/3}\\Big]^{1\/2},\n$$\nand\n$$\n\\deltat\\sum_{n=1}^m\\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/6}\\le \\sqrt T\n\\Big[k\\sum_{n=1}^m\\Big(\\sum_{\\sigma\\in {\\cal E}}|\\sigma| h|\\bu_\\sigma^n|^6\\Big)^{1\/3}\\Big]^{1\/2}\n$$\n}\nComing back to (\\ref{R6.1b}) we deduce that\n\\begin{equation}\\label{R6.1}\n|R_{6,1}|\n\\le \\sqrt h \\; c(M_0,E_0,\\underline r,\\overline r, |p'|_{C([\\underline r,\\overline r])}, \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)})\n\\end{equation}\nprovided $\\gamma\\ge 12\/11$, where we use estimate (\\ref{dissipative1}), estimates (\\ref{est1}), (\\ref{est3}) of Corollary \\ref{Corollary1} and equivalence relation (\\ref{norms1}).\nIn the case $\\gamma>2$, the same final bound may be obtained by a similar argument, replacing the estimate (\\ref{dissipative1}) by (\\ref{dissipative2}).\n\n\\vspace{2mm}\nLet us now decompose the term $T_{6,1}$ as\n{\n\\begin{equation*}\n\\begin{aligned}\n\t&T_{6,1}=T_{6,2}+ R_{6,2}, \\mbox{ with } T_{6,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}\n|\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1}) [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}],\n \\\\ & R_{6,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal K}}\\sum_{\\sigma\\in {\\cal E}(K) }\nR_{6,2}^{n,\\sigma,K},\\;\n\\mbox{and }\\\\ & R_{6,2}^{n,\\sigma,K}=|\\sigma|\\vr^n_K \\Big(H'(r^{n-1}_K)-H'(r^{n-1}_\\sigma)-H''(r^{n-1}_K)(r^{n-1}_K-r^{n-1}_\\sigma)\\Big)\n[\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]\n \\end{aligned}\n\\end{equation*}\n}\nTherefore, by virtue of the { second order Taylor formula applied to function H'}, H\\\"older's inequality, (\\ref{sob1}), (\\ref{norms1}),\nand (\\ref{est0}), (\\ref{est3}) in Corollary \\ref{Corollary1}, we have, provided\n$\\gamma\\ge 6\/5$,\n{\n\\begin{align}\n| R_{6,2}| &\\le h c \\Big(|H''|_{C([\\underline r,\\overline r])}+|H'''|_{C([\\underline r,\\overline r])}\\Big)\\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} \\|\\vr\\|_{L^\\infty(0,T;L^\\gamma(\\Omega))} \\|\\bu\\|_{L^2(0,T;V_h^2(\\Omega;\\Rm^3))}\n\\nonumber \\\\\n\\label{R6.2}\n& \\le h\\;\nc(M_0,E_0,\\underline r, \\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} ),\n\\end{align}\n}\nwhere in the first line we have used notation (\\ref{notation1}).\n\n\\vspace{2mm}\n\nLet us now deal with the term $T_{6,2}$.\nNoting that $ \\displaystyle\n\\int_K\\nabla r^{ n-1} \\dx = \\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|(r_\\sigma^{ n-1}-r_K^{ n-1})\\bn_{\\sigma,K},$ we may write\n\\begin{multline*}\n\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1}) [\\bu^n_\\sigma\\cdot\\bn_{\\sigma,K}]\n\\\\= -\\int_K\\vr^n_K H''(r_K^{ n-1}) \\bu^n_K\\cdot\\nabla r^{ n-1}\\dx +\n\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1}) (\\bu^n_\\sigma-\\bu^n_K)\\cdot\\bn_{\\sigma,K}.\n\\end{multline*}\nConsequently, $T_{6,2}= T_{6,3}+ R_{6,3},$ with\n$$\n\\begin{aligned}\n& T_{6,3}=\n -\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\int_K{\\vr^n_K} H''(r_K^{ n-1}) \\bu^n \\cdot\\nabla r^{ n-1}\\dx, \\\\\n& R_{6,3} = \\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}} \\int_K\\vr^n_K H''(r_K^{ n-1})(\\bu^n- \\bu^n_K)\\cdot\\nabla r^{ n-1}\\dx\n \\\\ & \\hspace{3cm}+\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} |\\sigma|\\vr^n_K H''(r_K^{ n-1})(r_K^{ n-1}-r_\\sigma^{ n-1})\n(\\bu^n_\\sigma-\\bu^n_K)\\cdot\\bn_{\\sigma,K},\n\\end{aligned}\n$$\n{\n$$\n|R_{6,3}|\\le c\\,\\|H''(r)\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}\\Big[ \\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\|\\vr^n_K\\|_{L^{\\gamma_0}(K)}\\|\\bu^n- \\bu^n_K\\|_{L^{\\gamma_0'}(K;\\Rm^3)}+\n$$\n$$\n\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} \\|\\vr^n_K\\|_{L^{\\gamma_0}(K)}\\|\\bu_\\sigma^n- \\bu^n_K\\|_{L^{\\gamma_0'}(K;\\Rm^3)}\\Big],\\quad \\gamma_0=\\min\\{\\gamma,2\\},\n$$\nwhere we have used the H\\\"older inequality, and also the Taylor formula applied to function $x\\mapsto r(t_{n-1},x)$\ntogether with equivalence relation (\\ref{reg1}) yielding $|\\sigma|h\\le |K|$, to treat the second term.\nConsequently, by virtue of H\\\"older's inequality, interpolation inequality (\\ref{interpol1}) (to estimate\n$\\|\\vc u^n-\\vc u^n_K\\|_{L^{\\gamma_0'}(K;\\Rm^3)}$ by $h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\|\\Grad\\vc u^n \\|_{L^2(K;\\Rm^9)}$, $\\gamma_0=\\min\\{\\gamma,2\\}$)\nin the first term, and by the the H\\\"older inequality\n and (\\ref{interpol1}--\\ref{interpol2}) (to estimate $\\|\\vc u_\\sigma^n-\\vc u^n_K\\|_{L^{\\gamma'}(K;\\Rm^3)}$ by $h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\|\\Grad\\vc u^n \\|_{L^2(K;\\Rm^9)}$) in the second term, we get\n$$\n|R_{6,3}|\\le c\\,h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\,\\|H''(r)\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} \\deltat\\sum_{n=1}^m\\Big(\\sum_{K \\in {\\cal T}}\\|\\vr^n_K\\|^2_{L^{\\gamma_0}(K)}\\Big)^{1\/2}\\Big(\\sum_{K \\in {\\cal T}}\\|\\Grad\\bu^n\\|^2_{L^{2}(K;\\Rm^9)}\\Big)^{1\/2}\n$$\n$$\n\\le\nc\\,h^{(5\\gamma_0-6)\/(2\\gamma_0)}\\,\\|H''(r)\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} \\deltat\\sum_{n=1}^m\\Big(\\sum_{K \\in {\\cal T}}\\|\\vr^n_K\\|^{\\gamma_0}_{L^{\\gamma_0}(K)}\\Big)^{1\/\\gamma_0}\\Big(\\sum_{K \\in {\\cal T}}\\|\\Grad\\bu^n\\|^2_{L^{2}(K;\\Rm^9)}\\Big)^{1\/2},\n$$\nprovided $\\gamma\\ge 6\/5$,\nwhere we have used the discrete H\\\"older inequality and the algebraic inequality (\\ref{dod1*}).\n}\nNow it remains to use (\\ref{est0}), (\\ref{est3}) in Corollary \\ref{Corollary1} in order to get\n\\begin{equation}\\label{R6.3}\n|R_{6,3}|\\le h^A \\;c(M_0,E_0,\\underline r, \\overline r, |p'|_{C^1([\\underline r,\\overline r])} \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)} ),\n\\end{equation}\nwhere $A$ is defined in \\eqref{A1}.\n\n{\nFinally we write\n $T_{6,3}= T_{6,4}+ R_{6,4},$ with\n\\begin{equation}\\label{T6}\n\\begin{aligned}\n& T_{6,4}=\n -\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\int_K{\\vr^n_K}\\frac{ p'(r_K^{n})}{r_K^n} \\bu^n\\cdot\\nabla r^{ n}\\dx, \\\\\n& R_{6,4} = \\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}} \\int_K\\vr^n_K \\Big(H''(r_K^{ n})\\nabla r^{n}-\nH''(r_K^{n-1})\\nabla r^{n-1}\\Big)\\cdot\\bu^n\\dx,\n\\end{aligned}\n\\end{equation}\n{ where by the same token as in (\\ref{R5.2}),\n\\begin{equation}\\label{R6.4}\n|R_{6,4}|\\le \\deltat\\; c (M_0,E_0, \\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|\\nabla r, \\partial_t r\\|_{L^\\infty(Q_T;\\Rm^4)}, \\|\\partial_t\\nabla r\\|_{L^2(0,T; L^{{6\\gamma}\/{(5\\gamma-6)}}(\\Omega;\\Rm^3))}).\n\\end{equation}\n}\n}\n\n\n\\vspace{2mm}\nWe are now in position to conclude the proof of Lemma \\ref{6.6}: we obtain the inequality \\eqref{relativeenergy-} by gathering the principal terms (\\ref{T2}), (\\ref{T3}), (\\ref{T4}), (\\ref{T5}), (\\ref{T6}) and the residual terms estimated in (\\ref{R2.1}), (\\ref{R2.2}), (\\ref{R3.1}), (\\ref{R3.2}), (\\ref{R3.3}), (\\ref{R5.1}), (\\ref{R5.2}), (\\ref{R6.1a}), \\eqref{R6.1b}, (\\ref{R6.2}), (\\ref{R6.3}), { (\\ref{R6.4}) at} the right hand side $\\sum_{i=1}^6 T_i$ of the discrete relative energy inequality (\\ref{drelativeenergy}).\n\n\\end{proof}\n\n\\section{ A discrete identity satisfied by the strong solution}\\label{7}\n{ This section is devoted to the proof of a discrete identity satisfied by any strong solution. This identity\nis stated in Lemma \\ref{strongentropy} below. It will be used in combination with the approximate relative energy inequality stated in Lemma \\ref{refrelenergy} to deduce the convenient form of the relative energy inequality verified by any function being a strong solution to the compressible Navier-Stokes system. This last step is performed in the next section.}\n\n\n\\begin{lm}[A discrete identity for strong solutions]\\label{strongentropy}\nSuppose that $\\Omega\\subset \\R^3$ is a bounded polyhedral domain and ${\\cal T}$ a regular triangulation { introduced in Section \\ref{3.1}}.\nLet the pressure $p$ be a $C^2(0,\\infty)$ function satisfying hypotheses (\\ref{hypp}) and (\\ref{pressure1}) with\n$\\gamma\\ge 3\/2$. Let $(r,\\bU)$ belong to the class (\\ref{dr,U}) satisfy equation \\eqref{pbcont} with the viscosity coefficients $\\mu$, $\\lambda$ obeying (\\ref{visc}).\n\nLet $(\\vr^0,\\bu^0) \\in L_h^{+}(\\Omega) \\times \\bW_h(\\Omega)$ and suppose that $(\\vr^n)_{1\\le n \\le N}\\in [L_h^{+}(\\Omega)]^N$, $(\\bu^n)_{1\\le n \\le N} \\in [\\bW_h(\\Omega)]^N$ is a solution of the discrete problem \\eqref{scheme}.\nThen there exists\n\\begin{align*}\n& c=c\\Big(M_0, E_0, \\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])},\n\\|(\\nabla r, \\partial_t r, \\bU, \\nabla\\bU, { \\nabla^2\\bU}, \\partial_t\\bU, \\partial_t^2\\bU,\\partial_t\\nabla\\bU,)\\|_{L^\\infty (Q_T;\\Rm^{58})})\\Big)> 0,\n\\end{align*}\nsuch that for any $m=1,\\ldots,N$, the following identity holds:\n\\begin{equation}\\label{strong1}\n\\begin{aligned}\n\t&\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K\\Big(\\mu\\nabla\\bU_h^n\\cdot\\nabla(\\bu^n-\\bU^n_{ h})+ (\\mu +\\lambda)\\dv \\bU^n_{ h}\\dv (\\bu^n-\\bU^n_{ h})\\Big)\\dx\n\\\\\n\t&\\qquad +\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr_K^{n-1}\\frac{\\bU^n_{ h,K}-\\bU^{n-1}_{ h,K}}{\\deltat}\\cdot (\\bu_K^{ n}-\\bU_{h,K}^{ n})\\dx\n\\\\\n\t &\\qquad+ { \\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| \\hat r_\\sigma^{n,{\\rm up}}\n[\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}})}\n\\\\\n\t&\\qquad +\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K p(r_K^n)\\dv \\bU^n\\dx\n+\\deltat \\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\int_K p'(r^n_K)\\bu^n\\cdot\\nabla r^n\\dx\n{ + {\\cal R}^m_{h,\\deltat} =0},\n\\end{aligned}\n\\end{equation}\n{ where\n\\[\n|{\\cal R}_{h, \\deltat}^m|\\le c\\Big( h + \\deltat\\Big)\n\\]\n { and where we have used notation (\\ref{notation2-}) for $r^n$, $\\bU^n$ and (\\ref{vhat}--\\ref{vtilde}) for $\\vc U^n_h$, $\\vc U^n_{h,K}$, $r^n_K$, $\\vc u^n_{\\sigma}$.}\n}\n\\end{lm}\n{ Before starting the proof we recall an auxiliary algebraic inequality whose straightforward proof is left to the reader, and introduce some notations.}\n\\begin{lm}\\label{LL1}\n\tLet $p$ satisfies assumptions (\\ref{hypp}) and (\\ref{pressure1}).\n\tLet $00$ such that for all $\\vr\\in [0,\\infty)$ and $r\\in [a,b]$ there holds\n\t\\[\n\t\tE(\\vr|r)\\ge c(a,b)\\Big(1_{R_+\\setminus [a\/2,2 b]}{ (\\vr)}+\\vr^\\gamma 1_{R_+\\setminus [a\/2,2 b]}{ (\\vr)}+ (\\vr-r)^2 1_{[a\/2,2 b]}{ (\\vr)}\\Big),\n\t\\]\n\twhere $E(\\vr|r)$ is defined in (\\ref{E}).\n\\end{lm}\n{ If we take in Lemma \\ref{LL1} $\\vr=\\vr^n(x)$, $\\vr^n\\in L^+_h(\\Omega)$, $r=\\hat r^n(x)$, $a=\\underline r$, $b=\\overline r$ (where r is a function belonging to class (\\ref{dr,U}) and $\\underline r$, $\\overline r$\nare its lower and upper bounds, respectively), we obtain\n\\begin{equation}\\label{added}\nE(\\vr^n(x)|\\hat r^n(x))\\ge c(\\underline r,\\overline r)\\Big(1_{R_+\\setminus [\\underline r\/2,2 \\overline r]}{ (\\vr^n(x))}+(\\vr^n)^\\gamma(x) 1_{R_+\\setminus [\\underline r\/2,2 \\overline r]}{ (\\vr^n(x))}+ (\\vr^n(x)-\\hat r^n(x))^2 1_{[\\underline r\/2,2 \\overline r]}{ (\\vr^n(x))}\\Big)\n\\end{equation}\n}\n Now, for fixed numbers $\\underline r$ and $\\overline r$ { and fixed functions $\\vr^n\\in L^+_h(\\Omega)$, $n=0,\\ldots, N$, we introduce the residual and essential subsets of $\\Omega$ (relative to $\\vr^n$) as follows:}\n\\begin{equation}\\label{essres}\nN_{\\rm ess}^n=\\{x\\in\\Omega\\,\\Big|\\,\\frac 12\\underline r\\le \\vr^n(x)\\le 2\\overline r\\},\\;\nN_{\\rm res}^n= \\Omega\\setminus N_{\\rm ess}^n.\n\\end{equation}\nand we set\n\\[\n[g]_{\\rm ess}{ (x)}= g{ (x)} 1_{N^n_{\\rm ess}}{ (x)},\\; [g]_{\\rm res}{ (x)}= g { (x)}1_{N^n_{\\rm res}}{ (x)},\\;\\; { x\\in \\Omega},\\;\\;g\\in L^1(\\Omega).\n\\]\n\n Integrating inequality (\\ref{added}) we deduce\n\\begin{equation}\\label{rentropy}\nc(\\underline r,\\overline r)\\sum_{K\\in{\\cal T}} \\int_K{\\Big(\\Big[1\\Big]_{\\rm res}+\\Big[(\\vr^n)^\\gamma\\Big]_{\\rm res}+\\Big[\\vr^n-\\hat r^n\\Big]^2_{\\rm ess}\\Big)}\\dx\\le{\\cal E}(\\vr^n,\\bu^n\\Big| r^n,\\bU^n).\n\\end{equation}\nfor any pair $(r,\\bU)$ belonging to the class (\\ref{dr,U}) and any $\\vr^n\\in L_h(\\Omega)$.\n\n{ We are now ready to proceed to the proof of Lemma \\ref{strongentropy}.}\n\n\\begin{proof}\n\nWe start by projecting the momentum equation to the discrete spaces.\\label{7.2}\nSince $(r,\\bU)$ satisfies \\eqref{pbcont} and belongs to the class (\\ref{dr,U}), Equation (\\ref{mov2}) can be rewritten in the form\n\\begin{equation}\\label{str1}\nr\\partial_t\\bU+r\\bU\\cdot\\nabla\\bU +\\nabla p(r)=\\mu\\Delta\\bU +(\\mu+\\lambda)\\nabla\\dv\\bU.\n\\end{equation}\nWe write equation (\\ref{str1}) at $t=t_n$, multiply scalarly by $\\bu^n-\\bU^n_{h}$, and integrate over $\\Omega$.\nWe get, after summation from $n=1$ to $m$,\n\\begin{equation}\\label{strong0}\n\\begin{aligned}\n& \\sum_{i=1}^5{\\cal T}_i=0, \\qquad\\qquad\\qquad \\mbox{ with } && {\\cal T}_1 = -\\deltat\\sum_{n=1}^m\\int_\\Omega\\Big(\\mu\\Delta \\bU^n+ (\\mu+\\lambda)\n\\nabla\\dv\\bU^n\\Big)\\cdot(\\bu^n-\\bU^n_h)\\dx, &&\\\\\n&{\\cal T}_2 =\\deltat\\sum_{n=1}^m\\int_\\Omega r^{n}[\\partial_t\\bU]^n\\cdot (\\bu^n-\\bU^n_h)\\dx, && {\\cal T}_3 = \\deltat \\sum_{n=1}^m\\int_\\Omega r^n\\bU^n\\cdot\\nabla\\bU^n\\cdot (\\bu^n-\\bU^n_h)\\dx \\\\\n&{\\cal T}_4 = \\deltat\\sum_{n=1}^m\\int_\\Omega\\nabla p(r^n)\\cdot\\bu^n \\dx, && {\\cal T}_5 =-\\deltat\\sum_{n=1}^m\\int_\\Omega\\nabla p(r^n)\\cdot\\bU^n_h\\dx.\n \\end{aligned}\n\\end{equation}\n\n\n\n In the steps below, we deal with each of the terms ${\\cal T}_i$.\n\n\\vspace{2mm}\n\\textbf{Step 1: }\\textit{Term ${\\cal T}_1$.}\\label{7.3}\n Integrating by parts, we get:\n {\n\\begin{equation}\\label{cT1}\n \\begin{aligned}\n&{\\cal T}_1={\\cal T}_{1,1} + {\\cal R}_{1,1},\n\\\\\n& \\mbox{with }{\\cal T}_{1,1} = \\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K\\Big(\\mu\\nabla\\bU_{h}^n:\\nabla(\\bu^n-\\bU_h^n)+\n(\\mu +\\lambda)\\dv \\bU_{ h}^n\\dv (\\bu^n-\\bU_h^n)\\Big)\\dx\n \\\\\n &\\mbox{and }\\;\n {\\cal R}_{1,1}=I_1+I_2,\\;\\mbox{with}\\\\\n &I_1=\n \\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_K\\Big(\\mu\\nabla(\\bU^n-\\bU_{ h}^n):\\nabla(\\bu^n-\\bU_h^n)+ (\\mu+\\lambda)\n \\dv((\\bU^n-\\bU_{ h}^n)\\dv(\\bu^n-\\bU_h^n)\\Big)\n \\dx\n \\\\\n &\n I_2=-\\deltat\\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)} \\int_{\\sigma} \\Big(\\mu\\bn_{\\sigma,K} \\cdot\\nabla\\bU^n \\cdot(\\bu^n- \\bU_h^n) + (\\lambda+\\mu)\\dv\\bU^n(\\bu^n-\\bU^n_h)\\cdot\\bn_{\\sigma,K}\\Big)\\dS\n \\\\\n & =\n -\\deltat\\sum_{n=1}^m\\ \\sum_{\\sigma\\in {\\cal E}} \\int_{\\sigma} \\Big(\\mu\\bn_{\\sigma} \\cdot\\nabla\\bU^n \\cdot\\Big[\\bu^n- \\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma} + (\\lambda+\\mu)\\dv\\bU^n\\Big[\\bu^n-\\bU^n_h\\Big]_{\\sigma,\\vc n_\\sigma}\\cdot\\bn_{\\sigma}\\Big)\\dS,\n\\end{aligned}\n\\end{equation}\n}\nwhere in the last line $\\vc n_\\sigma$ is a unit normal to $\\sigma$ and $[\\cdot]_{\\sigma,\\vc n_\\sigma}$ is the jump over sigma (with respect to $\\vc n_\\sigma$) defined in Lemma \\ref{Lemma6}. Employing estimate (\\ref{L1-3})\nwe easily verify that\n{\n$$\n|I_1|\\le h\\;c(M_0,E_0, \\|\\nabla\\vc U\\|_{L^\\infty(0,T; W^{1,\\infty}(\\Omega))}).\n$$\nSince the integral over any face $\\sigma$ of the jump of a function from $V_h(\\Omega)$ is zero, we may write\n$$\nI_2=-\\deltat\\sum_{n=1}^m\\ \\sum_{\\sigma\\in {\\cal E}} \\int_{\\sigma} \\Big(\\mu\\bn_{\\sigma} \\cdot\\Big(\\nabla\\bU^n -[\\nabla\\bU^n]_\\sigma\\Big)\\cdot\\Big[\\bu^n- \\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma}\n$$\n$$\n+ (\\lambda+\\mu)\\Big(\\dv\\bU^n-[\\dv\\bU^n]_\\sigma\\Big)\\Big[\\bu^n-\\bU^n_h\\Big]_{\\sigma,\\vc n_\\sigma}\\cdot\\bn_{\\sigma}\\Big)\\dS;\n$$\nwhence by using the { first order Taylor formula applied to functions $x\\mapsto \\nabla \\vc U^n(x)$ to evaluate the differences $\\nabla\\bU^n -[\\nabla\\bU^n]_\\sigma$, $\\dv\\bU^n-[\\dv\\bU^n]_\\sigma$, }\n and H\\\"older's inequality,\n$$\n\\begin{aligned}\n&|I_2| \\le \\deltat\\, h\\; c\\, \\|\\nabla^2\\bU\\|_{L^\\infty(Q_T;\\Rm^{27})} \\sum_{n=1}^m\\sum_{\\sigma\\in {\\cal E}} \\sqrt{|\\sigma|}\\sqrt h\\;\\Big(\\frac 1{\\sqrt h}\\,\\Big\\|\\Big[\\bu^n-\\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma}\\Big\\|_{L^2(\\sigma;\\Rm^3)}\\Big)\\\\\n&\\le \\deltat\\, h\\; c\\, \\|\\nabla^2\\bU\\|_{L^\\infty(Q_T;\\Rm^{27})} \\sum_{n=1}^m\\sum_{\\sigma\\in {\\cal E}}\\Big(|\\sigma|h+\n\\frac 1h\\,\\Big\\|\\Big[\\bu^n-\\bU_h^n\\Big]_{\\sigma,\\vc n_\\sigma}\\Big\\|_{L^2(\\sigma;\\Rm^3)}^2\\Big).\n\\end{aligned}\n$$\nTherefore,\n\\begin{equation}\\label{cR1.1}\n|{\\cal R}_{1,1}|\\le h\\, c(M_0, E_0,\\|\\vc U, \\nabla\\bU, \\nabla^2\\bU\\|_{L^\\infty(Q_T,\\Rm^{39})}),\n\\end{equation}\nwhere we have employed Lemma \\ref{Lemma6} and\n (\\ref{est0}) in Corollary \\ref{Corollary1}.\n\n\n\n}\n\n\\vspace{2mm}\n\\textbf{Step 2:}\\textit{ Term ${\\cal T}_2$.}\\label{7.4}\nLet us now decompose the term ${\\cal T}_2$ as\n\\begin{align*}\n\t & {\\cal T}_2={\\cal T}_{2,1}+ {\\cal R}_{2,1},\\\\\n\t& \\mbox{with } {\\cal T}_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr^{n-1}\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\cdot (\\bu^n-\\bU^n_h)\\dx,\\quad {\\cal R}_{2,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,1}^{n,K}, \\\\\n\t& \\mbox{and }{ {\\cal R}_{2,1}^{n,K}=\\int_K(r^n-r^{n-1})[\\partial_t\\vc U]^n\\cdot(\\bu^n-\\bU^n_h)\\dx }+ \\int_Kr^{n-1}\\Big([\\partial_t \\bU]^n-\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big) \\cdot(\\bu^n-\\bU^n_h)\\dx.\n\\end{align*}\nThe remainder ${\\cal R}_{2,1}^{n,K}$ can be rewritten as follows\n$$\n{\\cal R}_{2,1}^{n,K}=\\int_K\\Big[\\int_{t_{n-1}}^{t_n}r(t,\\cdot){\\rm d}t\\Big][\\partial_t\\vc U]^n\\cdot(\\bu^n-\\bU^n_{h})\\dx\n $$\n $$\n + \\frac 1k\\int_Kr^{n-1}\\Big[\\int_{t_{n-1}}^{t_n}\\int_s^{t_n}\\partial^2_t \\bU(z,\\cdot){\\rm d}z{\\rm d}s\\Big] \\cdot(\\bu^n-\\bU_{h}^n)\\dx;\n$$\n{ whence,\n\\[\n|{\\cal R}_{2,1}^{n,K}|\\le \\deltat\\Big[ (\\|r\\|_{L^\\infty(Q_T)}+\\|\\partial_t r\\|_{L^\\infty(Q_T)}) (\\|\\partial_t\\bU\\|_{L^\\infty(Q_T;\\Rm^3)}|K|^{5\/6}(\\|\\bu^n\\|_{L^6(K)}+ \\|\\bU^n_h\\|_{L^6(K)})\n \\]\n \\[\n +\\|\\partial^2_t \\bU^n\\|_{L^{6\/5}(\\Omega;\\Rm^3))} (\\|\\bu^n\\|_{L^6(K)}+ \\|\\bU^n_h\\|_{L^6(K)}).\n\\]\nConsequently, by the same token as in (\\ref{R5.2}) or (\\ref{R6.4}),\n\\begin{equation}\\label{cR2.1}\n|{\\cal R}_{2,1}|\\le \\deltat\\, c\\Big(M_0, E_0,\\overline r,\\|(\\partial_t r, \\bU, \\partial_t\\bU, \\nabla\\bU , )\\|_{L^\\infty(Q_T;\\Rm^{16})}, \\|\\partial^2_t \\bU\\|_{L^2(0,T; L^{6\/5}(\\Omega;\\Rm^3))}\\Big),\n\\end{equation}\n where we have used the H\\\"older and Young inequalities, the estimates (\\ref{L1-2}), (\\ref{L1-4}), (\\ref{L1-6}), (\\ref{sob1}), and to the energy bound (\\ref{est0}) from Corollary \\ref{Corollary1}.\n }\n\n\n\n\\vspace{2mm}\n{\\bf Step 2a:} {\\it Term ${\\cal T}_{2,1}$.} We decompose the term ${\\cal T}_{2,1}$ as\n\\begin{align*}\n&{\\cal T}_{2,1}={\\cal T}_{2,2}+ {\\cal R}_{2,2}, \\\\\n&\\mbox{with } {\\cal T}_{2,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr_K^{n-1}\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\cdot (\\bu^{n}-\\bU^{n}_h)\\dx, \\;\n {\\cal R}_{2,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,2}^{n,K}, \\\\\n&\\mbox{and }{\\cal R}_{2,2}^{n,K}=\\int_K(r^{n-1}-r_K^{n-1})\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\cdot(\\bu^{n}-\\bU_h^n)\\dx;\n\\end{align*}\ntherefore,\n\\[\n|{\\cal R}_{2,2}^{n}|= |\\sum_{K\\in {\\cal T}}{\\cal R}_{2,2}^{n,K}| \\le h \\, c\\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}\\|\\partial_t\\bU\\|_{L^\\infty(Q_T;\\Rm^{3})}\\|\\bu^n-\\bU^n_h\\|_{L^6(\\Omega;\\Rm^3)}.\n\\]\nConsequently, by virtue of formula (\\ref{est1}) in Corollary \\ref{Corollary1} and estimates (\\ref{sob1}), (\\ref{L1-5}),\n\\begin{equation}\\label{cR2.2}\n|{\\cal R}_{2,2}|\\le h \\, c(M_0, E_0, \\|(\\nabla r, \\bU,\\partial_t\\bU,\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{18})}).\n\\end{equation}\n\n{\n\\vspace{2mm}\n{\\bf Step 2b:} {\\it Term ${\\cal T}_{2,2}$.} We decompose the term ${\\cal T}_{2,2}$ as\n\\begin{align*}\n&{\\cal T}_{2,2}={\\cal T}_{2,3}+ {\\cal R}_{2,3}, \\\\\n&\\mbox{with } {\\cal T}_{2,3}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}\\int_Kr_K^{n-1}\\frac{\\bU^n_{h,K}-\\bU^{n-1}_{h,K}}{\\deltat}\\cdot (\\bu^{n}-\\bU^{n}_h)\\dx, \\;\n {\\cal R}_{2,3}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,3}^{n,K}, \\\\\n&\\mbox{and }{\\cal R}_{2,3}^{n,K}=\\int_K r_K^{n-1}\\Big(\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}-\\Big[\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big]_h\\Big)\n\\cdot(\\bu^{n}-\\bU_h^n)\\dx\\\\ &\n+\n\\int_K r_K^{n-1}\\Big(\\Big[\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big]_h-\\Big[\\frac{\\bU^n-\\bU^{n-1}}{\\deltat}\\Big]_{h,K}\\Big)\n\\cdot(\\bu^{n}-\\bU_h^n)\\dx { =I_1^K+I_2^K}.\n\\end{align*}\n{ We have\n$$\n|I_2^K|=\\frac 1 \\deltat r_K^{n-1}\\Big|\\int_K \\Big(\\Big[\\int_{t_{n-1}}^{t_n}\\partial_t\\bU(z){\\rm d}z\\Big]_h- \\Big[\\int_{t_{n-1}}^{t_n}\\partial_t\\bU(z){\\rm d}z\\Big]_{h,K}\\Big)\n\\cdot(\\bu^{n}-\\bU_h^n)\\dx\\Big|\n$$\n$$\n\\le \\frac h \\deltat r_K^{n-1}\\int_{t_{n-1}}^{t_n}\\Big\\|\\Grad\\Big[\\partial_t\\bU(z)\\Big]_h\\Big\\|_{L^{6\/5}(K;\\Rm^3)}\n\\|\\bu^n-\\bU^n_h\\|_{L^6(K;\\Rm^3)}\n$$\nwhere we have used the Fubini theorem, H\\\"older's inequality and (\\ref{L2-1}), (\\ref{L1-3})$_{s=1}$.\nFurther, employing the Sobolev inequality on Crouzeix-Raviart space (\\ref{sob1}), H\\\"older's and Young's inequalities and estimate (\\ref{L1-3})$_{s=1}$, we get\n\n$$\n\\sum_{K\\in {\\cal T}}|I_2^K|\\le \\frac h \\deltat r_K^{n-1} \\|\\bu^n-\\bU^n_h\\|_{L^6(\\Omega;\\Rm^3)}\\int_{t_{n-1}}^{t_n}\\Big\\|\\Grad\\partial_t\\bU(z)\\Big\\|_{L^{6\/5}(\\Omega;\\Rm^3)}\n{\\rm d}z\n$$\n$$\n\\le\nc r_K^{n-1} \\Big(h|\\bu^n-\\bU^n_h|^2_{V^2_h(\\Omega;\\Rm^3)}+\\frac h \\deltat \\int_{t_{n-1}}^{t_n}\n\\Big\\|\\Grad\\partial_t\\bU(z)\\Big\\|^2_{L^{6\/5}(\\Omega;\\Rm^3)}\\Big)\n$$\n\nWe reserve the similar treatment to the term $I_1^K$. Resuming these calculations we get by using Corollary \\ref{Corollary1}\n\\begin{equation}\\label{cR2.3}\n|{\\cal R}_{2,3}|\\le h \\, c(M_0, E_0, \\|(r, \\bU,\\nabla\\bU, \\partial_t\\bU)\\|_{L^\\infty(Q_T;\\Rm^{16})},\\|\\partial_t\\nabla\\bU\\|_{L^2(0,T;L^{6\/5}(\\Omega;\\Rm^{9}))}).\n\\end{equation}\n}\n}\n{ \\bf Step 2c:} {\\it Term ${\\cal T}_{2,3}$.}\n{ We rewrite this term in the form\n\\begin{equation}\\label{cT2}\\begin{aligned}\n& {\\cal T}_{2,3}={\\cal T}_{2,4}+ {\\cal R}_{2,4},\\; {\\cal R}_{2,4}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}}{\\cal R}_{2,4}^{n,K}, \\\\\n&\\mbox{with } {\\cal T}_{2,4}=\\deltat\\sum_{n=1}^m\\sum_{K\\in {\\cal T}} \\int_Kr_K^{n-1} \\frac {\\bU^n_{h,K}-\\bU^{n-1}_{h,K}} {\\deltat}\\cdot (\\bu^{n}_K-\\bU^{n}_{h,K})\\dx,\n\\\\ &\\mbox{and }{\\cal R}_{2,4}^{n,K}=\\int_Kr_K^{n-1}\\frac{\\bU_{h,K}^n-\\bU_{h,K}^{n-1}}{\\deltat}\\cdot\\Big((\\bu^{n}-\\bu^n_K) -(\\bU_h^n-\\bU_{h,K}^n)\\Big)\\dx.\n\\end{aligned}\n\\end{equation}\n{\nFirst we write, as in (\\ref{R2.1}),\n$$\n\\Big|\\frac{[{\\bU}^{n}-{\\bU}^{n-1}]_{h,K}}{\\deltat}\\Big|=\n\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\Big[\\int_{t_{n-1}}^{t_n} \\partial_t\\vc U(z,x) {\\rm d} z\\Big]_h\\Big]{\\rm d}x\\Big|\n$$\n$$\n=\\Big|\\frac 1{|K|}\\int_K\\Big[\\frac 1\\deltat \\int_{t_{n-1}}^{t_n} [\\partial_t\\vc U(z) \\Big]_h(x){\\rm d} z\\Big]{\\rm d}x\\Big|\\le \\|[\\partial_t\\vc U ]_h\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))}\\le \\|\\partial_t\\vc U \\Big\\|_{L^\\infty(0,T;L^\\infty(\\Omega;\\Rm^3))},\n$$\nNext we evaluate $\\bu^n-\\bu^n_K$ employing (\\ref{L2-1})$_{p=2}$, and $\\bU_h^n-\\bU_{h,K}^n$ by using\n(\\ref{L2-1})$_{p=\\infty}$, (\\ref{L1-3})$_{s=1}$. Finally we employ the H\\\"older inequality to get\n\\begin{equation}\\label{cR2.4}\n|{\\cal R}_{2,4}|\\le h \\; c(M_0,E_0,\\overline r, \\|(\\partial_t\\bU, \\bU, \\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{15})}, \\|\\partial_t\\nabla\\bU )\\|_{L^2(0,T;L^{6\/5}(\\Omega;\\Rm^{9}))}).\n\\end{equation}\n}\n}\n\n\\vspace{2mm}\n\\textbf{Step 3:} \\textit{ Term ${\\cal T}_3$.}\nLet us first decompose ${\\cal T}_3$ as\n\\begin{align*}\n& {\\cal T}_3={\\cal T}_{3,1} + {\\cal R}_{3,1}, \\\\\n&\\mbox{with } {\\cal T}_{3,1}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\int_K r_K^n\\bU_{h,K}^n\\cdot\\nabla\\bU^n\\cdot (\\bu_K^n-\\bU_{h,K}^n)\\dx,\n \\quad {\\cal R}_{3,1}=\\deltat\\sum_{n=1}^m \\sum_{K\\in{\\cal T}}{\\cal R}_{3,1}^{n,K},\n\\\\ &\\mbox{and }{\\cal R}_{3,1}^{n, K}=\\int_K(r^n-r_K^n)\\bU^n\\cdot\\nabla\\bU^n\\cdot(\\bu^n-\\bU^n_h)\\dx\n+ \\int_K r_K^n(\\bU^n-\\bU^n_h)\\cdot\\nabla\\bU^n\\cdot(\\bu^n-\\bU^n_h)\\dx \\\\\n& \\phantom{\\mbox{and }{\\cal R}_{3,1}^{n, K}=} +\\int_K r_K^n(\\bU_h^n-\\bU^n_{h,K})\\cdot\\nabla\\bU^n\\cdot(\\bu^n-\\bU^n_h)\\dx +\\int_K r_K^n\\bU^n_{h,K}\\cdot\\nabla\\bU^n\\cdot\\Big(\\bu^n-\\bu^n_K-(\\bU^n_h-\\bU^n_{h,K})\\Big)\\dx.\n\\end{align*}\nWe find that\n\\begin{align*}\n&|{\\cal R}_{3,1}^{n,K}| \\le h\\,\\Big[ |K|^{1\/2} (\\|\\bu^n\\|_{L^2(K;\\Rm^3)}+ \\|\\bU_h^n\\|_{L^2(K;\\Rm^3)})+|K|^{ 1\/2} (\\|\\nabla\\bu^n\\|_{L^2(K;\\Rm^3)}+ \\|\\nabla\\bU_h^n\\|_{L^2(K;\\Rm^3)})\\Big]\n\\\\ & \\qquad\\qquad\\qquad\\qquad\\qquad \\times\n\\Big(\\|r\\|_{L^\\infty(Q_T)}+\\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}\\Big)\\,\\Big(\\|\\bU\\|_{L^\\infty(Q_T;\\Rm^3)} +\\|\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^{9})}\\Big)^2,\n\\end{align*}\nwhere we have used several times H\\\"older's inequality and the standard first order Taylor formula ({ to evaluate\n$r^n-r^n_K$)}, along with the estimates\n(\\ref{L1-2}) (to evaluate $\\bU^n-\\bU^n_h$), (\\ref{L2-1}), (\\ref{L1-3})$_{s=1}$ (to evaluate\n$\\bU^n_h-\\bU^n_{h,K}$), (\\ref{L2-1}) (to evaluate $\\bu^n-\\bu^n_K$).\n\nConsequently, using again (\\ref{L1-3})$_{s=1}$ (to estimate $\\|\\nabla\\bU_h^n\\|_{L^2(K;\\Rm^3)}$), the definition\n of $ |\\cdot|_{V^2_h(\\Omega)}$ norm, the Sobolev inequality (\\ref{sob1}) and the energy\nbound (\\ref{est0}) from Corollary \\ref{Corollary1}, we conclude that\n\\begin{equation}\\label{cR3.1}\n|{\\cal R}_{3,1}|\\le h\\; c(M_0, E_0,\\overline r, \\|(\\nabla r, \\bU, \\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{15})}).\n\\end{equation}\n\n\nNow we shall deal wit term ${\\cal T}_{3,1}$. Integrating by parts, we get:\n\n\n\n\n\n\n\n\n\n\\begin{align*}\n\t\\int_K r_K^n\\bU_{h,K}^n\\cdot\\nabla\\bU^n\\cdot (\\bu_K^n-\\bU_{h,K}^n)\\dx &=\\sum_{\\sigma\\in {\\cal E}(K)}{ |\\sigma|} r_K^n [\\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}]\\bU_\\sigma^n\\cdot(\\bu_K^n-\\bU_{h,K}^n)\n\t\\\\\n\t&=\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| r_K^n [\\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\bu_K^n-\\bU_{h,K}^n),\n\\end{align*}\nthanks to the the fact that $\\sum_{\\sigma\\in {\\cal E}(K)}\\int_\\sigma \\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}{\\rm d} S=0$.\n\n{ Next we write\n\\begin{align*}\n& {\\cal T}_{3,1}= {\\cal T}_{3,2}+ {\\cal R}_{3,2}, \\quad {\\cal R}_{3,2}=\\deltat\\sum_{n=1}^m {\\cal R}_{3,2}^{n},\n\\end{align*}\n\\begin{align}\n&\n{\\cal T}_{3,2}=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| \\hat r_\\sigma^{n,{\\rm up}}\n[\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\hat\\bu_\\sigma^{n,{\\rm up}}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}}), \\label{cT3}\n\\end{align}\n\\begin{align*}\n& \\mbox{and }{\\cal R}_{3,2}^{n}=\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|(r_K^n-\\hat r^{n,{\\rm up}}_\\sigma) [\\bU_{h,K}^n\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\bu_K^n-\\bU^n_{h,K})\n\\\\\n&+\n\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\hat r^{n,{\\rm up}}_\\sigma \\Big[\\Big(\\bU_{h,K}^n-\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\Big)\\cdot{\\vc n}_{\\sigma,K}\\Big](\\bU_\\sigma^n-\\bU^n_{h,K})\\cdot(\\bu_K^n-\\bU^n_{h,K})\\\\\n&\n+\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma|\\hat r^{n,{\\rm up}}_\\sigma [\\hat \\bU_{h,\\sigma}^{n,{\\rm up}}\\cdot{\\vc n}_{\\sigma,K}](\\bU_\\sigma^n-\\bU^n_{h,K})\n\\cdot\\Big((\\bu_K^n -\\hat\\bu_{h,\\sigma}^{n,{\\rm up}}) -(\\bU^n_{h,K}-\\hat\\bU_{h,\\sigma}^{n,{\\rm up}})\\Big).\n\\end{align*}\nWe may use several times the { Taylor formula (in order to estimate $r_K^n-\\hat r_\\sigma^{n,{\\rm up}}$,\n $\\vc U_\\sigma^n-{\\vc U}_{h,K}^n$, $\\vc U_{h,K}^n-\\hat{\\vc U}_{h,\\sigma}^{n,{\\rm up}}$)} to get the bound\n$$\n|{\\cal R}_{3,2}^n|\\le h\\, c \\|r\\|_{W^{1,\\infty}(\\Omega)}\\Big(1+ \\|\\bU\\|_{W^{1,\\infty}(\\Omega;\\Rm^3)}\\Big)^3\n\\sum_{K\\in {\\cal T}} h|\\sigma||\\bu^n_K|\n$$\n$$\n+ c \\|r\\|_{W^{1,\\infty}(\\Omega)}\\Big(1+ \\|\\bU\\|_{W^{1,\\infty}(\\Omega;\\Rm^3)}\\Big)^2\\sum_{K\\in{\\cal T}}\n\\sum_{\\sigma\\in {\\cal E}(K)} h|\\sigma| |\\bu_K^n-\\bu_\\sigma^n|,\n$$\nwhere by virtue of H\\\"older's inequality, (\\ref{sob4}), (\\ref{sob3}), (\\ref{L2+-1}) (\\ref{L2+-2}),\n$$\n\\sum_{K\\in {\\cal T}} h|\\sigma||\\bu^n_K|\\le c\\Big(\\sum_{\\sigma\\in {\\cal T}} h|\\sigma||\\bu^n_K|^6\\Big)^{1\/6}\n\\le c\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\|\\bu^n-\\bu_K^n\\|^6_{L^6(K;\\Rm^3)}\\Big)^{1\/6}\n$$\n$$\n+\\Big(\\sum_{K\\in {\\cal T}}\\|\\bu^n\\|^6_{L^6(K;\\Rm^3)}\\Big)^{1\/6}\\Big]\n\\le c\\Big(\\sum_ {K\\in {\\cal T}}\\|\\nabla\\bu_n\\|^2_{L^2(K;\\Rm^9)}\\Big)^{1\/2},\n$$\n$$\n\\sum_{K\\in{\\cal T}}\n\\sum_{\\sigma\\in {\\cal E}(K)} h|\\sigma| |\\bu_K^n-\\bu_\\sigma^n|\\le c\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\|\\bu^n-\\bu_K^n\\|^2_{L^2(K;\\Rm^3)}\\Big)^{1\/2}\n$$\n$$\n+ \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|\\bu^n-\\bu_\\sigma^n\\|^2_{L^2(K;\\Rm^3)}\\Big)^{1\/2}\\Big]\n\\le h\\,c\\Big(\\sum_{K\\in {\\cal T}} \\|\\nabla\\bu_n\\|^2_{L^2(K;\\Rm^9)}\\Big)^{1\/2},\n$$\nConsequently, we may use (\\ref{est0}) to conclude\n\\begin{equation}\\label{cR3.2}\n|{\\cal R}_{3,2}|\\le h\\, c\\Big(M_0, E_0, \\overline r, \\|\\nabla r, \\bU, \\nabla\\bU\\|_{L^{\\infty}(Q_T;\\Rm^{15})}\\Big).\n\\end{equation}\n}\n\n\n\\vspace{2mm}\n\\textbf{Step 4:}\\textit{ Terms ${\\cal T}_4$ and ${\\cal T}_5$.} \\label{7.6}\nWe decompose ${\\cal T}_4$ as\n\n\\begin{equation}\\label{cT4}\n\\begin{aligned}\n\t& {\\cal T}_4= {\\cal T}_{4,1}+ {\\cal R}_{4,1},\n\t\\\\\n\t& \\mbox{with }{\\cal T}_{4,1}=\\deltat \\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\int_K p'(r^n_K)\\bu^n\\cdot\\nabla r^n\\dx,\n\\quad\n{\\cal R}_{4,1}= \\deltat \\sum_{n=1}^m \\sum_{K\\in{\\cal T}}\\int_K \\Big( p'(r^n)-p'(r^n_K)\\Big)\\bu^n\\cdot\\nabla r^n\\dx;\n\\end{aligned}\n\\end{equation}\nwhence\n\\begin{equation}\\label{cR4.1}\n\t|{\\cal R}_{4,1}|\\le h\\, c(M_0, E_0,\\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])}, \\|\\nabla r\\|_{L^\\infty(Q_T;\\Rm^3)}).\n\\end{equation}\n\nEmploying integration by parts, we infer\n\n\\begin{equation}\\label{cT5}\n\\begin{aligned}\n\t& {\\cal T}_5= {\\cal T}_{5,1} + {\\cal R}_{5,1}, \\mbox{ with }{\\cal T}_{5,1}=\n-\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K \\nabla p(r^n)\\cdot \\bU^n\\dx,\n\t\\\\ &\n{\\cal R}_{5,1}=\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K\\nabla p(r^n)\\cdot\\Big(\\bU^n-\\bU^n_h\\Big)\\dx\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\\label{cR5.1}\n|{\\cal R}_{5,1}|\\le h\\, c(\\overline r, | p'|_{C([\\underline r,\\overline r])},\\|{ \\nabla r},\\nabla \\bU\\|_{L^\\infty(Q_T;\\Rm^{12})}).\n\\end{equation}\n\\label{7.7}\nIntegrating by parts, we obtain\n$$\n{\\cal T}_{5,1}= \\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K p(r^n) \\dv \\bU^n\\dx;\n$$\nwhence\n$$\n\\begin{aligned}\n\t& {\\cal T}_{5,1}= {\\cal T}_{5,2} + {\\cal R}_{5,2}, \\mbox{ with }{\\cal T}_{5,2}=\n\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K p(r_K^n)\\dv \\bU^n\\dx,\n\t\\\\ &\n{\\cal R}_{5,2}=-\\deltat \\sum_{n=1}^m \\sum_{K\\in {\\cal T}}\\int_K(p(r_K^n)-p(r^n))\\dv\\bU^n\\dx\n\\end{aligned}\n$$\nand\n\\begin{equation}\\label{cR5.2}\n|{\\cal R}_{5,2}|\\le h\\, c(\\overline r, |p'|_{[\\underline r,\\overline r]},\\|\\nabla r,\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^{12})}).\n\\end{equation}\n\nGathering the formulae (\\ref{cT1}), (\\ref{cT2}), (\\ref{cT3}), (\\ref{cT4}), (\\ref{cT5}) and estimates for the residual terms (\\ref{cR1.1}), (\\ref{cR2.1}--\\ref{cR2.4}), (\\ref{cR3.1}--\\ref{cR3.2}), (\\ref{cR4.1}), (\\ref{cR5.1}), (\\ref{cR5.2}) concludes the proof of Lemma \\ref{strongentropy}.\n\\end{proof}\n\n\\section{{ End} of the proof of the error estimate (Theorem \\ref{Main})}\n\n{ In this Section we put together the relative energy inequality (\\ref{relativeenergy-}) and the identity (\\ref{strong1}) derived in the previous section. The final inequality resulting from this manipulation is formulated in the following lemma.}\n\\begin{lm}\\label{Gronwall}\nUnder assumptions of Theorem \\ref{Main} there exists a positive number\n\\[\n c=c\\Big(M_0, E_0, \\underline r,\\overline r, |p'|_{C^1([\\underline r,\\overline r])},\n \\|(\\nabla r, \\partial_t r, \\partial_t\\nabla r, \\partial^2_t r, \\bU, \\nabla\\bU, { \\nabla^2\\bU}, \\partial_t\\bU, \\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{65})}\\Big)\n\\]\n(depending tacitly also on $T$, $\\theta_0$, $\\gamma$, ${\\rm diam} (\\Omega)$, $|\\Omega|$),\nsuch that for all $m=1,\\ldots,N,$ there holds:\n\\[{\\cal E}(\\vr^m,\\bu^m|r^m,\\bu^m)\n{ +\\deltat\\frac \\mu 2\\sum_{n=1}^m\\sum_{{K}\\in {\\cal T}}\\int_K|\\Grad(\\vc u^n-\\vc U^n_h)|^2{\\rm d x}}\n\\]\n\\[\n\\le c\\Big[h^A+\\sqrt{\\deltat} + {\\cal E}(\\vr^0,\\bu^0|r^0,\\bU^0)\\Big] + c\\,\\deltat\\sum_{n=1}^m {\\cal E}(\\vr^n,\\bu^n|r^n,\\bu^n),\n\\]\nwhere $A$ is defined in (\\ref{A1}).\n\\end{lm}\n\n\n\\begin{proof}\nGathering the formulae (\\ref{relativeenergy-}) and (\\ref{strong1}), one gets\n\\begin{equation}\\label{relativeenergy-1}\n{\\cal E}(\\vr^m,\\bu^m\\Big| r^m, U^m)- {\\cal E}(\\vr^0,\\bu^0\\Big|r(0),\\bU(0)) + \\mu\\deltat\\sum_{n=1}^m\\Big|\\bu^n-\\bU^n_h\\Big|^2_{V^2_h(\\Omega;\\Rm^3)}\n\\end{equation}\n$$\n\\le{\\cal P}_1+ {\\cal P}_2 +{\\cal P}_3 + {\\cal Q}\n$$\nwhere\n\\begin{align*}\n&{\\cal P}_1=\n\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}|K|(\\vr_K^{n-1}-r^{n-1}_K)\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n} - \\bu_K^{n}\\Big),\n\\\\\n&{ {\\cal P}_2=\\deltat\\sum_{n=1}^m\\sum_{K\\in{\\cal T}}\\sum_{\\sigma=K|L\\in {\\cal E}(K)}|\\sigma|\\Big(\\vr_\\sigma^{n,{\\rm up}}-\\hat r_\\sigma^{n,{\\rm up}}\\Big)\\Big({\\hat\\bU}^{n,{\\rm up}}_{h,\\sigma}-\n{\\hat\\bu}^{n,{\\rm up}}_\\sigma\\Big)\\cdot\\Big(\\bU^n_\\sigma-\\bU^n_{h,K}\\Big) \\hat \\bU_{h,\\sigma}^{n, {\\rm up}}\\cdot\\bn_{\\sigma,K},}\n\\\\\n&{\\cal P}_3=\n\\deltat \\sum_{n=1}^m\\sum_{K\\in {\\cal T}} \\int_K \\Big(p(r_K^n)-p(\\vr_K^n)\\Big)\\dv\\bU^n\\dx\n\\\\\n&\\qquad \\qquad+\\deltat\\sum_{n=1}^m\\sum_{K \\in {\\cal T}}\\Big[\\int_K\\frac{r_K^n-\\vr^n_K}{r^n_K} p'(r^n_K) \\bu^n\\cdot\\nabla r^n\\dx\n+\\int_K \\frac{ r^n_K-\\vr^n_K} {r_K^n}p'(r_K^n) [\\partial_t r]^n\\dx\\Big],\n\\\\\n&{ {\\cal Q}=\n{\\cal R}^m_{h,\\deltat} +R^m_{h,\\deltat} +{ G}^m. }\n\\end{align*}\n\n\nNow, we estimate conveniently the terms ${\\cal P}_1$, ${\\cal P}_2$, ${\\cal P}_3$ in three steps.\n\n\\vspace{2mm}\n\n{\\bf Step 1:} {\\it Term ${\\cal P}_1$.}\n{ { We have\n$$\n\\Big|{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}\\Big| \\le \\int_{t_{n-1}}^{t_n}\\|[\\partial_t\\bU(z)]_h\\|_{L^\\infty(K;\\Rm^9)}{\\rm d z}\n\\le c \\int_{t_{n-1}}^{t_n}\\|\\partial_t\\bU(z)\\|_{L^\\infty(K;\\Rm^9)}{\\rm d z},\n$$\nwhere we have used (\\ref{ddd}).\n\nAccording to Lemma \\ref{LL1},\n$$\n|\\vr-r|^\\gamma 1_{R_+\\setminus[\\underline r\/2,2\\overline r]}(\\vr)\\le c(p) E^p(\\vr|r)\n$$\nwith any $p\\ge 1$. In particular, \n\\begin{equation}\\label{aaa}\n|\\vr-r|^{6\/5} 1_{R_+\\setminus[\\underline r\/2,2\\overline r]}(\\vr)\\le c E(\\vr|r)\n\\end{equation}\nprovided $\\gamma\\ge 6\/5$.\n\nWe get by using the H\\\"older inequality,\n$$\n\\Big|\\sum_{K\\in{\\cal T}}|K|(\\vr_K^{n-1}-r^{n-1}_K)\\frac{{\\bU}_{h,K}^{n}-{\\bU}_{h,K}^{n-1}}{\\deltat}\\cdot \\Big({\\bU}_{h,K}^{n} - \\bu_K^{n}\\Big)\\Big|\\le c\\Big(\\|\\partial_t\\bU \\|_{L^\\infty(Q_T;\\Rm^3)}+\\|\\partial_t\\nabla\\bU \\|_{L^\\infty(Q_T;\\Rm^9)}\\Big)\\times\n$$\n$$\n\\Big[\\Big(\\sum_{K\\in{\\cal T}}|K||\\vr^{n-1}_K-r^{n-1}_K|^2 1_{[\\underline r\/2,2\\overline r]}(\\vr_K)\\Big)^{1\/2}\n+\n\\Big(\\sum_{K\\in{\\cal T}}|K||\\vr^{n-1}_K-r^{n-1}_K|^{6\/5} 1_{R_+\\setminus [\\underline r\/2,2\\overline r]}(\\vr_K)\\Big)^{5\/6}\\Big]\n\\times\n$$\n$$\n\\Big(\\sum_{K\\in{\\cal T}}|K| \\Big|{\\bU}_{h,K}^{n} - \\bu_K^{n}\\Big|^6\\Big)^{1\/6}\n\\le c(\\|(\\partial_t\\bU,\\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})})\\Big({\\cal E}^{1\/2}(\\vr^{n-1},\\bu^{n-1}|r^{n-1},\\bU^{n-1})\n$$\n$$\n+\n{\\cal E}^{5\/6}(\\vr^{n-1},\\bu^{n-1}|r^{n-1},\\bU^{n-1})\\Big)\\,\\Big(\\sum_{K\\in {\\cal T}}\\| {\\bU}_{h,K}^{n} - \\bu_K^{n}\\|_{L^6(K;\\Rm^3)}^6\\Big)^{1\/6},\n$$\nwhere we have used (\\ref{aaa}) and estimate (\\ref{est4}) to obtain the last line.\n\n Now, by the Minkowski inequality,\n }\n$$\n\\Big(\\sum_{K\\in {\\cal T}}\\| {\\bU}_{h,K}^{n} - \\bu_K^{n}\\|_{L^6(K;\\Rm^3)}^6\\Big)^{1\/6}\\le\n\\Big(\\sum_{K\\in {\\cal T}}\\| ({\\bU}_{h,K}^{n} - \\bu_K^{n})-(\\bU_h^n-\\bu^{n})\\|_{L^6(K;\\Rm^3)}^6\\Big)^{1\/6}\n$$\n$$\n+\n\\|\\bU_h^n-\\bu^{n}\\|_{L^6(\\Omega;\\Rm^3)}\\le c \\Big|\\bu^n-\\bU^n_h\\Big|_{V^2_h(\\Omega;\\Rm^3)},\n$$\nwhere we have used estimate (\\ref{sob4-}) and the Sobolev inequality (\\ref{sob1}). Finally, employing\nYoung's inequality, and estimate (\\ref{est4}), we arrive at\n\\begin{multline}\\label{cP1}\n|{\\cal P}_1| \\le \\; c({ \\delta},M_0,E_0,\\underline r,\\overline r,\\|(\\bU,\\nabla\\bU,\\partial_t\\bU,\\partial_t\\nabla\\bU)\\|_{L^\\infty(Q_T,\\Rm^{15})} )\n\\\\\n\\times \\Big(\\deltat {\\cal E}(\\vr^0,\\bu^0|r^0,\\bU^0)+ \\deltat\\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n)\\Big)\n+ \\delta \\deltat\\sum_{n=1}^m\\Big|\\bu^n-\\bU^n_h\\Big|^2_{V^2_h(\\Omega;\\Rm^3)}\n\\end{multline}\nwith any $\\delta>0$.\n}\n\n\n\n{\\bf Step 2:} {\\it Term ${\\cal P}_2$.}\n{ We write ${\\cal P}_2=\\deltat \\sum_{n=1}^m{\\cal P}_2^n$ where Lemma \\ref{LL1} and the H\\\"older inequality yield, similarly as in the previous step,\n\\begin{equation*}\n\\begin{aligned}\n|{\\cal P}^n_2| & \\le c(\\underline r,\\overline r, \\|\\nabla\\bU\\|_{L^\\infty(Q_T;\\Rm^9)}) \\times\n\\\\ &\n\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\n\\Big(E^{1\/2}(\\vr_\\sigma^{n,{\\rm up}},\\hat r_\\sigma^{n,{\\rm up}})+ E^{2\/3}(\\vr_\\sigma^{n,{\\rm up}},\\hat r_\\sigma^{n,{\\rm up}}\\Big)\\,|\\hat\\bU^{n,{\\rm up}}_{h,\\sigma}|\\,|{\\hat\\bU}_{h,\\sigma}^{n,{\\rm up}} -\\hat\\bu_\\sigma^{n,{\\rm up}}|\n\\\\\n& \\le c(\\underline r,\\overline r, \\|(\\bU,\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})})\\Big[\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h \\Big(E(\\vr_\\sigma^{n,{\\rm up}}|\\hat r_\\sigma^{n,{\\rm up}})\\Big)^{1\/2}\n\\\\\n&\n+\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\nE(\\vr_\\sigma^{n,{\\rm up}}|\\hat r_\\sigma^{n,{\\rm up}})\\Big)^{2\/3}\\Big]\n \\times\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h\\Big|{\\hat\\bU}_{h,\\sigma}^{n,{\\rm up}} -\\hat\\bu_\\sigma^{n,{\\rm up}}\\Big|^6\\Big)^{1\/6}\n\\end{aligned}\n\\end{equation*}\nprovided $\\gamma\\ge 3\/2$.\nNext, we observe that the contribution of the face $\\sigma=K|L$ to the sums\n$\n\\sum_{K\\in {\\cal T}}$ $\\sum_{\\sigma\\in {\\cal E}(K)}$ $|\\sigma| h\nE(\\vr_\\sigma^{n,{\\rm up}}|\\hat r_\\sigma^{n,{\\rm up}})$\nand $ \\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}|\\sigma| h|{\\hat\\bU}_{h,\\sigma}^{n,{\\rm up}} -\\hat\\bu_\\sigma^{n,{\\rm up}}|^6$\nis less or equal than\n$2|\\sigma| h ( E(\\vr_K^{n}|\\hat r_K^{n}) + E(\\vr_L^{n}|\\hat r_L^{n})),\n$ and\nthan $2|\\sigma| h (|{\\bU}_{h,K}^{n} -\\bu_K^{n}|^6+ |{\\bU}_{h,L}^{n} -\\bu_L^{n}|^6)$, respectively. Consequently,we get\nby the same reasoning\nas in the previous step, under assumption $\\gamma\\ge 3\/2$,\n\\begin{equation}\\label{cP2}\n|{\\cal P}_2|\\le c(\\delta, M_0, E_0, \\underline r,\\overline r, \\|(\\bU,\\nabla\\bU)\\|_{L^\\infty(Q_T;\\Rm^{12})})\\,\\deltat \\sum_{n=1}^m {\\cal E}(\\vr^n,\\bu^n|\nr^n,\\bU^n) + \\delta\\; \\deltat \\sum_{n=1}^m |\\bu^n-\\bU_{h}^n)|_{V^2_h(\\Omega;\\Rm^3)}^2.\n\\end{equation}\n}\n{\\bf Step 3:} {\\it Term ${\\cal P}_3$.}\nSince the pair $(r,\\bU)$ satisfies continuity equation (\\ref{cont2}) in the classical sense, we have for all $n=1,\\ldots,N$,\n\\[\n[\\partial_t r]^n +\\bU^n\\cdot\\nabla r^n=-r^n\\dv\\bU^n,\n\\]\nwhere we recall that $[\\partial_t r]^n(x)=\\partial_t r(t_n,x)$ in accordance with (\\ref{notation2-}).\nUsing this identity we write\n\\begin{align*}\n&{\\cal P}^n_3= {\\cal P}_{3,1} +{\\cal P}_{3,2},\\quad {\\cal P}_{3,i}=\\deltat\\sum_{n=1}^m {\\cal P}_{3,i}^n,\\\\\n& \\mbox{with } {\\cal P}_{3,1}^n=\n-\\sum_{K\\in {\\cal T}} \\int_K \\Big(p(\\vr_K^n)-p'(r_K^n)(\\vr_K^n-r_K^n)-p(r_K^n)\\Big)\\dv\\bU^n\\dx \\\\\n& \\mbox{and } {\\cal P}_{3,2}^n= \\sum_{K \\in {\\cal T}}\\Big[\\int_K\\frac{r_K^n-\\vr^n_K}{r^n_K} p'(r^n_K)( \\bu^n-\\bU^n)\\cdot\\nabla r^n\\dx.\n\\end{align*}\nNow, we apply Lemma \\ref{LL1} in combination with assumption (\\ref{pressure1}) to deduce\n\\begin{equation}\\label{cP3}\n|{\\cal P}_{3,1}|\\le c\\|\\dv \\bU\\|_{L^\\infty(Q_T)}\\deltat \\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n).\n\\end{equation}\nFinally, the same reasoning as in Step 2 leads to the estimate\n\\begin{equation}\\label{cP4}\n\\begin{aligned}\n|{\\cal P}_{3,2}|& \\le h\\;c(M_0, E_0,\\underline r, \\overline r, |p'|_{C([\\underline r,\\overline r])} \\|(\\nabla r,\\nabla\\bU)\\|_{L^\\infty(\\Omega;\\Rm^9)})\n\\\\ & + c(\\delta, \\|\\underline r, \\overline r, |p'|_{C([\\underline r,\\overline r])} \\|\\nabla r\\|_{L^\\infty(\\Omega;\\Rm^3)})\\;\n\\deltat \\sum_{n=1}^m{\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n) +\\delta \\,\\deltat \\sum_{n=1}^m|\\bu^n-\\bU^n_h|_{V^2_h(\\Omega;\\Rm^3)}^2.\n \\end{aligned}\n\\end{equation}\nGathering the formulae (\\ref{relativeenergy-1}) and (\\ref{cP1})-(\\ref{cP4}) with $\\delta$ sufficiently small (with respect to $\\mu$), we conclude the proof of Lemma \\ref{Gronwall}.\n\\end{proof}\n\n{ Finally, Lemma \\ref{Gronwall} in combination with the bound (\\ref{est4}) yields\n\\[{\\cal E}(\\vr^m,\\bu^m|r^m,\\bU^m)\\le c\\Big[h^A+\\sqrt{\\deltat}+\\deltat + {\\cal E}(\\vr^0,\\bu^0|r^0,\\bU^0)\\Big] + c\\,\\deltat\\sum_{n=1}^{m-1} {\\cal E}(\\vr^n,\\bu^n|r^n,\\bU^n);\n\\]\nwhence Theorem \\ref{Main} is a direct consequence of the standard discrete version of Gronwall's lemma. Theorem \\ref{Main} is thus proved.}\n\n\\section{Appendix: Fundamental auxiliary lemmas and estimates}\\label{3}\n\nIn this section we report several results related to the properties of the Sobolev spaces on tetrahedra\nand of the Crouzeix-Raviart (C-R) space. { We refer to the book Brezzi, Fortin \\cite{BRFO} for the general introduction to the subject.}\n\n\n\n\n\n\n\n\n\n{ We start with the inequalities that can be obtained by rescaling from the standard inequalities on a reference tetrahedron of size equivalent to one.}\n\n\\begin{lm}[{ Poincar\\'e, Sobolev and interpolation inequalities on tetrahedra}]\\label{Lemma2}\nLet $1\\le p\\le \\infty$. Let $\\theta_0 > 0$ and ${\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in (\\ref{reg}). Then we have:\n\\begin{description}\n\\item{\\it (1) Poincar\\'e type inequalities on tetrahedra}\n\nLet $1\\le p\\le\\infty$. There exists $c=c(\\theta_0,p)>0$ such that for all\n$K\\in {\\cal T}$ and for all $v\\in W^{1,p}(K)$ we have\n\\begin{equation}\\label{L2-1}\n\\|v-v_K\\|_{L^p(K)}\\le c h\\|\\nabla v\\|_{L^p(K)},\n\\end{equation}\n\\begin{equation}\\label{L2-2}\n\\forall \\sigma\\in {\\cal E}(K), \\;\\|v-v_\\sigma\\|_{L^p(K)}\\le c h\\|\\nabla v\\|_{L^p(K)}.\n\\end{equation}\n\\item{\\it (2) Sobolev type inequalities on tetrahedra}\n\nLet $1\\le p0$ such that for all\n$K\\in {\\cal T}$ and for all $v\\in W^{1,p}(K)$ we have\n\\begin{equation}\\label{L2-3}\n\\|v-v_K\\|_{L^{p^*}(K)}\\le c \\|\\nabla v\\|_{L^p(K)},\n\\end{equation}\n\\begin{equation}\\label{L2-4}\n\\forall \\sigma\\in {\\cal E(}K), \\;\\|v-v_\\sigma\\|_{L^{p^*}(K)}\\le c \\|\\nabla v\\|_{L^p(K)},\n\\end{equation}\nwhere $p^*=\\frac{dp}{d-p}$.\n\\item{\\it (3) Interpolation inequalities on the tetrahedra}\n\nLet $1\\le p0$ such that for all\n$K\\in{\\cal T}$ and $v\\in W^{1,p}(K)$ we have\n\\begin{equation}\\label{interpol1}\n\\|v-v_K\\|_{L^{q}(K)}\n\\le c h^\\beta\\|\\nabla v\\|_{L^p(K;\\Rm^d)},\n\\end{equation}\n\\begin{equation}\\label{interpol2}\n\\|v-v_\\sigma\\|_{L^{q}(K)}\n\\le ch^\\beta \\|\\nabla v\\|_{L^p(K;\\Rm^d)},\n\\end{equation}\nwhere $\\frac 1q=\\frac \\beta p+\\frac {1-\\beta}{p^*}$.\n\\end{description}\n\\end{lm}\n{ Combining estimates (\\ref{L2-1}--\\ref{interpol2}) with the algebraic inequality}\n\\begin{equation}\\label{dod1*}\n\t\\Big(\\sum_{i=1}^L|a_i|^p\\Big)^{1\/p}\\le \\Big(\\sum_{i=1}^L|a_i|^q\\Big)^{1\/q}\n\\end{equation}\nfor all $(a=(a_1,\\ldots,a_L)\\in \\Rm^L$, $1\\le q\\le p<\\infty$, we obtain the following corollaries.\n\\begin{cor}[Poincar\\'e and Sobolev type inequalities on the Sobolev spaces]\\label{cor1}\nUnder the assumptions of Lemma \\ref{Lemma2}, we have:\n\\begin{description}\n\\item{\\it (1) Poincar\\'e type inequalities on the domain $\\Omega$}\n\n Let $1\\le p\\le \\infty$. There exists $c=c(\\theta_0,p)>0$ such that for all $v\\in W^{1,p}(\\Omega)$ we have\n\\begin{align}\\label{L2-5}\n\t\\|v-\\hat v\\|_{L^p(\\Omega)}\\equiv\\Big(\\sum_{K\\in {\\cal T}}\\|v-v_K\\|^p_{L^p(K)}\\Big)^{1\/p}\\le c h\\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)},\\\\\n\t\\label{L2-6}\n\t\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|^p_{L^p(K)}\\Big)^{1\/p}\\le c h\\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}\n\\end{align}\nwhere $\\hat v$ and $v_\\sigma$ are defined by \\eqref{vhat} and \\eqref{vtilde}.\n\\item{\\it (2) Sobolev type inequalities on the domain $\\Omega$}\n\nLet $1\\le p0$ such that for all\n$v \\in W^{1,p}(\\Omega)$ we have\n\\begin{equation}\\label{L2-7}\n\\|v-\\hat v\\|_{L^{p^*}(\\Omega)}\n\\le c \\|\\nabla v\\|_{L^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{L2-8}\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|^{p^*}_{L^{p^*}(K)}\\Big)^{1\/p^*}\n\\le c \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}.\n\\end{equation}\n\\item{\\it (3) Interpolation inequalities on the domain $\\Omega$}\n\nLet $1\\le p0$ such that for all\n$v\\in W^{1,p}(\\Omega)$ we have\n\\begin{equation}\\label{L2-9}\n\\|v-\\hat v\\|_{L^{q}(\\Omega)}\n\\le c h^\\beta\\|\\nabla v\\|_{L^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{L2-10}\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|^{q}_{L^{q}(K)}\\Big)^{\\frac 1 q}\n\\le ch^\\beta \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)},\n\\end{equation}\nwhere $\\frac 1q=\\frac \\beta p+\\frac {1-\\beta}{p^*}$.\n\\end{description}\n\\end{cor}\n\n\\begin{cor}[Poincar\\'e and Sobolev type inequalities on $V_h$]\\label{cor2}\nUnder assumptions of Lemma \\ref{Lemma2}, there holds:\n\\begin{description}\n\\item {\\it (1) Poincar\\'e type inequality in $V_h(\\Omega)$:}\nLet $1\\le p<\\infty$. There exists $c=c(\\theta_0,p)$ such that for all $v \\in V_h,$\n\\begin{equation}\\label{sob4-}\n\\|v-\\hat v\\|_{L^{p}(\\Omega)}\n\\le c h |v|_{V_h^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{sob5-}\n \\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|_{L^p(K)}^p\\Big)^{\\frac 1 p}\n\\le c h | v|_{V_h^p(\\Omega)}.\n\\end{equation}\n\n\\item {\\it (2) Sobolev type inequality in $V_h(\\Omega)$:}\nLet $1\\le p0$ such that for all\n$v\\in V_h(\\Omega)$ we have\n\\begin{equation}\\label{L2+-1}\n\\|v-\\hat v\\|_{L^{q}(\\Omega)}\n\\le c h^\\beta|v|_{V_h^p(\\Omega)},\n\\end{equation}\n\\begin{equation}\\label{L2+-2}\n\\Big(\\sum_{K\\in {\\cal T}}\\sum_{\\sigma\\in {\\cal E}(K)}\\|v-v_\\sigma\\|_{L^q(K)}^q\\Big)^{\\frac 1 q}\n\\le ch^\\beta |v|_{V_h^p(\\Omega)},\n\\end{equation}\nwhere $\\frac 1q=\\frac \\beta p+\\frac {1-\\beta}{p^*}$.\n\\end{description}\n\\end{cor}\n\n{ The next fundamental lemma deals with the properties of the projection $v_h$ defined by (\\ref{vh})}.\n\n\\begin{lm} [Projection on $V_h$]\\label{Lemma1}\nLet $\\theta_0 > 0$ and ${\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in (\\ref{reg}).\n\\begin{description}\n\\item{\\it (1) Approximation estimates on the tetrahedra}\n\nLet $1\\le p\\le \\infty$. There exists $c=c(\\theta_0,p)>0$ such that\n{\n\\begin{equation}\\label{ddd}\n\\forall v\\in W^{1,p}_0(\\Omega)\\cap L^\\infty(\\Omega), \\forall K \\in {\\cal{T}},\\;\\|v_h\\|_{L^\\infty(K)}\\le c \\|v\\|_{L^\\infty(K)},\n\\end{equation}\n}\n\\begin{equation}\\label{L1-2}\n\\forall v \\in W^{1,p}_0\\cap W^{s,p}(\\Omega), \\forall K \\in {\\cal{T}},~ || v-v_h||_{L^p(K)} \\le c h^{s}\\|\\nabla^s v\\|_{L^p(K;\\Rm^{d^s})},\n\\end{equation}\n\\begin{equation}\\label{L1-3}\n|| \\nabla( v- v_h) ||_{L^p(K;\\Rm^d)} \\le c h^{s-1} \\|\\nabla^s v\\|_{L^p(K;\\Rm^{d^s})}, s=1,2.\n\\end{equation}\n\n\\item{\\it (2) Preservation of divergence}\n\n\\begin{equation}\\label{L1-1}\n\\forall \\bv \\in W^{1,2}_{0}(\\Omega,\\R^d), \\forall q \\in L_h(\\Omega), \\sum_{K\\in {\\cal T}}\\int_K q ~\\dv\\bv_h \\dx = \\int_{\\Omega} q ~\\dv \\bv \\dx\n\\end{equation}\n\n\\item{(3) Approximation estimates of the Poincar\\'e type on the whole domain}\n\nLet $1\\le p<\\infty$. There exists $c=c(\\theta_0,p)>0$ such that for all $v\\in W^{1,p}_0(\\Omega)$,\n\\begin{equation}\\label{L1-4}\n\\|v-v_h\\|_{L^p(\\Omega)}\\le c h \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}.\n\\end{equation}\n\n\n\\item{ (4) Approximation estimates of the Sobolev type on the whole domain}\n\nLet $1\\le p0$ such that for all $v\\in W^{1,p}_0(\\Omega)$,\n\\begin{equation}\\label{L1-5}\n\\|v-v_h\\|_{L^{p^*}(\\Omega)}\\le c \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)}.\n\\end{equation}\n\\end{description}\n\\end{lm}\n\n{ Statement {\\it (2)} of Lemma \\ref{Lemma1} is proved in\n\\cite{cro-73-con}, where one can find also the proof of item {\\it (1)} for $p=2$. We present here the proof of statements {\\it (1), (3), (4)} for arbitrary $p$ for the reader's convenience, since a straightforward reference is not available.\n\\\\ \\\\\n\\begin{proof}\n\n{\\textbf{ Step 1:}} We start with some generalities. First we complete the Crouzeix-Raviart basis (\\ref{CRB}) by functions $\\phi_\\sigma$ indexed also with $\\sigma\\in {\\cal E}_{\\rm ext}$ saying\n$$\n\\frac 1{|\\sigma'|}\\int_{\\sigma'}\\phi_\\sigma {\\rm d} S=\\delta_{\\sigma,\\sigma'},\\; \\;(\\sigma,\\sigma')\\in {\\cal E}^2\n$$\nand observe that\n\\begin{equation}\\label{+1}\n\\sum_{\\sigma\\in {\\cal E}(K)}\\phi_\\sigma(x)= 1\\;\\mbox{ for any $x\\in K$.}\n\\end{equation}\nA scaling argument yields\n\\begin{equation}\\label{+2}\n\\|\\phi_\\sigma\\|_{L^\\infty(\\Omega)}\\le c(\\theta_0),\\; h\\|\\nabla\\phi_\\sigma\\|_{L^\\infty(\\Omega;\\Rm^d)}\\le c(\\theta_0).\n\\end{equation}\nSecond, we define the projection $v\\to v_h$ for any $v\\in W^{1,p}(\\Omega)$ by saying\n$$\nv_h=\\sum_{\\sigma\\in {\\cal E}} v_\\sigma\\phi_\\sigma.\n$$\nWe notice that if $v\\in W^{1,p}_0(\\Omega)$ then $v_h$ coincides with (\\ref{CRP}). Moreover,\n\\begin{equation}\\label{+3}\nv_h=v\\;\\mbox{for any affine function $v$.}\n\\end{equation}\nThird, due to the density argument, it is enough to show the remaining statements {\\it (1), (3), (4)} for\n$v\\in W^{1,p}_0(\\Omega)\\cap W^{s,\\infty}(\\Omega)$, $s=1,2$, according to the case.\n\\\\ \\\\\n{\\textbf{Step 2:}} { We realize that ${\\rm supp }\\phi_\\sigma=K\\cup L$ and derive (\\ref{ddd}) directly by employing representation (\\ref{vh}), definition of $v_\\sigma$ and estimate (\\ref{+2}).}\n\nWe denote by $x_K=\\frac 1{|K|}\\int_K x{\\rm d}x$ the center of gravity of the tetrahedron $K$. We calculate by using (\\ref{+3}) and the first order Taylor formula\n$$\nv(x)-v_h(x)= v(x)-v(x_K) - [v-v(x_K)]_h(x)\n$$\n$$\n=(x-x_K)\\cdot\\int_0^1\\nabla v(x_K+t(x-x_K)){\\rm d} t\n- \\sum_{\\sigma\\in {\\cal E}(K)}\\phi_\\sigma(x) \\frac 1{|\\sigma|}\\int_\\sigma\\Big[(x-x_K)\\cdot\\int_0^1\\nabla v(x_K+t(x-x_K)){\\rm d} t\\Big]{\\rm d} S,\n$$\nwhere $x\\in K$.\nThis formula yields immediately the upper bound stated in (\\ref{L1-2})$_{s=1}$ if $p=\\infty$. If $1\\le p<\\infty$\nwe calculate the upper bound of the $L^p$-norm of each term at the right-hand side separately by using (\\ref{+2}), Fubini's theorem, H\\\"older's inequality and the change of variables $y= x_K+t(x-x_K)$ together with the convexity of $K$.\n\nThe same reasoning can be applied to prove (\\ref{L1-2})$_{s=2}$. Indeed, we observe\nthat\n$$\nv(x)-v_h(x)= v(x)-(x-x_K)\\cdot\\nabla v(x_K)-v(x_K) - [v -(x-x_K)\\cdot\\nabla v(x_K)-v(x_K)]_h(x)\n$$\nby virtue of (\\ref{+3}). Now we apply to the right hand side of the last expression the second order Taylor formula in the integral form, and proceed exactly as described before.\n\nFinally, one applies the same straightforward argumentation to get (\\ref{L1-3}). This completes the proof of statement {\\it (1)}.\n\\\\ \\\\\n{\\textbf{Step 3:}} Statement {\\it (3)} follows easily from (\\ref{L1-2})$_{s=1}$ and the algebraic inequality (\\ref{dod1*}).\n\\\\ \\\\\n{\\textbf{ Step 4:}} We use (\\ref{+1}) and (\\ref{+3}) to write\n$$\nv(x)-v_h(x)= \\sum_{\\sigma\\in {\\cal E}(K)} (v(x)-v_\\sigma)\\phi_\\sigma(x),\\;\\; x\\in K;\n$$\nwhence\n$$\n\\|v-v_h\\|_{L^{p^*}(K)}\\le c\\|\\nabla v\\|_{L^p(K;\\Rm^d)}\n$$\nwhere we have used the Sobolev inequality (\\ref{L2-4}) on the tetrahedron $K\\in {\\cal T}$ and the $L^\\infty$-bound (\\ref{+2}). We conclude the proof of statement {\\it (4)} by using the relation (\\ref{dod1*}). The proof of Lemma \\ref{Lemma1} is complete.\n\\end{proof}\n}\n\n\nThe following corollary is a direct consequence of (\\ref{L1-3}).\n\n\\begin{cor}[Continuity of the projection onto $V_h$]\\label{cor3}\nUnder assumptions of Lemma \\ref{Lemma1}, there exists $c=c(\\theta_0,p)>0$ such that\n\\begin{equation}\\label{L1-6}\n\\forall v \\in W^{1,p}_0(\\Omega),\\;|v_h|_{V_h^p(\\Omega)}\\le c \\|\\nabla v\\|_{L^p(\\Omega;\\Rm^d)},\n\\end{equation}\nwhere $1\\le p<\\infty$.\n\\end{cor}\n\nAlthough the non conforming finite element space $V_h$ is not a subspace of any Sobolev space, its elements enjoy the Sobolev type inequalities.\nThis important fact is formulated in the next lemma.\n\n\\begin{lm}[Sobolev inequality on $V_h$]\\label{Lemma2+}\nLet $\\Omega$ be a bounded domain of $\\R^d$.\nLet ${\\cal{T}}$ be a triangulation of the domain $\\Omega$ in simplices such that $\\theta \\ge \\theta_0 >0$ where $\\theta$ is defined in $(\\ref{reg})$.\nThen we have:\n\\begin{description}\n\\item{ (1) Sobolev inequality in $V_h(\\Omega)$ (case $1\\le p0$ such that forall $v \\in V_h(\\Omega)$,\n\\begin{equation}\\label{sob3}\n\\|v\\|_{L^q(\\Omega)} \\le c|v|_{V_h^p(\\Omega)}\n\\end{equation}\n\n\\end{description}\n\\end{lm}\n\\begin{proof}\n\n\\textbf{Step 1}\nLet $1 \\le r \\le \\alpha < \\infty$. Let $ u \\in V_h$. We call $v$ the element of $V_h$ such that $ v_\\sigma = |u_\\sigma|^\\alpha $. Then there exists $C$ only depending on $d,r,\\alpha$ such that\n\\begin{equation}\\label{leman3}\n ||u||_{L^r(\\Omega)}^\\alpha \\le || u ||_{L^{\\frac{r}{\\alpha}}(\\Omega)}.\n\\end{equation}\n\nTo prove ($\\ref{leman3}$) we remark that, using a change of variable, it is enough to show to prove the existence of $C$ for only the unit symplex $ \\hat{K}$. Let $u \\in \\mathbb{P}_1(\\hat{K}) $ and we call $v$ the element of $\\mathbb{P}_1(\\hat{K}) $ such that $ v_\\sigma = |u_\\sigma|^\\alpha $. Let $T(u) = ||u||_{L^r(\\hat{K})} $ and $ S(u) = || u ||_{L^{\\frac{r}{\\alpha}}(\\hat{K})}\n^{\\frac{1}{\\alpha}}$. These two functions are continuous, homogeneous of degree $1$ and non zero if $ u \\ne 0$. Since $\\mathbb{P}_1(\\hat{K}) $ is a finite dimensional space, we can choose a norm on $\\mathbb{P}_1(\\hat{K}) $ and take $C = (\\frac{M}{m})^\\alpha $ where $ M = \\max \\{ T(u), ||u||_{\\mathbb{P}_1(\\hat{K})}=1 \\} $ and $ m = \\min \\{ T(u), ||u||_{\\mathbb{P}_1(\\hat{K})}=1 \\} $.\n\\vspace{2em}\n\n\n\n\\textbf{ Step 2: Proof for $p=1.$} \\\\\nWe set $u=0$ outside $\\Omega$. For $\\sigma \\in {\\cal{E}_{\\intt}}, \\sigma=K|L$, we set $ |[u(x)]|=|u_K(x)-u_L(x)|$ for $x \\in \\sigma$. For $ \\sigma \\in {\\cal{E}}_{\\extt}\\cap {\\cal{E}}(K) $, we set $|[u(x)]| =|u_K(x)|$ for $x \\in \\sigma$. We first remark that there exists $C_{1,1}$ and $C_{1,2}$ only depending on $d$ such that\n$$ ||u||_{L^{\\frac{d}{d-1}}(\\Omega)} \\le C_{1,1}||u||_{BV(\\R^d)} \\le C_{1,2} || \\nabla_h u||_{L^1(\\Omega)} + C_{1,2} \\sum_{\\sigma \\in {\\cal{E}}} \\int_\\sigma |[u]| {\\rm d}S. $$\nWe now prove that there exits $C_{1,3}$ only depending on $d$ and $\\theta_0$ such that\n$$ \\sum_{\\sigma \\in {\\cal{E}}} \\int_\\sigma |[u]| {\\rm d}S \\le C_{1,3} || \\nabla_h u||_{L^1(\\Omega)}. $$\nLet $K \\in \\T $ and $ \\sigma \\in {\\cal{E}}(K) $. Let $x_\\sigma $ be the center of mass of $\\sigma$. We have, with$ u_K =u $ in $K$,\n$$ u_K(x) -u(x_\\sigma) = \\int_0^1 \\nabla u_K \\cdot (x-x_\\sigma) \\dx.$$\nThen if $ \\sigma=K|L$ we have\n$$ |u_K(x)-u_L(x)| \\le h_\\sigma \\Big( |\\nabla u_K| + |\\nabla u_L| \\Big). $$\nIntegrating this inequality on $\\sigma$ gives\n$$ \\int_\\sigma |[u]| {\\rm d}S \\le |\\sigma|h_\\sigma \\Big( |\\nabla u_K| + |\\nabla u_L| \\Big) \\le \\frac{2}{\\theta_0^d} \\Big( || \\nabla u||_{L^1(K)} +|| \\nabla u||_{L^1(L)} \\Big).$$\nSimilarly for $ \\sigma \\in \\E_{\\extt}\\cap \\E(K) $ we have\n$$ \\int_\\sigma |[u]| {\\rm d}S \\le \\frac{2}{\\theta_0^d} || \\nabla u||_{L^1(K)} $$\nThen there exists $C_{1,3}=C(d,\\theta_0) $ such that\n$$ \\stie \\int_\\sigma |[u]| {\\rm d}S \\le C_{1,3} || \\nabla_h u ||_{L^1(\\Omega)}. $$\nand then,\n$$ ||u||_{L^{1^*}(\\Omega)} \\le c(d,\\theta_0) || \\nabla_h u||_{L^1(\\Omega)}. $$\n\\textbf{ Step 3: Proof for $1 1$ and $\\alpha 1^*=p^* $. We call $v$ the element of $V_h$ such that $ v_\\sigma = |u_\\sigma|^\\alpha $ for $ \\sigma \\in \\E $. One has $v \\ne |u|^\\alpha $ but there exits $C_{2,1} $ only depending on $d$ and $p$ (see lemma $\\ref{leman3}$) such that\n$$ || u||_{L^{p^*}(\\Omega)}^\\alpha \\le C_{2,1} ||v||_{L^{1^*}(\\Omega)} \\le c(d,p,\\theta_0) ||\\nabla_h v||_{L^{1}(\\Omega)}. $$\nMoreover using a scalling argument we obtain\n$$ || \\nabla_h v ||_{L^1(K)} \\le c(d,p,\\theta_0) \\stiek |u_\\sigma|^{\\alpha-1}| \\nabla u_K | |K|. $$\nThen, using H\\\"older Inequality, we have, with $q=\\frac{p}{p-1}$ (so that $q( \\alpha-1) =p^*$),\n$$ || \\nabla_h v ||_{L^1(K)} \\le c(d,p,\\theta_0) || \\nabla u ||_{L^p(K)} || u||_{L^{p^*}(K)}^{\\frac{p^*}{q}}.$$\nSumming on $ K \\in \\T$ we obtain\n$$||u||_{L^{p^*}(\\Omega)} \\le C_2 || \\nabla_h u||_{L^p(\\Omega)}.$$\n\\textbf{Step 4: Proof for $p\\ge d.$} \\\\\nLet $ 1 \\le q < \\infty $. There exists $r=r(d,q)$ such that $r 0$ and $ {\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in $ (\\ref{reg}) $. Then\nthe norms\n\\begin{equation}\\label{norms1}\n\\Big(\\sti |\\sigma| h |v_\\sigma|^p\\Big)^{1\/p}\\quad\\mbox{and}\\quad ||v||^p_{L^p(\\Omega)}\n\\end{equation}\nare equivalent on $V_h(\\Omega)$ uniformly with respect to $h>0$.\n\\end{lm}\n\nThe last lemma in this overview deals with the estimates of jumps over faces.\nThe reader can consult \\cite[Lemma 3.32]{ern-04-the} or \\cite[Lemma 2.2]{GHL2009iso} for its proof.\n\\begin{lm}[Jumps over faces in the Crouzeix-Raviart space]\\label{Lemma6}\nLet $ \\theta_0 > 0$ and $ {\\cal{T}} $ be a triangulation of $ \\Omega $ such that $ \\theta \\ge \\theta_0 $ where $ \\theta $ is defined in $ (\\ref{reg}) $. Then there exists $ c=c(\\theta_0)>0 $ such that for all $ v \\in V_h(\\Omega)$,\n\\begin{equation}\\label{tbound}\n\\sum_{\\sigma \\in {\\cal{E}}} \\frac{1}{h} \\int_\\sigma [v]_{\\sigma,\\bn_\\sigma}^2 \\dS \\le c |v|_{V_h^2(\\Omega)}^2,\n\\end{equation}\nwhere $[v]_{\\sigma,\\bn_\\sigma}$ is a jump of $v$ with respect to a normal $\\bn_\\sigma$ to the face $\\sigma$,\n\\[\n\\forall x\\in\\sigma=K|L\\in{\\cal E}_{\\rm int},\\quad [v]_{\\sigma,\\bn_\\sigma}(x)=\\left\\{\n\\begin{array}{c}\n v|_K(x)-v|_L(x)\\;\\mbox{if $\\bn_\\sigma=\\bn_{\\sigma,K}$}\\\\\nv|_L(x)-v|_K(x)\\;\\mbox{if $\\bn_\\sigma=\\bn_{\\sigma,L}$}\n\\end{array}\\right.\n\\]\nand\n\\[\n\\forall x\\in\\sigma\\in{\\cal E}_{\\rm ext},\\quad [v]_{\\sigma,\\bn_\\sigma}(x)=v(x),\\;\\mbox{with $\\bn_\\sigma$ an exterior normal to $\\partial\\Omega$}.\n\\]\n\\end{lm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{} \n\n\\section{Introduction}\nFor many years the active galaxy phenomenon was regarded as an interesting sideline\nin astrophysics that related solely to our understanding of black holes. However,\nwith the recognition that every massive galaxy in the local universe has a \nsupermassive black hole at its heart \n\\citep{magorrian98, richstone98, ferrarese00, gebhardt00}\nwe now recognise that black hole formation is an integral part of galaxy formation,\nwith relationships between black hole and host galaxy that are consistent for\ngalaxies with or without AGN \\citep{onken04, barth05}.\nMoreover, it now seems possible that feedback from black hole growth may have a\nsignificant effect in shutting off star formation and defining the colour-luminosity \ndistribution of massive galaxies \\citep{benson03, croton06, bower06, delucia06}.\nHowever, our understanding of both the physics and the prevalence of AGN feedback is\nextremely limited, as reviewed in the next section. We also have only a partial\nunderstanding of the cosmological dependence of black hole growth within galaxies,\nyet if we wish to understand AGN feedback in the cosmological context this is\nsurely a crucial aspect of the problem.\nIn later sections of this article I discuss one particular scenario for black hole\ngrowth, namely that black holes and galaxies and their dark halos grow coevally,\nand compare with observation.\n\n\\section{AGN feedback - fact or fiction?}\nThe physics of supermassive black hole growth in galaxies and its cosmological\nevolution has recently been receiving wide attention because of the possible\neffects of AGN feedback on the formation of the host galaxy itself \\citep[e.g.][]{dimatteo05}.\nSince we now recognise that every massive galaxy (at least) has a black hole, such\nfeedback may be an integral part of the general galaxy-formation process.\n\\citet{benson03} have argued that something like AGN feedback is required to shut\ndown star formation in the most massive galaxies in order to recreate the observed\ngalaxy mass function, and it has been argued that such feedback\nis required to reproduce the observed colour-luminosity distribution of galaxies\n\\citep{bower06, croton06, delucia06}.\nHowever, the feedback process has to operate in every galaxy and in\nthe relatively recent universe,\nso rather than feedback from the luminous QSO or AGN phase of black hole growth,\nthese authors envisage a ``radio mode'' in which a radiatively-quiet, \nbut kinetically-powerful, jet or outflow provides the feedback in the recent\nuniverse. \n\nIt has long been suggested that radio sources may provide significant heat input to\ncluster gas \\citep[e.g.][]{miller88,pedlar88,pedlar90} but this idea wasn't really taken\nseriously until the discovery both of a lack of cooling gas in ``cooling flow'' clusters\nand of a possible link with radio structures in massive clusters such as Perseus\n(\\citealt{peterson_fabian06} and references therein). But most galaxies at the present\nday don't have powerful radio sources associated with them, so we need to consider\nthe evidence for ``quiet'' outflows in nearby galaxies.\n\nNearby Seyfert active galaxies do have low density outflowing gas in their nuclear regions,\nbut so far these do not appear energetically significant: in NGC\\,5548 for example the\nmass outflow rate is $\\dot{M} \\sim 0.3$\\,M$_{\\odot}$\\,year$^{-1}$ \nwith a maximum outflow velocity $\\sim 1000$\\,km\\,s$^{-1}$ \\citep{steenbrugge05}, \nso it seems unlikely that this\namount of feedback would significantly affect the host galaxy. Evidence for \nhigher velocities and therefore\nsignificantly higher mass and momentum outflow rates has been seen in a few QSOs\n\\citep[e.g.][]{pounds03}. High velocity QSO outflows are also seen in UV absorption lines,\noccurring in $\\sim 26$\\,percent of QSOs \\citep{trump06}, and it has been supposed that\nthese might be associated with an energetic disc wind \\citep[e.g.][]{proga_kallman04},\nbut if such outflows are restricted to the rare (at $z=0$) luminous QSO phase of\nblack hole activity they won't do the job. But kinetic jet output may be more significant\nthan outflowing winds \\citep[e.g.][]{omma04}, and radiatively-inefficient flows\nmay produce such outflows \\citep{narayan94, ho02}, or indeed rotating black holes may \nproduce output electrodynamically \\citep{blandford_znajek77,reynolds06} \nperhaps independently of\nvisible radiation. But a requirement of all these pictures is that, regardless of\nwhether or not we can see it, the source of the feedback energy is gravitational - accretion\nof matter onto the black hole. Despite many attempts over the past four decades,\nwe still have limited understanding\nof how black holes form in galaxies and why their luminous accretion phase shows such\nstrong cosmological evolution.\nIn the remainder of this article we discuss what we know about\nblack hole accretion and its cosmological history, and in particular discuss the\nhypothesis that black holes and galaxies grew coevally.\n\n\\section{QSO and AGN evolution}\nTwenty years ago a striking picture emerged of the cosmological evolution of luminous QSOs:\nthey had an optical luminosity function that had a broken power-law form and that\nevolved steadily to lower luminosity at lower redshifts without changing in normalisation\n\\citep{marshall83, boyle88}. A natural interpretation of this was that supermassive black\nholes formed relatively early in the universe, and that accretion onto them steadily\ndeclined with time to produce the apparent ``pure luminosity evolution''. More recently\nhowever this picture has fallen into disfavour, for two reasons. \n\nFirst, it is widely\nbelieved that dark matter haloes and their galaxies have grown hierarchically, with massive\nstructures continuing to build up in the relatively recent universe. It is often hypothesised\nthat QSOs are triggered by mergers between galaxies in such a hierarchical universe, and\nmodels seeking a cosmological explanation for QSO evolution have been based on a merger-driven\nbuild-up of black holes \\citep[e.g.][]{kauffmann_haehnelt00}. This link to galaxy build-up\nwould explain the qualitative similarity in the cosmological evolution of star formation \nand nuclear black hole activity \\citep[e.g.][]{dunlop97, boyle98, percival_miller99}.\n\nSecond, a more detailed look at the luminosity function has revealed departures from \npure luminosity evolution, with evolution in the slope of the luminosity function at both\nbright and faint magnitudes \\citep{hewett93, goldschmidt98, hopkins06a, fan06}.\nMore significantly, it seems that the peak in the comoving space density of AGN\/QSOs of\na given luminosity shifts systematically to lower redshifts with decreasing luminosity\n(\\citealt{steffen03}, \n\\citealt{cowie03}, \n\\citealt{ueda03}, \n\\citealt{zheng04}, \n\\citealt{barger05}, \n\\citealt{lafranca05}, \n\\citealt{hasinger05}, and \n\\citealt{hopkins06b}),\na phenomenon that has become known as cosmic downsizing. The downsizing is often\ninterpreted as reflecting an increasing prevalence of lower-mass black hole growth\nwith decreasing redshift (but note that, unlike the case of galaxy downsizing, \nthe black hole masses are not well determined at the redshifts covered by the above\nsurveys, so this interpretation can only be regarded as preliminary).\n\nDespite these concerns, the basic picture of twenty years ago must nonetheless be correct:\nthe comoving irreducible mass in black holes cannot decrease with cosmic time, so the\ndecrease observed in the integrated AGN luminosity density (and in the differential luminosity\nfunction) must indeed arise from a mean accretion rate that decreases with cosmic time.\nThe departures from pure luminosity evolution are then most likely indicating that we are\nnot observing a single fixed population of long-lived black holes but rather the \nstatistical changes in a relatively (compared to the Hubble time) short-lived\npopulation.\n\nConfirmation that luminous QSO lifetimes are shorter than the Hubble time comes from\nmeasurement of their clustering properties: the lack of clustering growth to\nlower redshift shows that QSOs cannot be a long-lived population of objects\n(\\citealt{croom05}, these proceedings). The upper limit on their mean lifetime\nis redshift dependent and does depend on the bias model assumed for QSOs, but\na reasonable model yields $2\\sigma$ limits on their existence\nwithin the 2QZ survey of $< 2$\\,Gyr and $< 1$\\,Gyr at $z=1$ and $z=2$ respectively.\n\nHence the picture that now emerges is that AGN evolution must be caused by a decline\nin overall accretion rate onto black holes, but with a characteristic timescale that\nindicates a cosmological influence on a population of objects whose active lives are\nshort. We can measure the characteristic timescale for the QSO population change \nfrom the optical luminosity function. If the characteristic break luminosity,\n$L^{\\star}$ varies as $(1+z)^{\\gamma}$, then the characteristic timescale \nmay be expressed as \n$$\n\\tau \\equiv \\frac{L^{\\star}}{|dL^{\\star}\/dt|} = \\frac{1+z}{\\gamma dz\/dt} = \\frac{1}{\\gamma H(z)}\n$$\nwhere for the optical LF, $\\gamma \\simeq 3$. In the next section we see whether\na cosmological origin for this timescale can be identified.\n\n\\section{The dark halo accretion rate}\nThe growth of dark halos can be calculated within the framework of hierarchical\ngrowth using the extended Press-Schechter \\citep{lc93,lc94}\napproach \\citep{miller06}. The timescale for growth of halos of mass $M_{\\rm H}$ is \n$$\n\\tau_{\\rm H} \\equiv \\frac{M_{\\rm H}}{\\left\\langle dM_{\\rm H}\/dt \\right\\rangle} = \\frac{1}{f(M_{\\rm H}) \\left | d\\delta_c\/dt \\right |},\n$$\nwhere $f(M_{\\rm H})$ is a slowly-varying function of mass, of order unity, \nand where $\\delta_c$ is the usual\nPress-Schechter redshift-dependent critical overdensity for collapse (see \\citealt{miller06}\nfor full details). For an Einstein-de-Sitter universe this has a simple form,\n$$\n\\tau_{\\rm H, EdS} = \\frac{1}{1.68 f(M_{\\rm H}) H(z) (1+z)},\n$$\nwhich has a similar value at $z \\sim 1$ to that observed in the QSO luminosity\nfunction. The timescale is insensitive to the choice of cosmology. It is tempting \nthen to suppose that the timescales for black hole growth and dark halo (and hence galaxy)\ngrowth are comparable and perhaps related.\n\n\\section{Coeval evolution of black holes and their hosts}\nOne of the striking features of the black hole\/bulge $M-\\sigma$ relation in\nthe local universe is its remarkably small scatter, with an intrinsic dispersion\nno larger than $\\sim 0.3$\\,dex \\citep{tremaine02}. It seems likely that feedback\nbetween black hole and galaxy growth is required to produce such a tight relation\n\\citep[e.g.][]{king05}, but whatever the mechanism it seems most natural to suppose\nthat the black hole has acquired its mass at the same time as the galaxy has acquired\nits mass: i.e., that black holes and galaxies have grown coevally. \\citet{miller06}\nhave investigated the hypothesis that the timescales for black hole and galaxy\ngrowth are the same and show the same cosmological dependence. The hypothesis\nis that, {\\em averaged across all galaxies at any given cosmological epoch}, \n\\begin{equation}\n\\tau_{\\rm BH}(z) \\equiv \\frac{M_{\\rm BH}}{\\left\\langle dM_{\\rm BH}\/dt \\right\\rangle}\n= \\frac{M_{\\rm H}}{\\left\\langle dM_{\\rm H}\/dt \\right\\rangle}\n\\equiv \\tau_{\\rm H}(z),\n\\label{eqn:pce}\n\\end{equation}\nwhere $M_{\\rm BH}$ and $M_{\\rm H}$ are the mass of a black hole and its dark\nhalo respectively. We call this\n``Pure Coeval Evolution'' (PCE). The hypothesis does not require\nthere to be a direct causal link between the two, but it may be that feedback processes\ndrive the system towards this behaviour.\n\nThere are two key predictions of this hypothesis. First, the mean Eddington ratio \nof black hole accretion, averaged over all galaxies, should\nrise dramatically to high redshifts, as shown in Babi\\'{c} et al. (these proceedings).\nNote that at any given epoch there is expected to be a wide range of individual\nEddington ratios: galaxies with the highest values, close to unity, would be those\nrecognised as AGN or QSOs. If Eddington ratios do have an upper bound around unity,\nwe do not expect the Eddington ratio of the most luminous\nAGN to show much cosmological evolution \\citep[see][]{kollmeier06}. Less luminous\nAGN and normal galaxies may show evidence for such evolution, however\n\\citep{netzer06}. Averaged over all galaxies, not just AGN, \nthe mean Eddington ratio should increase at higher redshift, implying a greater\nprevalence of visibly accreting black holes at higher $z$.\n\n\\begin{figure}\n \\begin{center}\n \\resizebox{10cm}{!}{ \n \\rotatebox{270}{\n \\includegraphics{LMiller_fig1.ps}\n }}\n \\end{center}\n \\caption{\nThe AGN bolometric luminosity density deduced from the best-fit model of\n\\citet{ueda03}, integrating over the range \n$10^{40}10^{11.5}$\\,M$_{\\odot}$\n(dashed curve). \nBoth curves assume average radiative efficiency $\\langle\\epsilon\\rangle=0.04$. \n}\n\\label{fig1}\n\\end{figure}\n\nSecond, we can predict the expected bolometric output from accreting black holes,\nas follows. The integrated bolometric luminosity density $\\rho_{\\rm L}$\nproduced by AGN depends on the\nnumber density of black holes, on the mean accretion rate onto those, and on the\nmean radiative efficiency \n$\\left\\langle\\epsilon\\right\\rangle$. If we adopt the simplest assumption, that\nequation\\,\\ref{eqn:pce} applies on average to black holes of all masses,\nand if we approximate the function $f(M_{\\rm H})$ as being independent of mass \n\\citep[see][]{miller06} then we can write\n\\begin{eqnarray}\n\\nonumber\n\\left\\langle\\rho_{\\rm L}\\right\\rangle \n&\\simeq &\nc^2 \\rho_{\\rm BH} \n\\left\\langle \n\\frac{\\epsilon}{\\left(1-\\epsilon\\right)M_{\\rm BH}}\n\\frac{dM_{\\rm BH}}{dt} \n\\right\\rangle\\\\\n&\\simeq &\nc^2 \\rho_{\\rm BH} \n\\left\\langle \n\\frac{\\epsilon f\\left(M_{\\rm H}\\right)}{\\left(1-\\epsilon\\right)} \n\\right\\rangle\n\\left |\n\\frac{d\\delta_{c}}{dt} \n\\right |,\n\\end{eqnarray}\nwhere $\\rho_{\\rm BH}$ is the cosmic black hole mass density. Here we adopt\nthe value for $\\rho_{\\rm BH}$ at $z=0$ estimated by \\citet{marconi04}.\nThe result of\nthis calculation is shown in Fig.\\,\\ref{fig1} for two different assumptions.\nThe dot-dashed curve shows the expected evolution if $\\rho_{\\rm BH}$ does not change\nwith redshift: the dashed curve shows the expected evolution if $\\rho_{\\rm BH}$\ntracks the mass density in massive halos, $M_{\\rm H} > 10^{11.5}$\\,M$_{\\odot}$,\ncalculated from the EPS mass function. We see remarkably good agreement with\nthe evolution in bolometric luminosity density, derived from hard X-ray surveys\n(solid curve and points with error bars),\nif the mean radiative efficiency has a constant value \n$\\left\\langle\\epsilon\\right\\rangle \\simeq 0.04$.\nThis is a promising result: there is no other free normalisation of the predicted\nluminosity density, and the value of \n$\\left\\langle\\epsilon\\right\\rangle$ required matches very well\nboth theoretical expectation ($\\epsilon \\la 0.06$ for accretion onto a Schwarzschild\nblack hole) and determinations based on a comparison of the X-ray background\nand the local relic black hole mass density \\citep[e.g.][]{marconi04}.\n\nWe can however also see that the prediction is too high at $z<0.5$, overproducing\nthe bolometric luminosity density by a factor 2 at $z=0$. This implies that\nblack hole growth and halo\/galaxy growth have decoupled by the present day.\nThis result may well be related to the phenomenon of downsizing noted\nat $z=0$ by \\citet{heckman04}. In that work it appears that the mean Eddington\nratio becomes a function of mass, a result that also appears to be confirmed\nby \\citet{netzer06}. Possible physical causes of decoupling could\nbe that either it is an apparent effect caused by a decrease in radiative\nefficiency at low Eddington ratios \\citep[e.g.][]{narayan95,beckert02}, or that \nblack hole growth really does slow down as a result of the changing environments\nof galaxies towards $z=0$, with an increasing prevalence of galaxies forming\ninto groups and clusters.\n\n\\section{Conclusions and further thoughts}\nIt seems that the hypothesis of Pure Coeval Evolution of black holes\nand dark halos reproduces rather well the observed bolometric luminosity\ndensity produced by AGN at $z<3$, implying that, broadly-speaking, black\nholes and their hosts grow together. The dramatic decrease in AGN activity\ntowards $z=0$ is thus seen to be a result of the slow-down in growth of galaxies\nthemselves. This parallel evolution does not necessarily imply a direct causal\nlink between them, but it is attractive to invoke such a causality, especially\nas this would also help explain the very tight $M-\\sigma$ relation between black\nholes and their hosts in the local universe \\citep{tremaine02}.\n\nOne of the main successes of this hypothesis is that it reproduces the observed\nbolometric luminosity density assuming a very reasonable value for the mean radiative\nefficiency, $\\langle\\epsilon\\rangle \\simeq 0.04$ - no previous model of AGN\nevolution has been able to predict the absolute value of the luminosity density.\nIt is worth noting that Pure Coeval Evolution produces a better match to the\nluminosity density evolution than current generations of semi-analytic models\n\\citep[e.g.][]{croton06}.\n\nA factor-two decoupling between black holes and galaxy\/halo growth does seem to occur\nat low redshift, more measurements of Eddington ratio as a function of black hole\nmass, environment and redshift should enable us to distinguish alternative explanations\nfor this effect.\n\nMore observational and theoretical work is needed to further test the scenario\nand to understand the physics of the coevolution process - it is not obvious\nwhy the growth of dark-matter-dominated halos with total mass $\\sim 10^{12}$\\,M$_{\\odot}$ should have\nsuch a direct influence on the growth of central black holes with mass\n$10^{6-8}$\\,M$_{\\odot}$. \nMeanwhile, other work in progress (Babi\\'{c} et al., these proceedings\nand in preparation) shows how both the AGN luminosity function and the X-ray\nbackground can be successfully reproduced within this framework, and it would be very interesting\nto extend this to higher redshifts.\n\nA consequence of the predicted high Eddington ratios at high redshifts is that we expect\na large fraction of galaxies to have nuclear outflows from their growing black holes\n\\citep[e.g.][]{king03,kingpounds03} - searching for evidence of this would be important\nboth for testing the hypothesis and for helping to understand the importance of feedback \nwhen galaxies formed.\n\n\\acknowledgements\nI thank Y.\\,Ueda for providing the data points used for the calculation shown\nin Fig.\\,\\ref{fig1}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCurrent fluctuations in non-equilibrium open systems has been a major area of research in statistical physics over the last\nfew years \\cite{bodineau-derrida-prl2004, derrida-gers, derrida-gers-sep, bertini-jstatphys2006,\nbertini-landim-prl2005, prahofer-spohn-2002, prolhac-mallick-08,derrida-lecomte-wijland-pre2008,KM12}. The probability distribution of\nthe current of particles across a given point in space typically admits a large deviation form and the corresponding large deviation\nfunction (rate function) is often interpreted as an analogue of a free energy in non-equilibrium systems. For example, it was shown in \\cite{bodineau-derrida-prl2004} that for a large one dimensional system in contact with reservoirs at unequal densities at the two ends, there exists an additivity principle obeyed by the large deviation function, much like the free energy in equilibrium systems. This additivity principle has been exploited further to compute\ncumulants of the current distribution \\cite{prahofer-spohn-2002,prolhac-mallick-08,derrida-gers,derrida-gers-sep,bodineau-derrida-prl2004,bertini-jstatphys2006,bertini-landim-prl2005,derrida-lecomte-wijland-pre2008}. Several theoretical tools have been developed to study current fluctuations, \nnotable among them is the Macroscopic Fluctuation Theory~\\cite{bertini-jstatphys2006, derrida-gers, bodineau-derrida-prl2004,KM12} and the Bethe Ansatz~\\cite{derrida-gers-sep}. Most of these studies have focused on driven diffusive systems, both interacting (as in the simple symmetric exclusion process) and non-interacting, typically in a one-dimensional setting. \n\n\nAnother inherently out-of-equilibrium system, much studied recently, is the so-called {\\it active} run-and-tumble particle (RTP)~\\cite{Berg_book,TC_2008,Martens2012,patterson-gopinath}, a new incarnation of the persistent random walk \\cite{Sta87,Weiss}.\nSuch motion has been observed in certain bacteria such as E. Coli where the bacterium moves in straight runs, undergoes tumbling at the end of a run and chooses randomly a new direction for the next run~\\cite{Berg_book,TC_2008,Martens2012,patterson-gopinath,Sta87,Weiss}. These motions are inherently out-of-equilibrium \nsince they consume energy directly from the environment and self-propel themselves without any external force. There has been an enormous\namount of work concerning the collective properties of an assembly of such RTPs \\cite{Fodor17,bechinger_active_2016,cates_motility-induced_2015,Cates_Nature,SEB_16}. Even at the single-particle level, RTP displays interesting behaviour and several single-particle observables have been studied recently. These include the position distribution for a free RTP \\cite{Sta87,Martens2012,gradenigo}, non-Boltzmann \nstationary states for an RTP in a confining potential \\cite{Cates_Nature, ad-sm-gh,Mallmin_18,Sevilla_19,HP95}, effects of disordered potentials \\cite{Kardar-disorder}, first-passage properties \\cite{km-ad-sm,maes,pierre-satya-greg,ADP_2014, A2015, MLW_86}, the distribution of the time at which an RTP reaches its maximum displacement \\cite{anupam_rtp} and RTP subjected to stochastic resetting \\cite{me-sm-reset, review_resetting}. \n\n\n\n\n\n\nHowever, as far as we are aware, current fluctuations, even in a system of noninteracting RTPs have not been systematically studied. The purpose of this paper is to study the current fluctuations in the simplest setup \nwhere RTPs are noninteracting and initially confined on one-half of the real line (i.e. step-function initial condition). Such a setup was used before\nby Derrida and Gerschenfeld for noninteracting diffusive particles \\cite{derrida-gers} and they were able to compute the large-deviation form of the current\nor flux $Q_t$ of particles through the origin {\\it up to} time $t$. In this paper, we use exactly the same setup, but for a more general class of noninteracting particles, which includes both diffusive as well as run-and-tumble particles, and compute analytically the flux distribution up to time $t$. \n\nThus the main observable of our interest is the flux $Q_t$ defined as the number of particles that crossed the origin (either from left or right) {\\it up to} time $t$, starting from the step initial condition where the particles are uniformly distributed over only the left side of the origin. Let us denote its probability distribution by $P(Q,t) = {\\rm Prob.}(Q_t=Q)$. Clearly $Q_t$ is a history dependent quantity, since it involves counting of all the crossings of the origin up to time $t$. Our exact results rely on a simple but crucial observation, valid for this special step initial condition: each particle, starting from the left side of the origin, that crosses the origin an even number of times up to time $t$ does not contribute to the flux $Q_t$. But if it crosses the origin an odd number of times, it contributes unity to the flux $Q_t$. Hence, the flux $Q_t$ is exactly equal to the number $N^+_t$ of particles present on the right side of \nthe origin {\\it at} time $t$, i.e. $Q_t = N^+_t$. Thus the history dependent observable $Q_t$ gets related, via this observation, to $N^+_t$ which is an instantaneous observable at time $t$. As we will see later, it is much easier to compute the probability distribution $P(N^+,t)={\\rm Prob.}(N^+_t=N^+)$, rather than the distribution of $Q_t$ directly. Hence, knowing the distribution of $N^+_t$, we can compute the flux distribution from $P(Q,t) ={\\rm Prob.}(N^+_t=Q)$. Note that this equivalence holds for arbitrary dynamics of the particles, for example it holds both for diffusive as well as RTP dynamics of the particles. \n\n\nBased on this connection $Q_t = N^+_t$, we can apply our results for the flux distribution to another interesting problem. Consider for instance an ideal gas of noninteracting particles in \na box. Imagine that the box is divided into two halves by a removable wall. Initially, all the particles are on the left half of the box and at $t=0$ we \nlift the wall and let the particles explore the full box freely. At time $t$, we take a snapshot of the system and observe the locations of \nthe particles. Of course, on an average, one expects that, at long times, the particles will be uniformly distributed throughout the box. We can \nask: what is the probability that, at time $t$, all the particles are again back to the left half of the box? Clearly this is an extremely rare event but\nwhat is the probability of this event? How does it decay with time? But note that this is exactly the probability ${\\rm Prob.}(N^+_t=0)=P(Q_t=0,t)$. Hence, our computation of the flux distribution with step initial condition gives access to the probability of this rare event (corresponding to all the particles coming back to the left half of the box at time $t$), in a one-dimensional setting. \n \n\nThe rest of the paper is organised as follows. In section~\\ref{sec:model} we discuss the model and present a summary of our main results. Then in Sec.~\\ref{general-setting-sec} we introduce the general setting and show how the single-particle Green's function plays the central role in the analysis. Next in Secs.~\\ref{annealed-sec} and \\ref{quen-sec}, we calculate the annealed and quenched averages, respectively, of the probability distribution of the flux for both diffusive and run-and-tumble particles.\nThen in Sec.~\\ref{numerics} we give the numerical verifications of our results and\nfinally in Sec.~\\ref{conclu} we summarize and conclude. \n \n\\section{The model and the main results}\\label{sec:model}\n\nWe consider a set of $N$ noninteracting particles initially distributed uniformly with a density $\\rho$ on the negative real axis, as in Fig.~\\ref{model}. Without loss of generality, we label the particles $i=1,2, \\dots, N$ with $x_i(t)$ denoting the position of the $i$-th particle at time $t$. Each $x_i(t)$ evolves independently by a stochastic (or deterministic) evolution rule (the same law of evolution for each particle). For example, each particle can undergo independent Brownian motion. Alternatively, each particle can undergo independent RTP dynamics in one-dimension. \nThis RTP dynamics for a single particle is defined as follows. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm]{dynamics.eps}\n\\caption{Schematic representation of an initial realization with all particles on the left of an arbitrary origin ($x=0$) on an infinite line $L \\rightarrow \\infty$. After time $t$ each particle undergoes some displacement depending on\nthe dynamics. The quantity of interest in our case is the number of particles on the right of the origin at time $t$.\n} \\label{model}\n\\end{center}\n\\end{figure}\n\n\\vspace{0.4cm}\n\\noindent {\\bf RTP dynamics}. The position of a single RTP\n$x(t)$ evolves via the Langevin equation\n\\begin{equation}\n\\frac{dx}{dt}= v_0\\, \\sigma(t)\\, \n\\label{RTP_evol.1}\n\\end{equation}\nwhere $v_0$ is the intrinsic speed during a run and $\\sigma(t)=\\pm 1$ is a dichotomous \ntelegraphic noise that flips from\none state to another with a constant rate $\\gamma$. The effective noise $\\xi(t)=v_0\\, \\sigma(t)$\nis coloured which is simply seen by computing its autocorrelation function\n\\begin{equation}\n\\langle \\xi(t) \\xi(t')\\rangle= v_0^2\\, e^{-2\\, \\gamma\\, |t-t'|}\\, .\n\\label{autocorr.1}\n\\end{equation}\nThe time scale $\\gamma^{-1}$ is the `persistence' time of a run that encodes the memory\nof the noise. In the limit $\\gamma\\to \\infty$, \n$v_0\\to \\infty$ but keeping the ratio $D_{\\rm eff}= v_0^2\/{2\\gamma}$ fixed, the noise $\\xi(t)$ reduces to\na white noise since\n\\begin{equation}\n\\langle \\xi(t) \\xi(t')\\rangle= \\frac{v_0^2}{\\gamma}\\, \\left[\\gamma\\, e^{-2\\gamma|t-t'|}\\right]\n\\to 2D_{\\rm eff}\\, \\delta(t-t')\\, .\n\\label{autocorr.2}\n\\end{equation}\nThus in this so called `diffusive limit', the persistent random walker $x(t)$ reduces to an\nordinary Brownian motion. For finite $\\gamma$ (i.e., persistence time scale of memory),\nthe RTP will be referred to as an ``active'' particle. In the diffusive limit, the active particle dynamics reduces\nto an ordinary Brownian motion, which we refer to as a ``passive'' motion.\n\nGiven the stochastic dynamics of the individual particles, starting from the step initial condition, our main object of\ninterest is the flux $Q_t$ of particles through the origin up to time $t$. If a trajectory crosses the origin from left to right,\nthis will contribute a $+1$ to the net current while if it crosses from right to left, its contribution is $-1$. The flux $Q_t$ is thus\nthe net contribution to the current up to time $t$. Let us denote by $P(Q,t,\\{x_i\\})$ the probability distribution ${\\rm Prob.}(Q_t=Q)$ \nfor a given initial condition where $x_i$'s denote the initial positions of the particles at time $t=0$. \nFollowing Derrida and Gerschenfeld, \nthe effect of the initial condition on the distribution can be studied in\ntwo alternative ways, in analogy with the disordered systems where the realisation of a disorder plays an analogous\nrole as the initial condition in our problem. \\textcolor{black}{It was indeed argued in \\cite{derrida-gers} that one has to distinguish between two different ways of \naveraging over the initial conditions: (i) the annealed average, where the probability distribution of the flux is averaged over all the realizations of the initial condition \nand (ii) the quenched average where the probability distribution is computed for the {\\it typical} initial configurations.} Instead of considering the distribution $P(Q,t,\\{x_i\\})$ directly, it turns out to be convenient to consider its \ngenerating function $\\langle e^{-p Q}\\rangle_{\\{x_i\\}}$, where the angular brackets $\\langle \\cdots \\rangle_{\\{x_i\\}}$ denote an average over the history, but with fixed initial condition $x_i$. The annealed and quenched averages are now defined as follows:\n\\begin{eqnarray}\n&&\\sum_{Q=0}^\\infty e^{-p Q} \\, P_{\\rm an}(Q,t) \\, = \\overline{\\langle e^{-p Q}\\rangle_{\\{x_i\\}}} \\;, \\label{def_ann} \\\\\n&&\\sum_{Q=0}^\\infty e^{-p Q} \\, P_{\\rm qu}(Q,t) \\, = \\exp{\\left[ \\overline{\\ln \\langle e^{-p Q}\\rangle_{\\{x_i\\}}} \\right]} \\;, \\label{def_quen}\n\\end{eqnarray}\nwhere $\\overline{\\cdots}$ denotes an average over the initial conditions. Note that in this problem $Q_t$ is always an integer. As mentioned in the introduction, for the step initial condition, we can compute both $P_{\\rm an}(Q,t)$ and $P_{\\rm qu}(Q,t)$ \\textcolor{black}{(see Fig.~\\ref{Pqu-compare} for a plot of these probability distributions)} for arbitrary dynamics of the particles by using the identity $Q_t = N^+_t$, where $N^+_t$ is the number of particles on the right side of the origin at time $t$. Indeed, the only quantity that enters the computation for independent particles is the single particle Green's function $G(x,x_0,t)$ denoting the probability density of finding the particle at position $x$ at time $t$, starting from $x_0$ at $t=0$. Let us first define a central object, that will appear in all our formulas \n\\begin{eqnarray}\\label{UzT}\nU(z,t) = \\int_0^\\infty G(x,-z,t) \\, dx \\quad, \\quad z\\geq 0 \\;,\n\\end{eqnarray}\nobtained by integrating the Green's function over the final position, with the initial position fixed at $x_0=-z \\leq 0$. If we can compute $U(z,t)$ for a given dynamics, we can express $P_{\\rm an}(Q,t)$ and $P_{\\rm qu}(Q,t)$ in terms of this central function $U(z,t)$. Our main results can now be summarised as follows. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.45\\hsize}\n\\includegraphics[width=\\hsize]{Pan-compare-gm-1-t40.eps}\n\\end{minipage}\n\\begin{minipage}{0.45\\hsize}\n\\includegraphics[width=\\hsize]{Pqu-diffusion-rtp-Deff0-5-t40-gmt40.eps}\n\\end{minipage}\n\\end{center}\n\\caption{{(a) Annealed case: Semi-log plot of $P_{\\rm an}(Q,t)$ vs $Q$ for RTPs (red solid line) using Eqs.~(\\ref{ac-mut}), (\\ref{Poisson_largedev}) and (\\ref{psi_an}) compared to the diffusive case (black dashed curve) given by Eq. (\\ref{Poisson_largedev}) with $\\mu(t)=\\rho\\sqrt{\\frac{Dt}{\\pi}}$ for $\\rho = 1$ and $t=40$. For the RTP, we set $v_0=\\gamma=1$, corresponding to an effective diffusion constant $D_{\\rm eff}=v_0^2\/(2\\gamma)= 1\/2$, while we use, accordingly, $D=1\/2$ for diffusion. For such a (large) time where the scaling form (\\ref{Poisson_largedev}) is expected to hold, the RTP and the diffusive cases are almost indistinguishable. Finally, the blue dashed curve corresponds to the typical Gaussian approximation with mean and variance $\\mu(t)$ given in (\\ref{ac-mut}). (b)~ Quenched case: Semi-log plot of $P_{\\rm qu}(Q,t)$ vs $Q$ for the same set of parameters for RTPs (red solid line), obtained from Eqs.~(\\ref{Uzt-exact}) and~(\\ref{def_tildeI}), compared to the diffusive case (black dashed line) obtained from Eqs.~(\\ref{phi-1eq}) and (\\ref{fq-bas}). The blue dotted line corresponds to the Gaussian approximation with mean $\\mu(t)$ given in (\\ref{ac-mut}) and variance given in (\\ref{var_qu}). While the active and passive cases remain\nindistinguishable at small $Q$, the quenched distribution, at variance with the annealed one, carries a clear signature of activity \nat large $Q$. For example, in the quenched case, the maximum possible flux for the RTP is $Q= \\rho v_0 t = 40$ (large blue dot). \n}}\\label{Pqu-compare}\n\\end{figure} \n\n\n\\vspace*{0.4cm}\n\\noindent\n{\\bf Annealed case}: In this case, we show that $P_{\\rm an}(Q,t)$ is always a Poisson distribution \n\\begin{equation}\\label{Pan-model-def}\nP_{\\rm an}(Q=n,t) = e^{-\\mu(t)} \\frac{\\mu(t)^n}{n!} \\;, \\; n=0,1,2,\\cdots \\;,\n\\end{equation}\nwhere \n\\begin{equation}\\label{muan-mod-def}\n\\mu(t) = \\rho \\, \\int_0^{\\infty} dz ~U(z,t) \\;.\n\\end{equation} \nThe mean and the variance of $Q_t$ are both given by $\\mu(t)$, which \ncan be explicitly evaluated for different types of particle motion. For example, for a Brownian motion with diffusion constant $D$, using $G(x,x_0,t) = e^{-(x-x_0)^2\/(4 Dt)}\/\\sqrt{4 \\pi D t}$ in Eq. (\\ref{UzT}) we get \n\\begin{eqnarray} \\label{UzT_BM}\nU(z,t) = \\frac{1}{2} {\\rm erfc}\\left(\\frac{z}{\\sqrt{4 D t}} \\right) \\quad, \\quad {\\rm and} \\quad \\mu(t) = \\rho \\sqrt{Dt\/\\pi} \\;,\n\\end{eqnarray}\nwhere ${\\rm erfc}(z) = (2\/\\sqrt{\\pi}) \\int_z^\\infty e^{-u^2} du$. Our result for $P_{\\rm an}(Q=n,t)$ for the diffusive case is consistent with the result of Derrida and Gershenfeld \\cite{derrida-gers} obtained by a different method. \n\n\n\nIn the case of the RTP dynamics, we find explicitly that at all $t$,\n\\begin{equation}\\label{ac-mut}\n\\mu(t) = \\frac{1}{2}\\rho \\, v_0 \\, t\\, e^{-\\gamma t} \\,[I_0(\\gamma t) + I_1(\\gamma t)],\n\\end{equation}\nwhere $I_0(z)$ and $I_1(z)$ are modified Bessel functions of the first kind. Its asymptotic behaviours are given by\n\\begin{eqnarray} \\label{asympt_mu}\n\\mu(t) \\approx\n\\begin{cases}\n&\\dfrac{\\rho \\,v_0}{2}\\, t \\;, \\;\\;\\; {\\rm as}\\;\\;\\; t \\to 0 \\;, \\\\\n&\\\\\n&\\rho \\sqrt{\\dfrac{D_{\\rm eff}\\,t}{\\pi}} \\;, \\;{\\rm as}\\; t \\to \\infty \\;,\n\\end{cases}\n\\end{eqnarray}\nwhere $D_{\\rm eff} = v_0^2\/(2 \\gamma)$. Thus at late times, the RTP behaves like a diffusive particle with an effective diffusion\nconstant $D_{\\rm eff}$. \n\nNote that the Poisson distribution in Eq. (\\ref{Pan-model-def}) in the limit $Q \\to \\infty$, $\\mu(t) \\to \\infty$, keeping\nthe ratio $Q\/\\mu(t)$ fixed, can be written in a large deviation form (using simply Stirling's formula)\n\\begin{eqnarray}\\label{Poisson_largedev}\nP_{\\rm an}(Q,t) \\sim \\exp{\\left[- \\mu(t) \\, \\Psi_{\\rm an} \\left( \\frac{Q}{\\mu(t)}\\right) \\right]} \\;,\n\\end{eqnarray} \nwhere the rate function $\\Psi_{\\rm an}(q)$ is universal, i.e., independent of the particle dynamics, and is given by\n\\begin{eqnarray} \\label{psi_an}\n\\Psi_{\\rm an}(q) = q \\,\\ln q - q + 1 \\;, \\; q \\geq 0 \\;.\n\\end{eqnarray}\nIt has the asymptotic behaviours\n\\begin{eqnarray} \\label{asympt_psi_an}\n\\Psi_{\\rm an}(q) \\approx\n\\begin{cases}\n& 1 \\;, \\;\\; {\\rm as} \\;\\; q \\to 0 \\\\\n& \\dfrac{1}{2} (q-1)^2 \\;, \\;\\; {\\rm as} \\;\\; q \\to 1 \\\\\n& q \\, \\ln q \\;, \\;\\; {\\rm as} \\;\\; q \\to \\infty \\;. \n\\end{cases}\n\\end{eqnarray}\nThe quadratic behavior near the minimum at $q=1$ indicates typical Gaussian fluctuations for $Q$, with mean and variance both equal to $\\mu(t)$. Note that the dependence on the particle dynamics in Eq. (\\ref{Poisson_largedev}) enters only through the parameter $\\mu(t)$ but the function $\\Psi_{\\rm an}(q)$ is universal. \n\nWe also note that, from our general result in Eq. (\\ref{Pan-model-def}), it follows that \n\\begin{eqnarray}\\label{Pq0}\n{\\rm Prob.}(N_t^+ =0) \\Big|_{\\rm an} = P_{\\rm an}(Q=0,t) = e^{-\\mu(t)} \\;.\n\\end{eqnarray} \nThis result is valid for all $t$ and gives the probability of the rare event that all the particles are back on the left side of the origin at time $t$, as discussed in the introduction. {In Fig.~\\ref{Pan-qu-compare} (a) we show a plot of $P_{\\rm an}(Q=0,t)$ as a function of time, both for RTP and for diffusive particles.}\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{P0-an-log.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{P0-qu-log.eps}\n\\end{minipage}\n\\end{center}\n\\caption{{(a) Semi-log plot of $P_{\\rm an}(Q=0,t) = e^{-\\mu(t)}$ vs $t$ for an RTP (red solid line) with $\\mu(t)$ given in Eq.~(\\ref{Pq0})\n compared to the diffusive case (black dashed line) corresponding to $\\mu(t)=\\rho \\sqrt{\\frac{D t}{\\pi}}$, for a density $\\rho=1$ in both cases. For the RTP, we set $v_0=\\gamma=1$, corresponding to an effective diffusion constant $D_{\\rm eff}=v_0^2\/(2\\gamma)= 1\/2$, while we set, accordingly, $D=1\/2$ in the case of diffusion. (b) Semi-log plot of $P_{\\rm qu}(Q=0,t)$ vs $t$ as given in Eq.~(\\ref{P_neq0_qu}) both for RTPs (red solid line) using the result for $U(z,t)$ in (\\ref{Uzt-exact}) and for Brownian particles (black dashed line) for which $U(z,t)$ is given in~(\\ref{UzT_BM}). In both annealed and quenched cases, the zero-net flux probabilities for active and passive systems differ at short times but do coincide in the large time limit.}}\\label{Pan-qu-compare}\n\\end{figure}\n\n\n\\vspace*{0.4cm}\n\\noindent\n{\\bf Quenched case}: In this case, the generating function in Eq. (\\ref{def_quen}), for arbitrary single particle dynamics, can again be expressed in terms \nof the central function $U(z,t)$ (\\ref{UzT}) as follows\n\\begin{eqnarray}\\label{P_qu_gen}\n\\sum_{Q=0}^\\infty P_{\\rm qu}(Q,t) e^{-p Q} \\, = \\exp{\\left[\\rho \\int_0^\\infty dz \\, \\ln{\\left[1-(1-e^{-p})U(z,t)\\right]} \\right]} \\;.\n\\end{eqnarray} \nThe quenched cumulants of $Q$ can then be extracted and expressed in terms of $U(z,t)$. For instance, \nthe quenched mean and the variance of $Q$ are given by \n\\begin{eqnarray}\n&&\\langle Q \\rangle_{\\rm qu} = \\rho \\, \\int_0^\\infty U(z,t) \\, dz \\;, \\label{mean_qu} \\\\\n&&\\sigma_{\\rm qu}^2 = \\langle Q^2 \\rangle_{\\rm qu} - \\langle Q \\rangle^2_{\\rm qu} = \\rho \\int_0^\\infty U(z,t)(1-U(z,t))\\, dz \\;. \\label{var_qu}\n\\end{eqnarray}\nIn addition, expanding the right hand side (rhs) in powers of $e^{-p}$ and matching with the left hand side (lhs), one can in principle obtain $P_{\\rm qu}(Q,t)$ for any integer $Q$ as a functional of $U(z,t)$. For example, \n\\begin{eqnarray}\\label{P_neq0_qu}\n{\\rm Prob.}(N_t^+ = 0) \\Big |_{\\rm qu} = P_{\\rm qu}(Q=0,t) = \\exp{\\left[\\rho \\int_0^\\infty \\, \\ln{\\left[1-U(z,t)\\right]} \\, dz\\right]} \\;,\n\\end{eqnarray}\nwhich is valid for all times $t \\geq 0$. However, the formula gets more complicated for higher values of $Q$. {In Fig.~\\ref{Pan-qu-compare} (b) we show a plot of $P_{\\rm qu}(Q=0,t)$ as a function of time, both for RTP and for diffusive particles.}\n\n\nFor the full quenched distribution, we first consider the diffusive motion of the particles. In this case, we obtain, in the scaling limit $Q \\to \\infty$, $t \\to \\infty$ keeping the ratio $Q\/\\sqrt{t}$ fixed, the same large deviation form as Derrida and Gerschenfeld~\\cite{derrida-gers},\n\\begin{equation}\\label{P-F-diff-largeQ}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\sqrt{D t} ~ \\Psi_{\\rm diff}\\left(\\frac{Q}{\\rho \\sqrt{D t}}\\right)\\right] \\;,\n\\end{equation}\nwhere the rate function $\\Psi_{\\rm diff}(q)$ has the following precise asymptotics\n\\begin{eqnarray} \\label{psi_diff_asympt}\n\\Psi_{\\rm diff}(q) \\approx\n\\begin{cases}\n&\\overline{\\alpha} - q + q \\ln(q\/\\overline{\\beta}) \\;, \\; {\\rm as} \\; q \\to 0 \\\\\n& \\\\\n& \\sqrt{\\dfrac{\\pi}{2}} \\left(q-\\dfrac{1}{\\sqrt{\\pi}} \\right)^2 \\;, \\; {\\rm as} \\; q \\to 1\/\\sqrt{\\pi} \\\\\n& \\\\\n& \\frac{1}{12} q^3 \\;, \\; {\\rm as} \\; q \\to \\infty \\;,\n\\end{cases} \n\\end{eqnarray}\nwith the two constants $\\overline{\\alpha}$ and $\\overline{\\beta}$ given explicitly by\n\\begin{eqnarray} \n&&\\overline{\\alpha} = -2 \\int_0^\\infty dz \\, \\ln\\left(1 - \\frac{1}{2} \\, {\\rm erfc}(z) \\right) = 0.675336 \\ldots \\label{C} \\\\\n&&{\\overline{\\beta}} = \\int_0^\\infty dz \\, \\frac{{\\rm erfc}(z)}{1-\\frac{1}{2} {\\rm erfc}(z)} = 0.828581 \\ldots \\label{A} \\;.\n\\end{eqnarray}\nNote that the large $q$ behavior $\\Psi_{\\rm diff}(q) \\approx q^3\/12$ coincides with the result of Derrida and Gerschenfeld \\cite{derrida-gers} obtained\nby a different method. The small $q$ behavior was not investigated in Ref. \\cite{derrida-gers}. Taking the $q \\to 0$ limit in Eq. (\\ref{P-F-diff-largeQ}) and using the small $q$ behavior in the first line of Eq. (\\ref{psi_diff_asympt}) implies that for large $t$ \n$P_{\\rm qu}(Q=0,t) \\sim \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D\\,t}\\right]$ where the constant $\\overline{\\alpha}$ is given in Eq. (\\ref{C}). In fact, this result is valid not just at large time but at all times. Indeed, substituting $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$ in our general formula (\\ref{P_neq0_qu}), it follows that at all times $t \\geq 0$,\n\\begin{eqnarray} \\label{P_Q0}\nP_{\\rm qu}(Q=0,t)\\Big \\vert_{\\rm diff} = \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D\\,t}\\right] \\;.\n\\end{eqnarray}\nInterestingly, exactly the same rate function $\\Psi_{\\rm diff}(q)$ also appeared in the completely different context, namely as a large deviation function characterising the distribution of the number of eigenvalues (in a disk of radius $R$) of a complex Ginibre ensemble of $N\\times N$ Gaussian random matrices \\cite{castillo,bertrand}. \n\n\n\nWe then consider the quenched flux distribution for the RTP dynamics. First, we show that, for small $Q \\ll \\sqrt{t}$, the flux distribution decays as a stretched exponential at late times, e. g. $P_{\\rm qu}(Q=0,t) $ is given, for large $t$, by\n\\begin{eqnarray}\\label{P_Q0-1_RTP}\nP_{\\rm qu}(Q=0,t) \\Big \\vert_{\\rm RTP} \\sim \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D_{\\rm eff}\\,t}\\right] \\;,\n\\end{eqnarray}\nwhere $\\bar{\\alpha}$ is the same constant as in Eq. (\\ref{C}) and $D_{\\rm eff} = v_0^2\/(2 \\gamma)$. One of the main results of this analysis is to find a new scaling limit $Q \\to \\infty$, $t \\to \\infty$, keeping the ratio $Q\/(\\rho\\,v_0 t)$ fixed where the quenched distribution admits a large deviation form [quite different from the diffusive case in Eq. (\\ref{P-F-diff-largeQ})]\n\\begin{equation}\\label{ac-sig}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\, v_0 \\, \\gamma \\, t^2 \\, \\Psi_{\\rm RTP}\\left(\\frac{Q}{\\rho \\, v_0 \\,t}\\right)\\right],\n\\end{equation}\nwhere the rate function $\\Psi_{\\rm RTP}(q)$ is given explicitly by\n\\begin{equation}\\label{rtp-ldf-model}\n\\Psi_{\\rm RTP}(q)=q-\\frac{q}{2}\\sqrt{1-q^2}-{\\rm sin}^{-1}\\left[ \\sqrt{\\frac{1-\\sqrt{1-q^2}}{2}} \\right]\\;, \\quad 0 \\leq q \\leq 1 \\;.\n\\end{equation}\nThe rate function has the asymptotic behavior\n\\begin{eqnarray}\\label{asympt_RTP}\n\\Psi_{\\rm RTP}(q) \\approx\n\\begin{cases}\n& \\dfrac{q^3}{6} \\quad,\\quad \\quad \\quad \\; q\\rightarrow 0 \\,\\\\ \n& \\\\\n& 1-\\dfrac{\\pi}{4} \\quad,\\quad ~ ~ ~ q = 1 \\;.\n\\end{cases}\n\\end{eqnarray}\nOne consequence of our result is the prediction of the probability of the rare event that the flux $Q$ up to time\n$t$ achieves its maximum possible value, namely $Q = \\rho \\, v_0\\, t$ -- this corresponds to the case where all\nthe particles move ballistically to the right up to time $t$. We find that the probability of this rare event is given by\n\\begin{eqnarray} \\label{P_qu_max}\nP_{\\rm qu}(Q = \\rho \\, v_0 \\, t,t) \\approx \\exp\\left[-\\left( 1 - \\frac{\\pi}{4}\\right) \\rho \\, v_0 \\, \\gamma \\, t^2 \\right] \\;.\n\\end{eqnarray}\nSuch a faster than exponential decay for the probability of this rare event is a nontrivial prediction of our\ntheory. \n\n\n\n\n\n\n\n\n\n\n\n\\section{The general setting and the single-particle Green's function}\\label{general-setting-sec}\n\nWe start with a step initial condition where $N$ particles are initially located on the negative half line at positions $\\{x_1, x_2, \\cdots, x_N \\}$ where \nall $x_i <0$. As stated before, for this step initial condition, the flux $Q_t$ up to time $t$ is identical in law to the number of particles $N^+_t$ to the\nright of the origin at time $t$. Let us introduce an indicator function ${\\cal I}_i(t)$ \nsuch that ${\\cal I}_i(t)=1$ if the $i{\\rm th}$ particle\nis to the right of the origin at time $t$, else ${\\cal I}_i(t)=0$. Hence we have\n\\begin{equation}\\label{N+defn}\n N_t^+= \\sum_{i=1}^N {\\cal I}_i(t) \\;.\n\\end{equation}\nFor fixed $x_i$'s the flux distribution is then given by\n\\begin{eqnarray}\\label{basic-defn}\nP(Q,t,\\{ x_i\\}) = {\\rm Prob.}(N_t^+ = Q) = \\left \\langle \\delta \\left[Q-\\sum_{i=1}^N {\\cal I}_i(t)\\right]\\right \\rangle_{\\{x_i\\}} \\;, \n\\end{eqnarray}\nwhere the angular brackets $\\langle \\cdots \\rangle_{\\{x_i\\}}$ denote an average over the history, but with fixed initial condition $x_i$. Taking the Laplace transform on both sides of Eq. (\\ref{basic-defn}) gives\n\\begin{equation}\\label{lpbasic}\n \\sum_{Q=0}^{\\infty} e^{-pQ} P(Q,t,\\{x_i\\}) = \\langle e^{-pQ}\\rangle_{\\{x_i\\}} = \\left \\langle \\exp[{-p \\sum_{i=1}^N {\\cal I}_i(t)}]\\right \\rangle_{\\{x_i\\}} \\;.\n \\end{equation}\nSince the ${\\cal I}_i$ can only take the values $0$ or $1$, one has the identity $e^{-p {\\cal I}_i} = 1 -(1-e^{-p}){\\cal I}_i$. Inserting this identity in Eq. (\\ref{lpbasic}) and using the independence of the random variables ${\\cal I}_i$'s we get\n \\begin{equation}\\label{sq-Ii}\n \\langle e^{-pQ}\\rangle_{\\{x_i\\}} = \\prod_{i=1}^N\\left[1- (1-e^{-p})\\langle {\\cal I}_i(t) \\rangle_{\\{x_i\\}} \\right],\n \\end{equation}\nwhere the right hand side (r.h.s.) implicitly depends on the $x_i$'s. The average $\\langle {\\cal I}_i(t) \\rangle_{\\{x_i\\}}$ is just the probability that the $i{\\rm th}$ particle is to the right of the origin at time $t$, starting initially at $x_i$ and hence we have \n\\begin{equation}\\label{I-propagator}\n \\langle {\\cal I}_i(t) \\rangle_{\\{x_i\\}} = \\int_0^{\\infty} G(x,x_i,t) dx = U(-x_i,t) \\;, \\quad x_i < 0 \\;,\n \\end{equation}\nwhere $G(x,x_i,t)$ is the single-particle Green's function, i.e., the propagator for a particle to reach $x$ at time $t$, starting initially at $x_i<0$. Note that $U(z,t)$ is defined in Eq. (\\ref{UzT}) and corresponds to the probability that a particle is on the positive side of the origin at time $t$, starting initially at $-z<0$. Inserting Eq.~(\\ref{I-propagator}) into Eq.~(\\ref{sq-Ii}), one obtains\n\\begin{equation}\\label{propagator}\n \\langle e^{-pQ}\\rangle_{\\{x_i\\}}= \\prod_{i=1}^N\\left[1- (1-e^{-p})U(-x_i,t)\\right] \\;, \\quad x_i < 0 \\;, \\quad \\forall i = 1, \\cdots, N \\;.\n \\end{equation}\nThis Eq.~(\\ref{propagator}) is general, i.e., valid for\n{\\em any} set of non-interacting particles undergoing a common dynamics in one-dimension. The information about the dynamics\nis entirely encoded in the function $U(z,t)$. \n\nFor instance, for simple diffusion, the single-particle Green's function is given by \n\\begin{equation}\\label{diff-pro2}\nG(x,x_i,t) = \\frac{1}{\\sqrt{4\\pi D t}}{{\\rm exp}\\left[-\\frac{(x-x_i)^2}{4Dt}\\right]} \\;,\n\\end{equation}\nwhich gives $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$. For the RTP, on the other hand, the Green's function is known explicitly~\\cite{Weiss,me-sm-reset} \n\\begin{equation}\\label{ac-fullpd3}\nG(x,x_i,t) = \\frac{e^{-\\gamma t}}{2}\\left\\{ \\delta(x-x_i-v_0t) + \\delta(x-x_i+v_0t) +\\frac{\\gamma}{v_0}\\left[I_0(\\omega) + \\frac{\\gamma t I_1(\\omega)}{\\rho} \\right] \\Theta(v_0t-|x-x_i|) \\right\\},\n\\end{equation}\nwhere $\\omega$ is given by\n\\begin{eqnarray}\\label{def_omega}\n\\omega = \\frac{\\gamma}{v_0}\\sqrt{v_0^2t^2 -(x-x_i)^2} \\;.\n\\end{eqnarray}\nIn Eq. (\\ref{ac-fullpd3}), $\\Theta(z)$ is the Heaviside Theta function, and $I_0(\\omega)$ and $I_1(\\omega)$ are modified Bessel functions. Computing $U(z,t) = \\int_0^\\infty G(x,-z,t)\\,dx$ explicitly using Eq. (\\ref{ac-fullpd3}) is complicated. It is however much more useful, as we will see later, to work with the Laplace transform of $G(x,x_i,t)$ with respect to $t$, which has a much simpler expression, namely \n\\begin{equation}\\label{ac-Lap3}\n\\tilde{G}(x,x_i,s) = \\int_0^{\\infty} dt~ e^{-st} G(x,x_i,t) = \\frac{\\lambda(s)}{2s}e^{\\lambda(s)\\,|x-x_i|} \\;, \\quad \\lambda(s) = \\frac{\\sqrt{s(s+2\\gamma)}}{v_0} \\;.\n\\end{equation}\n\n\nThe relation in Eq. (\\ref{propagator}) is the central result of this section and we will analyse the annealed and the quenched cases separately in the next two sections. \n\n\n\n\\section{Flux distribution in the annealed case}\\label{annealed-sec}\n\nThe annealed distribution $P_{\\rm an}(Q,t)$ is defined in Eq. (\\ref{def_ann}) where the $\\overline{\\cdots}$ denotes an average over\nthe initial conditions. Performing this average in Eq. (\\ref{propagator}) gives\n\\begin{equation}\\label{ann-G-avg}\n\\langle\\overline{e^{-pQ}\\rangle_{\\{x_i\\}}}= \\prod_{i=1}^N\\left[1- (1-e^{-p})\\, \\overline{U(-x_i,t)}\\right] \\;,\n\\end{equation}\nwhere $U(-x_i,t)$ is defined in Eq. (\\ref{I-propagator}). To perform the average over the initial conditions with a fixed uniform density $\\rho$, \nwe assume that each of the $N$ particles is distributed independently and uniformly over a box $[-L,0]$ and then eventually take the limit \n$N \\to \\infty$, $L \\to \\infty$ keeping the density $\\rho = N\/L$ fixed. For this uniform measure, each $x_i$ is uniformly distributed in the box\n$[-L,0]$. Using the independence of the $x_i$'s we then get \n\\begin{equation}\\label{x_i-avg}\n\\langle\\overline{e^{-pQ}\\rangle_{\\{x_i\\}}} = \\prod_{i=1}^N \\left[1- (1-e^{-p}) \\int_{-L}^{0} U(-x_i,t) \\frac{dx_i}{L} \\right] = \\left[1- \\frac{1}{L}(1-e^{-p}) \\int_0^L U(z,t) dz \\right]^N \\;,\n\\end{equation}\nwhere, in the last equality, we made the change of variable $z=-x_i$. Taking now the limit $N \\to \\infty$, $L \\to \\infty$ keeping $\\rho = N\/L$ fixed gives\n\\begin{eqnarray}\\label{lap-pan}\n\\sum_{Q=0}^{\\infty} e^{-pQ} P_{\\rm an}(Q,t) = \\langle\\overline{e^{-pQ}\\rangle_{\\{x_i\\}}} = \\exp\\left[-\\mu(t) ~ (1-e^{-p})\\right] \\;, \\quad {\\rm where} \\;\\quad \\mu(t)= \\rho \\int_0^{\\infty} dz ~ U(z,t) \\;.\n\\end{eqnarray}\nBy expanding $\\exp\\left[-\\mu(t) ~ (1-e^{-p})\\right]$ in powers of $e^{-p}$ and comparing to the left hand side, we see that $Q$ can take only integer values $Q=n=0,1,2,\\cdots$ and the probability distribution is simply a Poisson distribution with mean $\\mu(t)$ as given in Eqs. (\\ref{Pan-model-def}) and (\\ref{muan-mod-def}). \n\nThis Poisson distribution, in the annealed case, is thus universal, i.e., holds for any dynamics. The details of the dynamics is encoded in the single\nparameter $\\mu(t)$ which can be computed explicitly for different types of dynamics. For example, for diffusing particles, using the explicit expression for the Brownian propagator, we get $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$ and, hence, $\\mu(t) = \\rho \\sqrt{Dt\/\\pi}$ as mentioned in Eq. (\\ref{UzT_BM}). In contrast, for the RTP dynamics, $\\mu(t)$ is nontrivial. As discussed earlier, computing $U(z,t)$ from the Green's function in Eq. (\\ref{ac-fullpd3}) is difficult. Consequently, calculating $\\mu(t) = \\rho\\int_0^\\infty U(z,t)\\,dz$ is also hard. However, it turns out that its Laplace transform is much easier to manipulate, due to the simple nature of the formula in Eq. (\\ref{ac-Lap3}). The Laplace transform of $\\mu(t)$ is given by\n\\begin{equation}\\label{mus-RTP}\n\\tilde{\\mu}(s) = \\int_{0}^{\\infty} dt~ e^{-st}~ \\mu(t) = \\rho \\int_0^\\infty dz ~\\tilde{U}(z,s) \\;, \\quad {\\rm with} \\quad \\tilde U(z,s) = \\int_{0}^{\\infty} dt~ e^{-st}~ U(z,t) \\;,\n\\end{equation}\nwhere we have used the relation $\\mu(t) = \\rho \\int_0^\\infty U(z,t)\\,dz$. The Laplace transform of $U(z,t)$ can be computed as follows\n\\begin{eqnarray}\\label{LaplaceU_1}\n\\tilde U(z,s) = \\int_{0}^{\\infty} dt~ e^{-st}~ U(z,t) = \\int_0^\\infty dt \\, e^{-st} \\, \\int_0^\\infty G(x,-z,t) \\, dx \\;.\n\\end{eqnarray}\nExchanging the integrals over $x$ and $t$, and using the relation in Eq. (\\ref{ac-Lap3}) and integrating over $x$ we get\n\\begin{eqnarray}\\label{LaplaceU_2}\n\\tilde U(z,s) = \\frac{e^{-\\lambda(s) z}}{2s} \\;, \\quad {\\rm where} \\quad \\lambda(s) = \\frac{\\sqrt{s(s+2\\gamma)}}{v_0} \\;.\n\\end{eqnarray}\nInserting this relation in Eq. (\\ref{mus-RTP}) and performing the integral over $z$, we get\n\\begin{equation}\\label{mu-ac-lap}\n\\tilde{\\mu}(s) = \\frac{1}{2 s \\, \\lambda(s)} = \\frac{v_0}{2s\\sqrt{s(s+2\\gamma)}} \\;.\n\\end{equation}\nThis Laplace transform can be explicitly inverted, yielding the result in Eq. (\\ref{ac-mut}). \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Flux distribution in the quenched case}\\label{quen-sec}\n\n\nAs stated in Section \\ref{sec:model}, the quenched flux distribution is defined as\n\\begin{equation}\\label{quenched-definition}\n\\sum_{Q=0}^{\\infty} P_{\\rm qu} (Q,t) e^{-pQ} = \\exp \\left[\\overline{{\\ln}\\left[\\langle e^{-pQ}\\rangle_{\\{x_i\\}} \\right]} \\right] ,\n\\end{equation}\nwhere $\\overline{\\cdots}$ once again represents an average over the initial positions $\\{x_i\\}$. Our starting point is again Eq. (\\ref{propagator}). Taking the logarithm on both sides of (\\ref{propagator}) gives\n\\begin{equation}\\label{Nlog-quenched}\n{\\ln}\\left[\\langle e^{-pQ}\\rangle_{\\{x_i\\}} \\right] = \\sum_{i=1}^N {\\ln}\\left[1-(1-e^{-p})U(-x_i,t) \\right].\n\\end{equation}\nWe now perform the average over the initial positions, as in the annealed case, i.e., choosing each $x_i$ independently and uniformly from the box $[-L,0]$ and finally taking the limit $N \\to \\infty$, $L \\to \\infty$ keeping $\\rho = N\/L$ fixed. This gives\n\\begin{equation}\\label{Nlog-qu-avg}\n\\overline{{\\rm log}\\left[\\langle e^{-pQ}\\rangle_{\\{x_i\\}} \\right]}= \\frac{N}{L}\\int_{-L}^0 dx_i ~{\\ln}\\left[1-(1-e^{-p})U(-x_i,t) \\right] \\longrightarrow \\rho \\int_0^\\infty dz\\, \\ln \\left[ 1 - (1-e^{-p}) U(z,t)\\right] \\;.\n\\end{equation} \nTherefore the Laplace transform of the quenched flux distribution is given by\n\\begin{equation}\\label{Pqu2}\n\\sum_{Q=0}^{\\infty} P_{\\rm qu} (Q,t) e^{-pQ} = \\exp\\left[I(p,t)\\right] \\;,\n\\end{equation}\nwhere \n\\begin{eqnarray}\\label{def_Ip}\nI(p,t) = \\rho \\int_0^\\infty dz\\, \\ln \\left[ 1 - (1-e^{-p}) U(z,t)\\right] \\;.\n\\end{eqnarray}\nBefore extracting the full distribution $P_{\\rm qu} (Q,t)$ from this Laplace transform, it is useful to study first the asymptotic behaviors of $I(p,t)$ \nin the two limits : (i) $p \\rightarrow 0$ and (ii) $p \\rightarrow \\infty$. \n\\begin{itemize}\n\\item[$\\bullet$] $p\\to 0$ limit: Expanding $e^{-p}$ in powers of $p$ in Eq. (\\ref{def_Ip}), we get\n\\begin{equation}\\label{Ip0}\nI(p,t) = - p ~\\rho \\int_0^{\\infty} dz~U(z,t) + \\frac{p^2}{2} ~\\rho \\int_0^{\\infty} dz~ U(z,t)\\left[1-U(z,t) \\right] + {\\cal O}(p^3) \\;.\n\\end{equation} \nSubstituting this in Eq. (\\ref{Pqu2}) and expanding both sides in powers of $p$ we immediately get the mean and the variance of the flux $Q_t$ for the quenched case\nas stated in Eqs. (\\ref{mean_qu}) and (\\ref{var_qu}) respectively. \n \n\\item[$\\bullet$] $p\\to \\infty$ limit: In this case we expand $I(p,t)$ in Eq. (\\ref{def_Ip}) in powers of $e^{-p}$. The two leading terms are given by\n\\begin{equation}\\label{Ip-pinf1}\nI(p,t) = A(t) + B(t) e^{-p} + {\\cal O}(e^{-2p}) \\;,\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{atbt-pinf1}\nA(t) &=& \\rho \\int_0^{\\infty} {\\ln} [1-U(z,t)] dz \\\\ \nB(t) &=& \\rho \\int_0^{\\infty} \\frac{U(z,t)}{1-U(z,t)} dz \\;.\n\\end{eqnarray}\nSubstituting this expansion (\\ref{Ip-pinf1}) on the rhs of Eq. (\\ref{Pqu2}) and matching the powers of $e^{-p}$ on both sides of Eq. (\\ref{Pqu2}) immediately gives\n\\begin{eqnarray}\nP_{\\rm qu}(Q=0,t) &=& e^{A(t)} = \\exp{\\left[\\rho\\int_0^\\infty \\ln(1-U(z,t))dz\\right]} \\label{Pqu_eq_0} \\\\\nP_{\\rm qu}(Q=1,t) &=& B(t) \\, e^{A(t)} \\;. \\label{Pqu_eq_1}\n\\end{eqnarray}\nThe first line yields the general result mentioned in Eq. (\\ref{P_neq0_qu}). \n\n \n\\end{itemize}\n\n\n\n\nThese results so far are quite general, i.e., they hold for any dynamics -- the dependence on the dynamics comes\nonly through the function $U(z,t)$. In the following, we focus on two interesting dynamics, namely the diffusive and \nthe RTP and extract the large time behavior of $P_{\\rm qu}(Q,t)$ using Eqs. (\\ref{Pqu2}) and (\\ref{def_Ip}). \n\n\n\n\n\n\\subsection{$P_{\\rm qu}(Q,t)$ for simple diffusion}\\label{quen-sec-diff}\n\nIn this case, using the explicit expression $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$, we get from Eq. (\\ref{def_Ip}) \n\\begin{equation}\\label{I-phi-correspondence}\nI(p,t) = \\rho \\sqrt{ 4 D t} \\int_0^{\\infty} dz~ {\\ln}\\left[1-\\frac{1}{2}(1-e^{-p}){{\\rm erfc}(z)}\\right]\n= -\\rho \\sqrt{D t}~ \\phi (p),\n\\end{equation}\nwhere \n\\begin{equation}\\label{phi-1eq}\n\\phi (p) = -2 \\int_0^{\\infty} dz~ {\\ln}\\left[1-\\frac{1}{2}(1-e^{-p}){{\\rm erfc}(z)}\\right ] \\;.\n\\end{equation}\nTherefore Eq. (\\ref{Pqu2}) reads for all time $t$\n\\begin{eqnarray} \\label{def_phi}\n\\sum_{Q=0}^{\\infty} e^{-pQ} \\,P_{\\rm qu} (Q,t) = \\exp\\left[-\\rho \\sqrt{D t}~ \\phi (p)\\right] \\;.\n\\end{eqnarray}\nIn the long time limit $t \\rightarrow \\infty$, we anticipate, and verify a posteriori, that \n$P_{\\rm qu}(Q,t)$ takes a large deviation form in the limit where $Q \\to \\infty$, $t \\to \\infty$ but\nwith the dimensionless ratio $q = Q\/(\\rho \\sqrt{D\\,t})$ fixed \n\\begin{equation}\\label{Pq}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\sqrt{D t}~ \\Psi_{\\rm diff} \\left(\\frac{Q}{\\rho \\sqrt{D t}}\\right)\\right], \n\\end{equation}\nwhere $\\Psi_{\\rm diff}(q)$ is a rate function that we wish to compute. Substituting this large deviation form (\\ref{Pq}) on the left hand side (lhs) of Eq. (\\ref{def_phi}) and replacing the discrete sum over $Q$ by an integral (which is valid for large $Q \\sim \\sqrt{t}$), we get\n\\begin{equation}\\label{Pq1}\n\\int_0^{\\infty} e^{-pQ} P_{\\rm qu}(Q,t) dQ \\sim \\rho \\sqrt{D t} \\int_0^{\\infty} e^{-\\rho \\sqrt{D t}\\, \\left[p \\, q+\\Psi_{\\rm diff}(q)\\right]} dq \\;.\n\\end{equation}\nFor large $t$, we can now evaluate the integral over $q$ in Eq. (\\ref{Pq1}) by a saddle point method, which gives\n\\begin{eqnarray}\\label{sp_diff}\n\\int_0^{\\infty} e^{-pQ} P_{\\rm qu}(Q,t) dQ \\sim \\exp{\\left[-\\rho \\sqrt{Dt}\\, \\underset{q}{\\min}[p\\,q+\\Psi_{\\rm diff}(q)] \\right]} \\;.\n\\end{eqnarray}\nComparing this with the rhs of Eq. (\\ref{def_phi}) we get\n\\begin{equation}\\label{legendre}\n\\underset{q}{\\min}\\left[p\\,q+\\Psi_{\\rm diff}(q)\\right]= \\phi(p) \\;.\n\\end{equation}\nInverting this Legendre transform one gets\n\\begin{equation}\\label{fq-bas}\n\\Psi_{\\rm diff}(q)=\\underset{p}{\\max}\\left[\\phi(p)-p\\,q\\right] \\;,\n\\end{equation}\nwhere $\\phi(p)$ is given in Eq. (\\ref{phi-1eq}). \n\nKnowing $\\phi(p)$ explicitly, one can plot the large deviation function $\\Psi_{\\rm diff}(q)$ using Eq. (\\ref{fq-bas}) -- see Fig. \\ref{diff-math-asymp}. Clearly,\n$\\Psi_{\\rm diff}(q)$ has a concave shape with a minimum at $q=q_{\\min}$, the value of $q_{\\min}$ will be computed shortly. The asymptotic behaviors of the\nrate function $\\Psi_{\\rm diff}(q)$ can also be extracted in the limits $q \\to q_{\\min}$, $q\\to 0$ and $q \\to \\infty$ by analysing $\\phi(p)$ respectively in the limits $p\\to 0$, $p\\to +\\infty$ and $p \\to -\\infty $ (where $\\phi(p)$ in Eq. (\\ref{phi-1eq}) has to be continued analytically to negative $p$). The results are summarised in Eqs. (\\ref{psi_diff_asympt}), (\\ref{C}) and (\\ref{A}) in section \\ref{sec:model}. Below we provide the derivation of these results. \n\n\n\n\n\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mathm-analytics-qtypandzero-diff.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mathm-analytics-qlarge-diff.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Large deviation function $\\Psi_{\\rm diff}(q)$ vs $q$ for the diffusive case with quenched initial conditions. On both panels, the dashed black lines correspond to \nevaluating via Mathematica $\\Psi_{\\rm diff}(q)$ from Eq. (\\ref{fq-bas}) with $\\phi(p)$ given in Eq. (\\ref{phi-1eq}). On the left panel (a), the solid yellow curve corresponds to the small $q$ asymptotic behavior of $\\Psi_{\\rm diff}(q)$ in Eq.~(\\ref{Fq-qzero-soln}) and the dashed-dotted green curve corresponds to the quadratic behavior in Eq.~(\\ref{fq_s0}). On the right panel (b), we zoom in on the large $q$ tail. The solid violet curve corresponds to the leading asymptotic behavior $\\Psi_{\\rm diff}(q) \\approx q^3\/12$, while the dashed-dotted green curve is the quadratic behavior as in Eq.~(\\ref{fq_s0}). The violet and the green curves clearly demonstrate the non-Gaussian tail of $\\Psi_{\\rm diff}(q)$.}\\label{diff-math-asymp}\n\\end{figure}\n\n\n\n\\subsubsection{Typical Fluctuations : $Q \\sim \\langle Q \\rangle_{\\rm qu}$}\\label{qmi}\n In order to derive the result for $\\Psi_{\\rm diff}(q \\rightarrow q_{\\min})$, we need to analyze $\\phi(p)$ for $p \\rightarrow 0$. We expand $\\phi(p)$ in Eq. (\\ref{phi-1eq}) up to order $p^2$ and get\n\\begin{eqnarray}\\label{phi_s0}\n\\phi(p) = \\alpha p - \\beta p^2 + {\\cal O}(p^3) \\;, \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n&&\\alpha=\\int_0^{\\infty} {\\rm erfc}(z) dz ~ = \\sqrt{\\frac{1}{\\pi}} \\label{alpha-defn} \\\\\n&&\\beta=\\frac{1}{4} \\int_0^{\\infty} (2\\, {\\rm erfc}(z) - {\\rm erfc}^2(z))\\, dz ~ = \\frac{1}{\\sqrt{8\\pi}}\\;.\n\\end{eqnarray}\nSubstituting $\\phi(p) = \\alpha \\, p - \\beta\\,p^2$ in Eq. (\\ref{fq-bas}) and maximising with respect to $p$ gives a quadratic form for the rate function \n \\begin{equation} \\label{fq_s0}\n \\Psi_{\\rm diff }(q) \\sim \\frac{(q-\\alpha)^2}{4\\beta}= \\sqrt{\\frac{\\pi}{2}} \\left(q-\\sqrt{\\frac{1}{\\pi}}\\right)^2 \\;.\n \\end{equation}\nThis form holds for $q$ close to $q_{\\min} = \\alpha = 1\/\\sqrt{\\pi}$ and gives the result in the second line in Eq. (\\ref{psi_diff_asympt}). Substituting this quadratic behavior in the large deviation form in Eq. (\\ref{Pq}) predicts\na Gaussian form for the quenched flux distribution for $q$ close to $q_{\\min}$\n\\begin{eqnarray}\\label{Gaussian_qu}\nP_{\\rm qu}(Q,t) \\sim \\exp{\\left[- \\frac{\\left(Q-\\langle Q\\rangle_{\\rm qu}\\right)^2}{2 \\sigma_{\\rm qu}^2}\\right]}\n\\end{eqnarray}\nwhere the mean the variance are given by \n \\begin{eqnarray}\n\\langle Q \\rangle_{\\rm qu} &=&\\rho \\sqrt{\\frac{D t}{\\pi}} \\label{diffusion-mean-quenched} \\\\\n\\sigma^2_{\\rm qu} &=& \\rho \\sqrt{\\frac{D t}{2\\pi}} \\label{diffusion-variance-quenched} \\;.\n\\end{eqnarray} \nNotice that these expressions for the mean and the variance, though derived here for large $t$, actually hold for all $t$, as one can verify directly from the formulae in Eqs. (\\ref{mean_qu}) and (\\ref{var_qu}) with $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4 Dt})$. Comparing with the annealed case, while the means in both cases are identical, both given by $\\mu(t) = \\rho \\sqrt{Dt\/\\pi}$, their variances and higher moments differ. For example, the variance in the annealed case is $\\mu(t)=\\rho \\sqrt{Dt\/\\pi}$ which differs by a factor $1\/\\sqrt{2}$ from the quenched case in Eq. (\\ref{diffusion-variance-quenched}). These results agree with those obtained in~\\cite{derrida-gers}. This typical quadratic behavior is shown by the dashed-dotted green curve in Fig. \\ref{diff-math-asymp}. \n\n\n \n\\subsubsection{Atypical fluctuations on the left of the mean: $Q \\ll \\langle Q \\rangle_{\\rm qu}$} \\label{qze}\n\nIn order to infer about the fluctuations of $P_{\\rm qu}(Q,t)$ around $Q \\rightarrow 0$, we need to evaluate how $\\Psi_{\\rm diff}(q)$ behaves when $q \\rightarrow 0$. This corresponds to the limit $p \\rightarrow \\infty$ for $\\phi(p)$ from Eq. (\\ref{fq-bas}). We use the large $p$ expansion in Eq. \\ref{Ip-pinf1}, and evaluate $A(t)$ and $B(t)$ from \nEq. (\\ref{atbt-pinf1}) using $U(z,t) = (1\/2) {\\rm erfc}(z\/\\sqrt{4Dt})$. This gives\n\\begin{eqnarray}\\label{At-diff}\nA(t) &=& -\\overline{\\alpha} \\, \\rho\\, \\sqrt{4Dt} \\;, \\quad {\\rm where} \\quad \\overline{\\alpha}= -2 \\int_0^{\\infty} {\\ln} \\left[1-\\frac{1}{2}{{\\rm erfc}(z)}\\right] dz = 0.675336\\ldots \\label{a-defn} \\\\ \nB(t) &=& \\overline{\\beta} \\, \\rho\\, \\sqrt{4Dt} \\;, \\quad {\\rm where} \\quad \\overline{\\beta} = \\int_0^{\\infty} \\frac{{\\rm erfc}(z)}{1-\\frac{1}{2}{\\rm erfc}(z)} dz = 0.828582 \\ldots \\;. \\label{b-defn}\n\\end{eqnarray}\nFrom Eqs. (\\ref{I-phi-correspondence}) and (\\ref{phi-1eq}) we get the two leading terms of $\\phi(p)$ for large $p>0$\n\\begin{equation}\\label{phip-zero-final}\n\\phi(p)= -\\frac{I(p,t)}{\\rho \\sqrt{D t}} \\approx \\overline{\\alpha}-\\overline{\\beta}~e^{-p} \\;.\n\\end{equation}\nPlugging this result for $\\phi(p)$ in Eq. (\\ref{fq-bas}) and maximizing with respect to $p$, we get the leading small $q$ behavior of $\\Psi_{\\rm diff}(q)$\n\\begin{equation}\\label{Fq-qzero-soln}\n\\Psi_{\\rm diff}(q)\\approx \\overline{\\alpha} - q +q ~ {\\ln}\\left(\\frac{q}{\\overline{\\beta}}\\right) \\;.\n\\end{equation}\nThis reproduces the first line of Eq. (\\ref{psi_diff_asympt}). In particular, for $q=0$, using $\\Psi_{\\rm diff}(q=0) = \\overline{\\alpha}$ in Eq. (\\ref{Pq}), we obtain $P_{\\rm qu}(Q=0,t) \\sim \\exp{(-\\overline{\\alpha} \\, \\rho \\sqrt{Dt})}$ as announced in Eq. (\\ref{P_Q0}). The small $q$ behavior of $\\Psi_{\\rm diff}(q)$ is shown by the solid yellow curve in Fig. \\ref{diff-math-asymp}(a). \n\n\n\n\\subsubsection{Atypical fluctuations on the right of the mean: $Q \\gg \\langle Q\\rangle_{\\rm qu}$}\\label{qla}\n\nIn order to derive the large $q$ asymptotics of $\\Psi_{\\rm diff}(q)$ from Eq. (\\ref{fq-bas}), we first need to continue $\\phi(p)$ in Eq. (\\ref{phi-1eq}) analytically to negative $p$\nand use its asymptotics in the limit $p \\to -\\infty$. For this, it is convenient to write first $p = -u$ where $u = |p|$. We write\n\\begin{equation}\\label{tildephi-diff}\n\\phi(p=-u) = \\tilde{\\phi}(u) = -2 \\int_0^{\\infty} dz ~{\\ln} \\left[1+ \\frac{\\left(e^u-1 \\right)}{2}{{\\rm erfc}(z)} \\right] \\quad \\underset{u \\to \\infty}{\\approx} \\quad -2 \\int_0^{\\infty} dz ~{\\ln} \\left[1+ \\frac{e^u}{2} {\\rm erfc}(z) \\right] \\;.\n\\end{equation}\nTo extract the large $u$ behavior of $\\tilde \\phi(u)$ from the integral on the rhs, it is convenient to take the derivative with respect to $u$\n\\begin{equation}\\label{qu-larphi}\n \\tilde{\\phi}'(u) \\approx -2\\int_0^{\\infty} dz \\, \\frac{~ \\frac{e^{u}}{2}\\,{\\rm erfc}(z)}{1 + \\frac{e^u}{2}\\,{\\rm erfc}(z)} \\;.\n\\end{equation} \nFor large $u$ the dominant contribution to this integral comes from large $z$ where ${\\rm erfc}(z) \\approx e^{-z^2}\/(z \\sqrt{\\pi})$. Hence we see that, for $z > \\sqrt{u}$ the integrand is essentially $0$ as $u \\to \\infty$, while, for $z < \\sqrt{u}$, the integrand is $1$ as $u \\to \\infty$. Hence, the integrand can be approximated by a Fermi function \n\\begin{eqnarray} \\label{phiprime2}\n\\tilde{\\phi}'(u) \\approx -2 \\int_0^{\\sqrt{u}} dz = - 2 \\sqrt{u} \\;.\n\\end{eqnarray}\nIntegrating it back, we get the leading order behavior for $\\tilde \\phi(u)$ for large $u$\n\\begin{equation}\\label{phi-large-diff}\n \\tilde{\\phi}(u) \\approx -\\frac{4}{3} u^{\\frac{3}{2}} \\;.\n\\end{equation}\nTherefore $\\phi(p) \\approx -(4\/3) (-p)^{3\/2}$ as $p \\to -\\infty$. Substituting this behavior in Eq. (\\ref{fq-bas}) and maximizing with respect to $p$ one gets $\\Psi_{\\rm diff}(q) \\approx q^3\/12$ as $q \\to \\infty$. This then gives the last line of the result in Eq. (\\ref{psi_diff_asympt}). As mentioned earlier, this leading large $q$ asymptotic behavior of $\\Psi_{\\rm diff}(q)$ coincides with the result of Ref. \\cite{derrida-gers} obtained by a different method. The large $q$ behavior of $\\Psi_{\\rm diff}(q)$ is shown solid the solid violet curve in Fig. \\ref{diff-math-asymp}(b). \n\n\n\n\n\n\\subsection{$P_{\\rm qu}(Q,t)$ for run-and-tumble particles}\\label{quen-act-sec}\n\nIn this case, our starting point again are Eqs. (\\ref{Pqu2}) and (\\ref{def_Ip}), except that the function $U(z,t)$ for the RTP is more complicated.\nIts Laplace transform is given in Eq. (\\ref{LaplaceU_2}). As shown in Appendix C of \\cite{pierre-satya-greg} it can be formally inverted to obtain $U(z,t)$ in real time\n\\begin{equation}\\label{Uzt-exact}\nU(z,t) = \\frac{1}{2}\\left[ e^{\\frac{-\\gamma z}{v_0}} + \\frac{\\gamma z}{v_0} \\int_1^{\\frac{v_0 t}{z}} dT \\frac{ e^{\\frac{-\\gamma z T}{v_0}}I_1(\\frac{\\gamma z}{v_0}\\sqrt{T^2-1})}{\\sqrt{T^2-1}}\\right] \\Theta(v_0 t-z) \\;.\n\\end{equation}\nHowever, it turns out that this expression is not very useful to extract the large deviation function at late times. \n\nBefore proceeding to compute the large deviation function at late times, it is useful to discuss the large $t$ behavior of $P_{\\rm qu}(Q,t)$ in different\nregimes of $Q$. In the following, we will first discuss the $Q \\to 0$ limit of $P_{\\rm qu}(Q,t)$, followed by the discussion of the typical\nfluctuations where $Q = {\\cal O}(\\sqrt{t})$. In this regime, we will recover the Gaussian fluctuations. When $Q\/(\\rho\\, \\sqrt{D\\,t}) \\gg 1$, we\nexpect to recover the large deviation regime for the diffusive behaviour discussed in the previous section. This is because, as explained in the \nintroduction, at late times, the RTP motion essentially reduces to that of a diffusive particle with an effective diffusion constant $D_{\\rm eff} = v_0^2\/(2 \\gamma)$.\nHowever, there exists yet another ``larger deviations regime'' where $Q \\sim {\\cal O}(t)$ where we will show that $P_{\\rm qu}(Q,t)$ carries the signature\nof activity and has a novel large deviation form\n\\begin{equation}\\label{ac-sig2}\nP_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\, v_0 \\, \\gamma \\, t^2 \\, \\Psi_{\\rm RTP}\\left(\\frac{Q}{\\rho \\, v_0 \\,t}\\right)\\right] \\;.\n\\end{equation}\nIn the following, we will indeed compute this rate function $\\Psi_{\\rm RTP}(q)$ and show that it is given by Eq. (\\ref{rtp-ldf-model}). \n \n\n\n\\subsubsection{Typical fluctuations: $Q \\sim \\langle Q\\rangle_{\\rm qu}$}\n\nIn order to extract the typical fluctuations of $Q_t$ around its mean value for the RTP case, we need to use the small $p$ expansion\nof $I(p,t)$ in Eq. (\\ref{Pqu2}). Quite generally, the small $p$ expansion of $I(p,t)$ is given in Eq. (\\ref{Ip0}). We use this expansion\non the rhs of Eq. (\\ref{Pqu2}) and approximate the sum on lhs by an integral. The resulting Laplace transform can be easily inverted\nand yields a Gaussian form \n\\begin{eqnarray}\\label{P_qu_Gauss}\nP_{\\rm qu}(Q,t) \\approx \\exp{\\left[-\\frac{(Q-\\langle Q \\rangle_{\\rm qu})^2}{2 \\, \\sigma_{\\rm qu}^2}\\right]} \\;,\n\\end{eqnarray} \nwhere $\\langle Q \\rangle_{\\rm qu}$ and $\\sigma^2_{\\rm qu}$ are given in Eqs. (\\ref{mean_qu}) and (\\ref{var_qu}) respectively where $U(z,t)$ is given in Eq. (\\ref{Uzt-exact}) --\nalternatively its Laplace transform is given by the simpler form in Eq. (\\ref{LaplaceU_2}). The mean value $\\langle Q \\rangle_{\\rm qu}$ can be computed explicitly. Indeed\n\\begin{eqnarray}\\label{av_qu_RTP}\n\\langle Q \\rangle_{\\rm qu} = \\rho \\int_0^\\infty U(z,t)\\, dz = \\mu(t)= \\frac{\\rho\\,v_0}{2} t\\,e^{-\\gamma t}\\left[ I_0(\\gamma\\,t) + I_1(\\gamma \\, t)\\right] \\;,\n\\end{eqnarray}\nwhere the last equality follows from Eq. (\\ref{ac-mut}). The variance $\\sigma_{\\rm qu}^2 = \\rho \\int_0^{\\infty} U(z,t)\\left[1-U(z,t)\\right] \\, dz$ is however difficult to compute explicitly using $U(z,t)$ from Eq. (\\ref{Uzt-exact}). However, it can be easily evaluated numerically. At large times, $\\langle Q \\rangle_{\\rm qu}$ and $\\sigma^2_{\\rm qu}$ converge to the diffusive limits given in Eqs. (\\ref{diffusion-mean-quenched}) and (\\ref{diffusion-variance-quenched}) respectively. \n\n\n\n\n\n\n\n\n\\subsubsection{Atypical fluctuations on the left of the mean: $Q \\ll \\langle Q \\rangle_{\\rm qu}$}\n\nExactly at $Q=0$ or $Q=1$, we have an exact\nexpression at all times $t$ for $P_{\\rm qu}(Q,t)$ in terms of $U(z,t)$, as given in Eqs. (\\ref{Pqu_eq_0})\nand (\\ref{Pqu_eq_1}). The function $U(z,t)$ for RTP appearing in these expressions is given in Eq. (\\ref{Uzt-exact}).\nGiven this rather complicated expression of $U(z,t)$, it is hard to obtain explicit formulae valid at all times for\n$P_{\\rm qu}(Q,t)$ even for $Q=0$ or $Q=1$. However, at late times, since $U(z,t)$ converges at late times\nto that of the diffusive limit in Eq. (\\ref{UzT_BM}) with an effective diffusion constant $D_{\\rm eff} = v_0^2\/(2 \\gamma)$, \nwe recover the diffusive results for this extreme left tail of $P_{\\rm qu}(Q,t)$.\nFor instance $P_{\\rm qu}(Q=0,t)$, which represents the probability of having no particle on the right side of the origin\nat time $t$, decays at late times as in the diffusive case \n\\begin{eqnarray} \\label{P_Q0_RTP}\nP_{\\rm qu}(Q=0,t)\\Big \\vert_{\\rm RTP} \\approx \\exp \\left[ - \\overline{\\alpha} \\, \\rho \\sqrt{D_{\\rm eff}\\,t}\\right] \\;,\n\\end{eqnarray} \nwhere $\\overline{\\alpha} = 0.675336\\ldots$ is given in Eq. (\\ref{C}). \n\n\n\\subsubsection{Atypical fluctuations on the right of the mean: $Q \\sim {\\cal O}(t) \\gg \\langle Q \\rangle_{\\rm qu}$}\n\nIn this section, we derive the result in Eq. (\\ref{ac-sig2}). We recall that in the diffusive case, the atypical \nfluctuations of $Q$ are encoded in the large deviation form in Eq. (\\ref{Pq}) with $Q \\sim \\rho \\sqrt{Dt}$. \nThe extreme fluctuations to the right of $\\langle Q \\rangle_{\\rm qu}$ in this case are described by the large\nargument behavior of the large deviation function ${\\Psi}_{\\rm diff}(q = Q\/(\\rho\\sqrt{D\\,t}))$, i.e., when $Q \\gg \\rho \\sqrt{Dt}$. \nThus, in the diffusive case, there is a single scale for the fluctuations of $Q$ at late times, namely $Q \\sim \\sqrt{t}$. \nIn contrast, for the RTP, in addition to the scale $\\sqrt{t}$ that describes the moderate large deviations around\nthe mean, there is yet another scale where $Q \\sim t$. This comes from the fact that each particle in time $t$ can move a maximum distance \n$v_0 t$, where $v_0$ is the velocity. So for an initial density $\\rho$, the maximum possible flux through the origin is $Q_{\\rm max} = \\rho v_0 t$. \nHence $Q \\sim t$ describes the scale of fluctuations at the very right tail of the distribution $P_{\\rm qu}(Q,t)$. \n\n\nTo extract this extreme right tail, we again start from Eqs. (\\ref{Pqu2}) and (\\ref{def_Ip}) with $U(z,t)$, for RTP, given by its Laplace transform \nin Eq. (\\ref{LaplaceU_2}). Before extracting the large deviation form of $P_{\\rm qu}(Q,t)$, we first analyse $U(z,t)$ in the limit $z \\sim t$. Inverting the Laplace transform of $U(z,t)$ in Eq. (\\ref{LaplaceU_2}) we get\n\\begin{eqnarray}\\label{U_Bromwich1}\nU(z,t) = \\int_\\Gamma \\frac{ds}{2\\pi i} \\exp{\\left[t\\left( s - \\sqrt{s(s+2\\gamma)} \\frac{z}{v_0\\,t}\\right)\\right]} \\;,\n\\end{eqnarray} \nwhere $\\Gamma$ represents the Bromwich contour in the complex $s$-plane. In the limit $t \\to \\infty$, $z \\to \\infty$ with the ratio $z\/t$ fixed,\nthe integral can be evaluated by the saddle-point method, which yields (up to pre-exponential factors)\n\\begin{equation}\\label{Uzt_saddle}\nU(z,t) \\approx \\exp \\left[-\\gamma t \\left(1-\\sqrt{1-\\frac{z^2}{v_0^2 t^2}}\\right)\\right] \\, \\Theta(v_0 \\,t-z ) ~.\n\\end{equation}\nWe have checked numerically that this approximation (\\ref{Uzt_saddle}) works very well, at large times, as one would expect. \n\nTo extract the large $Q \\sim t \\gg \\langle Q\\rangle_{\\rm qu}$ behavior from Eq. (\\ref{Pqu2}) we need to analytically continue\n$I(p,t)$ to $p$ negative and study the limit $p \\to -\\infty$, as in the diffusive case. Setting $p=-u$ with $u>0$, and approximating the discrete sum\non the lhs of Eq. (\\ref{Pqu2}) by an integral, we get\n\\begin{eqnarray}\\label{laplace1}\n\\int_0^\\infty P_{\\rm qu}(Q,t)\\, e^{u Q} \\,dQ \\approx e^{\\tilde I(u,t)} \\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{def_tildeI}\n\\tilde I(u,t) = \\rho \\int_0^\\infty dz\\, \\ln \\left( 1+ (e^u-1)\\, U(z,t)\\right) \\underset{u \\to \\infty}{\\approx} \\rho \\int_0^\\infty dz\\, \\ln \\left( 1+ e^u\\, U(z,t)\\right) \\;.\n\\end{eqnarray}\nTo extract the large $u$ behavior of $\\tilde I(u,t)$ we follow the same procedure as in the diffusive case and take a derivative with respect to $u$. \nWe get\n\\begin{eqnarray}\\label{derivative_1}\n\\frac{d\\tilde I(u,t)}{du} \\approx \\rho \\int_0^\\infty \\frac{dz}{1+e^{-u} \\frac{1}{U(z,t)}} \\;.\n\\end{eqnarray}\nFor large $u$, this integral is dominated by the region where $z \\sim t$ where we can use the approximate form of $U(z,t)$ given in Eq. (\\ref{Uzt_saddle}). Substituting this form for $U(z,t)$ in Eq. (\\ref{derivative_1}), we get\n\\begin{eqnarray}\\label{derivative_2}\n\\frac{d\\tilde I(u,t)}{du} \\approx \\rho \\int_0^{v_0\\,t} \\frac{dz}{1+\\exp{\\left[-\\left(u-\\gamma\\,t + \\gamma\\,t \\sqrt{1-\\frac{z^2}{v_0^2\\,t^2}}\\right)\\right]}} \\;.\n\\end{eqnarray}\nWe now analyse this integral in two different cases, assuming $u \\sim t \\gg 1$:\n\n\\vspace*{0.3cm}\n\\noindent $\\bullet$ If $u > \\gamma t$: in this case as $t \\to \\infty$ it is clear that the integrand in Eq. (\\ref{derivative_2}) is always $1$ for any $z$. Hence \n\\begin{eqnarray}\\label{derivative_3}\n\\frac{d\\tilde I(u,t)}{du} \\approx \\rho v_0 t \\quad \\quad {\\rm if} \\quad u > \\gamma t \\;.\n\\end{eqnarray}\n\n \\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{wx-plot.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{wpr-plot.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{active-large-deviation-full.eps}\n\\end{minipage}\n\\end{center}\n\\caption{ (a) Plot of $W(x)$ vs $x$ with $W(x)$ given by Eq.~(\\ref{phi-full}). The non-analytical point at $x=1$ is shown by a black dot\nwhere $W(x)$ and its first two derivatives are continuous, while the third derivative is discontinuous, as in Eq. (\\ref{third-der-phi}). \n(b) Plot of $W'(x)$ vs $x$ where $W'(x)$ can be read off from Eq. (\\ref{def_W}). Note that $W'(x) <1$ for $x<1$ and saturates at $1$ at $x=1$. (c) Large deviation function $\\psi_{\\rm {RTP}}(q)$ vs $q$ for the run-and-tumble case. Black solid line represents the analytical expression given in Eq.~(\\ref{rtp-ldf-model}) valid at very large time. Dashed lines ($t=100$ (red-dashed), $t=400$ (violet-dotted) and $t=600$ (blue dashed-dotted)) correspond to the finite time evaluation of $\\underset{x}{\\max}[\\frac{qu}{\\gamma t}-\\frac{\\tilde{I}(u,t)}{\\rho v_0 \\gamma t^2}]$ using Eq. (\\ref{def_tildeI}) -- \ntogether with Eq.~(\\ref{def_tildeI}) with $\\rho=v_0=1, \\gamma=0.5$ -- with Mathematica.\n}\\label{phi-full-plot}\n\\end{figure}\n \n \n\\vspace*{0.3cm}\n\\noindent $\\bullet$ If $u < \\gamma t$: this case is a bit more complicated to analyze. Since $u<\\gamma t$ the argument of the exponential in Eq. (\\ref{derivative_2}) can be either positive or negative. Accordingly, the integrand will either $0$ or $1$ for large $u \\sim t \\gg 1$. The value\nof $z$ for which the argument of the exponential changes sign is given by\n\\begin{eqnarray}\\label{zstar}\nz^*(u) = \\frac{v_0}{\\gamma} \\sqrt{u(2\\gamma\\,t - u)} \\;.\n\\end{eqnarray}\nThus, if $z>z^*(u)$ the integrand is $0$ while for $z1 \\;.\n\\end{cases}\n\\end{eqnarray}\nPerforming these integrals explicitly, we get\n\\begin{eqnarray}\\label{phi-full}\nW(x) =\n\\begin{cases}\n \\frac{(x-1)}{2}\\sqrt{x(2-x)}+{\\rm sin}^{-1}(\\sqrt{x\/2})\\quad,\\quad x<1\\\\ \n\\\\\n\\frac{\\pi}{4}+x-1\\quad,\\quad x>1 \\;.\n\\end{cases}\n\\end{eqnarray}\nThe function $W(x)$ is plotted vs $x$ in Fig.~\\ref{phi-full-plot} (a). Interestingly, while $W(x)$ and its first two derivatives are continuous \nat $x=1$, its third derivative is discontinuous. Indeed one has\n\\begin{eqnarray}\\label{third-der-phi}\nW^{'''}(x \\rightarrow 1^-) &=& -1 \\nonumber \\\\\nW^{'''}(x \\rightarrow 1^+) &=& 0 \\;.\n\\end{eqnarray}\nUsing this result for $W(x)$ in Eq. (\\ref{phi-full}) gives us an expression for $\\tilde I(u,t)$ in (\\ref{full-intI_1}). We then substitute this expression for $\\tilde I(u,t)$ in Eq. (\\ref{laplace1}) to get\n\\begin{eqnarray}\\label{PqW}\n\\int_0^\\infty P_{\\rm qu}(Q,t) e^{uQ} dQ \\sim \\exp{\\left[\\rho v_0 \\gamma t^2 W\\left(\\frac{u}{\\gamma t}\\right)\\right]} \\;.\n\\end{eqnarray}\nInverting formally this Laplace transform, we obtain\n\\begin{eqnarray} \\label{Inverse1}\nP_{\\rm qu}(Q,t) \\sim \\int_{\\Gamma} \\frac{du}{2 \\pi i} \\exp{\\left[- u Q + \\rho v_0 \\, \\gamma t^2 W\\left( \\frac{u}{\\gamma t}\\right) \\right]} \\;.\n\\end{eqnarray}\nRescaling $u\/(\\gamma t)= x$ we get, up to pre-exponential factors,\n\\begin{eqnarray} \\label{Inverse2}\nP_{\\rm qu}(Q,t) \\sim \\int_{\\Gamma} \\frac{dx}{2\\pi i} \\exp{\\left[ - \\rho v_0 \\gamma t^2 \\left(-W(x) + x q \\right) \\right]} \\;, \\quad {\\rm where} \\; q = \\frac{Q}{\\rho v_0 t} \\;.\n\\end{eqnarray}\nwhere $\\Gamma$ is the Bromwich contour in the complex $x$-plane. Performing this integral by using a saddle-point for large $t$, we get\n\\begin{eqnarray} \\label{Inverse3}\nP_{\\rm qu}(Q,t) \\sim \\exp{\\left[- \\rho v_0 \\gamma t^2 \\Psi_{\\rm RTP} \\left( q = \\frac{Q}{\\rho v_0 t}\\right) \\right]} \\;,\n\\end{eqnarray}\nwith the rate function given by\n\\begin{eqnarray}\\label{Psi_RTP}\n\\Psi_{\\rm RTP}(q) = \\underset{x}{\\max} \\left[q\\,x - W(x) \\right] \\;,\n\\end{eqnarray}\nwhere $W(x)$ is given explicitly in Eq. (\\ref{phi-full}). It is easy to verify that the maximum of the function $q\\,x-W(x)$ occurs at $x=x^* = 1-\\sqrt{1-q^2} < 1$. Since $x^*<1$ we use the branch of $W(x)$ in the first line of Eq. (\\ref{phi-full}). Substituting this value of $x^*$ in Eq. (\\ref{Psi_RTP}) we get the result in Eq. (\\ref{rtp-ldf-model}). The asymptotic behaviours of this function $\\Psi_{\\rm RTP}(q)$ are given in Eq. (\\ref{asympt_RTP}) and a plot of this function is shown in Fig. \\ref{phi-full-plot} (c). Note that for small $q$, i.e. $Q \\ll \\rho v_0 t$, $\\Psi_{\\rm RTP}(q)$ behaves as $\\Psi_{\\rm RTP}(q) \\sim q^3\/6$. Substituting this behavior in Eq. (\\ref{Inverse3}) gives\n\\begin{eqnarray}\\label{matching1}\nP_{\\rm qu}(Q,t) \\Big \\vert_{\\rm RTP} \\sim \\exp{\\left( - \\frac{\\gamma Q^3}{6 \\rho^2 v_0^2 t}\\right)} \\sim \\exp{\\left( - \\frac{Q^3}{12 \\rho^2 D_{\\rm eff}t}\\right)} \\;, \\quad {\\rm where} \\quad D_{\\rm eff} = \\frac{v_0^2}{2 \\gamma} \\;.\n\\end{eqnarray}\nOn the other hand, for the diffusive case, from Eq. (\\ref{P-F-diff-largeQ}), using $\\Psi_{\\rm diff}(q) \\approx q^3\/12$ for large $q$, i.e. $Q \\gg \\rho \\sqrt{D\\,t}$, one gets \n\\begin{eqnarray}\\label{matching2}\nP_{\\rm qu}(Q,t) \\Big \\vert_{\\rm diff} \\sim \\exp{\\left( - \\frac{Q^3}{12 \\rho^2 D_{}t}\\right)}\n\\end{eqnarray}\nComparing these two tails (\\ref{matching1}) and (\\ref{matching2}), one sees that for the RTP, on a scale where $\\rho \\, \\sqrt{D_{\\rm eff}t} \\ll Q \\ll \\rho \\, v_0 \\, t$ these two behaviors match perfectly, supporting the expectation that, at large $t$, even for moderately large fluctuations to the right of the mean, the flux distribution for the RTP and the diffusive case coincide, once one identifies the effective diffusion constant as $D_{\\rm eff} = v_0^2\/(2\\gamma)$. \n\n\n\n\n\n\\section{Numerical Results}\\label{numerics}\n\n\nThis section is dedicated to Monte Carlo simulations. We check numerically the analytical results previously obtained and characterize the properties of the physical realizations corresponding to large $Q$ values. \n\n\\subsubsection{The Importance Sampling strategy}\\label{IS-strategy}\n\n\n \\noindent In order to obtain the tails of $P_{an}(Q,t)$ and $P_{qu}(Q,t)$ we have employed Importance Sampling, a general method used to reduce the variance of observables whose expectation value is dominated by rare realizations, in this case rare trajectories. In the context of the evaluation of large deviation functions, very popular implementations of importance sampling ideas are cloning algorithms or transition path sampling \\cite{giardina2011simulating}.\n Here we use an implementation of Importance Sampling, similar to transition path sampling, the details of the technique can be found in \\cite{hartmann-epjb-2011, alberto-importance}.\nBasically we sample realizations with an exponential\nbias on their flux, $e^{-\\theta Q}$. The adjustable parameter $\\theta$ allows to explore atypical realizations:\n a negative $\\theta$ favours realizations with large $Q$, while a positive $\\theta$ favours small $Q$. The sampling is done using a standard Metropolis algorithm as discussed in \\cite{hartmann-epjb-2011, alberto-importance} and error bars are smaller than the symbol size.\n\n\nTo proceed we note that the flux depends only on the particle positions at time $t$ :\n\\begin{equation}\\label{dis-diff}\n x_i(t) = x_i(0) + \\Delta x_i(t), \\quad \\forall i \\;.\n\\end{equation}\nWe wrote this quantity as the sum of two contributions: (i) the initial (negative) position,\n $x_i(0) <0$ and (ii) the total displacement, $\\Delta x_i(t)$. The latter depends on the stochastic process we are considerning: for the diffusive particles it is a Gaussian number of zero mean and standard deviation $\\sqrt{2 D t}$.\nFor active particles, it can be expressed as \n\\begin{equation}\\label{rtp-pos-velo}\n \\Delta x_i(t) =\\pm v_0 (T_1-T_2),\n\\end{equation}\nwhere $T_1$, is the total time spent moving in the initial particle direction, $T_2=t-T_1$ the time spent in the opposite direction.\nThe signs $+$ or $-$ correspond to the initial direction and they are chosen with equal probability. The times $T_1$ and $T_2$ are determined as follows: the run times $\\tau_1, \\tau_2,\\ldots, \\tau_n$ are drawn from an exponential distribution of rate $\\gamma$, the last run being the first time interval for which $\\sum_{i=1}^{n} \\tau_i > t$. Then $\\tau_n$ is replaced by $t-\\sum_{i=1}^{n-1} \\tau_i$ and $T_1, T_2$ are computed.\n\nThe choice of the initial conditions is the delicate point that makes the annealed case different from the quenched one: in the annealed case, averages are performed over all initial conditions while in the quenched case the initial condition is fixed and typical. We first study the annealed case for active particles. At large times, their behavior is statistically equivalent to the one of diffusive particles with the effective constant $D_{\\text{eff}}=v_0^2\/(2 \\gamma)$. Then we discuss the quenched case and recover the exact results of previous sections. At variance with the annealed case one expects that when $Q \\simeq \\rho v_0 t $ active particles should have a clear non-diffusive nature even at long times. Unfortunately, the biased Monte Carlo used in this paper does not allow to sample configurations with $Q \\simeq \\rho v_0 t$.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{snapshot-active-diffusion-annealed-initial-histo-new.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{winners-displacement-annealed-new.eps}\n\\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n\\includegraphics[width=\\hsize]{mcs-psiann-ac-new-com.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Monte Carlo results for the annealed case. RTP versus Diffusion. The initial particle position is sampled with $\\rho=1$ and $L=2000$. Particles evolve up to time $t=400$ with $v_0=1, \\gamma=0.05$ for RTP (points), $D_{\\text{eff}}=10$ for diffusion (lines). We performed $10^7$ realizations.\n (a) Initial particle concentration. For typical value of $Q$ (violet circles\/black solid lines) the profile is flat, as expected. It has been obtained by setting $\\theta =0$ as importance sampling parameter which corresponds to $Q \\simeq \\langle Q \\rangle \\approx 35.45$. For large values of $Q$ (red squares\/yellow dashed lines) we observe an accumulation of particle close to origin. There we set $\\theta =-1$ as importance sampling parameter which corresponds to $ Q \\approx 93 \\gg \\langle Q \\rangle$.\n(b) Histogram (normalized to the number of realizations) of the displacement for particles with a positive final position with $\\theta=0$, and $\\theta=-1$. \n (c) $P_{an}(Q,t)$ vs $Q$; points represent Monte Carlo results (with $L=500$ (red squares) and $L=2000$ (green circles)) while the dashed line is the Poisson distribution with mean $\\mu=35.4573$ (Eq.~\\ref{ac-mut}).\nIn the tails the agreement improves with increasing $L$. \\label{annealed-ac-ld-causes}} \n\\end{figure}\n\n\n\n\\subsection{Annealed case} \n\n \n \\noindent The flux of RTPs depends on the evolution of the particles with an initial position $x(0)$ in $[-v_0 t, 0]$. But what is their number?\n Typically we expect $N \\sim \\rho v_0 t$, but, in the annealed case, rare realizations with large or small $N$ can occur. To capture them\nwe consider a large interval $[-L,0]$ with $L \\gg \\rho v_0 t$ and draw $L \\rho$ initial positions evenly distributed in the interval. Then only the partciles with $x(0) > -v_0 t$ are evolved.\n \n \\noindent Here we study RTP with $\\rho=1, v_0=1, \\gamma=0.05$. At time $t=400$, the average flux predicted by Eq.~\\ref{ac-mut} is $\\mu=35.4573$, very close to the one predicted in diffusive limit $\\sim \\sqrt{t\/(2 \\gamma \\pi)} =35.6825$. Then typical realizations ($\\theta=0$) are expected to be similar to the diffusive ones with $D_\\text{eff}=v_0^2\/(2 \\gamma)=10$. When the bias is applied ($\\theta =1$) the sampled realizations have a larger flux, $Q \\approx 93 $ and it is instructive to characterize their statistical properties for RTP and diffusion.\n \n \n\\noindent In Fig.~\\ref{annealed-ac-ld-causes}(a) we show the initial profile of particles. While for $\\theta=0$ it is flat as expected, for $\\theta=-1$ it displays an accumulation of particles around the origin and total number of particles in the interval $[-v_0 t, 0]$ is much larger than $\\rho v_0 t$. In Fig.~\\ref{annealed-ac-ld-causes}(b) we show the histogram (normalized to the number of realizations) of displacement for particles with a positive final position. The peak for $\\theta=0$ has the same location of the peak for $\\theta =-1$. The only difference is that more particles have a positive $x(t)$ for $\\theta =-1$ than for $\\theta=0$.\nThis suggests that, in the annealed case, larger values of the flux are essentially due to rare fluctuations of the initial conditions that are completly insensitive to the nature of the particle motion. \n\n \n \n\n\\noindent For this reason the non-gaussian tails of the flux are Poissonian both for diffusive and active particles. In Fig.~\\ref{annealed-ac-ld-causes}(c) \\footnote{Note that to obtain a reliable histogram one has to re-weight the sampled values of $Q$ by a factor $Z(\\theta) e^{\\theta Q}$. $Z(\\theta)$ is a normalization constant that is determined following the method explained in \\cite{hartmann-epjb-2011, alberto-importance}. } we observe that the agreement between simulations and Poissonian tails increases when $L$ is large, this confirms that the origin of anomalous fluctuations of the flux is in the rare realizations with large initial concentrations of particles close to the origin. This mechanism cannot work for the quenched case where the initial concentration is always flat.\n\n \n \n \n\n\\subsection{Quenched case} \nIn the quenched case the initial condition is fixed and the number of particles with an initial position in $[-v_0 t, 0]$ is always $N =\\rho v_0 t$.\nIn practice, we fix the position of the first particle\n$x_1$ using a uniform random number between $0$ and $-1\/(2 \\rho)$ and the positions of all the other $N-1$ particles are then slaved to $x_1$ according to\n\\begin{equation}\\label{eqp-inc}\n x_i (0) = x_1 (0) - i,\n\\end{equation}\n\nWe first check the agreement between our Monte Carlo simulations and the analytical predictions.\nIn Fig.~\\ref{diffusion-quenched-plots} we compare\nthe exact large deviation function $\\Psi_{\\rm diff}(\\frac{Q}{\\rho\\sqrt{D t}})$ and the exact probability distribution function $P_{qu}(Q,t)|_{\\rm diff}$ with the ones obtained from our Monte Carlo simulations. \nFor the quenched active case, the results are shown in Fig~\\ref{active-quenched-final-plots} where we also plot the large deviation function $\\Psi_{\\rm RTP}(\\frac{Q}{\\rho v_0 t})$, using for the Monte Carlo data the definition $\\Psi_{\\rm RTP}(q) = -\\frac{{\\rm log} P_{\\rm qu}(Q,t)}{\\rho v_0 \\gamma t^2}$ with $q=\\frac{Q}{\\rho v_0 t}$. We note that the Monte Carlo results perfectly match the predictions for both RTP and diffusion.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mcs-fq-com2.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{prob-diff-quen.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Quenched case for passive dynamics. Monte Carlo simulations (red points) and exact results obtained from Mathematica (dashed black line) using Eqs.~\\ref{phi-1eq} and \\ref{fq-bas}. Simulations were perfomed using $\\rho=1, D=0.2, t=10^4, N=2000$.\n(a) $\\Psi_{\\rm diff}(q)$ vs $q$.\n(b) $P_{qu}(Q,t)|_{\\rm diff}$ vs $Q$. The two functions are related via $P_{qu}(Q,t)|_{\\rm diff}=\\exp[-\\rho \\sqrt{D t} \\Psi_{\\rm diff}(\\frac{Q}{\\rho\\sqrt{D t}})] $.\n }\n\\label{diffusion-quenched-plots}\n\\end{figure}\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{mathm-analytics-mcs-t400et100active.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{unscaled-active-prob-t400-norm.eps}\n\\end{minipage}\n\\end{center}\n\\caption{Quenched case for active dynamics. Monte Carlo simulations (points) and exact results obtained from Mathematica (dashed black line) using Eqs.~\\ref{Uzt-exact} and~\\ref{def_tildeI}. Simulations were perfomed using $\\rho=v_0=1, \\gamma=0.5$ and $t=100$ (green squares) and $t=400$ (red circles). (a) $\\Psi_{\\rm RTP}(q)$ vs $q$. The violet solid curve represents $\\frac{q^3}{6}$, indicating that $t=400$ is also not large enough to see the ${q^3}$ behavior; also see Fig.~\\ref{phi-full-plot}(c) which shows that Eq.~\\ref{rtp-ldf-model} becomes a good fit to the exact results only at very large times.\n(b) $P_{qu}(Q,t)|_{\\rm RTP}$ vs $Q$ for $t=400$ in the semi-log scale. Note that $P_{qu}(Q,t)|_{\\rm RTP}$ decays slower than the Gaussian expected for the typical fluctuations (green solid line). The two functions are related via\n$P_{qu}(Q,t)|_{\\rm RTP} = \\exp [-\\rho v_0 \\gamma t^2 \\Psi_{\\rm RTP}(\\frac{Q}{\\rho v_0 t})]$.}\\label{active-quenched-final-plots}\n\\end{figure}\n\n\n\n\n\n\nThe difference between RTP and diffusive behaviour in the quenched case is shown is Fig.~\\ref{comp-rtp-diff-gmt40} (a). There we compare the exact distribution of RTP with $\\rho=v_0=1, \\gamma=0.5$ and $t=80$ (red) with the diffusive one with $D_\\text{eff}=v_0^2\/(2 \\gamma)=1$.\n Both distributions deviate from the Gaussian (solid green) but they become separable from each other only at extremely small probabilities, when $Q \\simeq 60 \\approx \\rho v_0 t$. The Importance Sampling strategy enables us to explore the non-Guassian tails of the distribution but not the extremely rare configurations where the finerprints of the activity are present. Indeed in Fig.~\\ref{comp-rtp-diff-gmt40} (b) we can hardly see a difference in histogram of the displacement of particles with positive final position between diffusion and RTP, both for $\\theta =0$ and $\\theta=-3$. This is because the displacements involved are still very small compare to $v_0t =80$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{quenched-diffusion-rtp-Deff1-t80-gmt40.eps}\n\\end{minipage}\n\\begin{minipage}{0.4\\hsize}\n\\includegraphics[width=\\hsize]{winners-displacement-quenched.eps}\n\\end{minipage}\n\\end{center}\n\\caption{ (a) $P_{\\rm qu}(Q,t)$ vs $Q$. RTPs with $\\rho=v_0=1, \\gamma=0.5$ and $t=80$ (long-dashed red lines, triangles) are compared with the diffusive ones with $D_\\text{eff}=v_0^2\/(2 \\gamma)=1$ (short-dashed black lines, circles). Exact results (lines) and Monte Carlo simulations (points). The solid green line represents the Gaussian valid at small $Q$.\n(b) Histogram (normalized to the number of realizations) of the displacement for particles with a positive final position with $\\theta=0$ (violet circles for RTP\/ black -solid lines for diffusion), and $\\theta=-3$ (red squares for RTP\/ yellow-dashed lines for diffusion).}\n\\label{comp-rtp-diff-gmt40}\n\\end{figure}\n\n\n\\section{Conclusion}\\label{conclu}\n\n\nIn this paper, we have presented a general framework to study current fluctuations for non-interacting particles executing a common random dynamics in one dimension and starting from a step initial condition. The probability distribution $P(Q,t, \\{x_i\\})$ depends on the initial positions of the particles $\\{x_i\\}<0$. The initial positions are distributed uniformly on the negative axis with a uniform density $\\rho$. There are two different ways to perform the average over the initial positions, namely (i) annealed and (ii) quenched averages, in analogy with disordered systems: here the initial condition plays the role of the disorder. In the annealed case, the distribution $P(Q,t,\\{x_i\\})$ is averaged directly over the initial positions. In contrast, in the quenched case, one considers the configurations of $x_i$'s that lead to the most likely current distribution (i.e., the {\\it typical} current distribution). In both cases, we have shown that, for noninteracting particles, the distribution can be fully characterized in terms of the single particle Green's function, which in general will depend on the dynamics of the particles. In this article, we have focused mostly on two different dynamics: a) when the single particle undergoes simple diffusion and b) when the single particle undergoes run-and-tumble dynamics (RTP). \n\nFor the annealed case, we have shown that $P_{\\rm an}(Q,t)$, at all times, is given by a Poisson distribution, with parameter $\\mu(t)$ given by the exact formula in Eq.~(\\ref{muan-mod-def}). We provide exact formula for $\\mu(t)$ in the RTP case (for the diffusive case this was known already\nfrom Ref. \\cite{derrida-gers}). For the quenched diffusive case we show that our formalism correctly recovers the large deviation result obtained in Ref. \\cite{derrida-gers}) using a different approach. For the RTP case, we showed that there is a new large deviation regime with $Q \\sim t$, where $P_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho \\, v_0 \\, \\gamma \\, t^2 \\, \\Psi_{\\rm RTP}\\left(\\frac{Q}{\\rho \\, v_0 \\,t}\\right)\\right]$. One of the main results of this paper is an explicit computation of the rate function $\\Psi_{\\rm RTP}(q)$ given by\n\\begin{equation}\\label{rtp-ldf-model_conclusion}\n\\Psi_{\\rm RTP}(q)=q-\\frac{q}{2}\\sqrt{1-q^2}-{\\rm sin}^{-1}\\left[ \\sqrt{\\frac{1-\\sqrt{1-q^2}}{2}} \\right]\\;, \\quad 0 \\leq q \\leq 1 \\;.\n\\end{equation}\n\n\nOur method gives access to another physical observable, namely the probability of an extremely rare event that \nthere is no particle on the right side of the origin at time $t$. We have shown that this is just the probability of having\nzero flux up to time $t$, i.e. $P(Q=0,t)$, both for the annealed and the quenched case. For the annealed case, this is just\n$P_{\\rm an}(Q=0,t) = e^{-\\mu(t)}$. For the quenched case, we have that, both for the diffusive and RTP cases, this probability \ndecays at late times as a stretched exponential $P_{\\rm qu}(Q=0,t) \\sim e^{-\\bar{\\alpha} \\sqrt{D_{\\rm eff}\\,t}}$, where we computed the constant $\\bar{\\alpha} = 0.675336\\ldots$ analytically [see Eq. (\\ref{C})]. For diffusive particles, $D_{\\rm eff} = D$ while for RTP's, $D_{\\rm eff} = v_0^2\/(2 \\gamma)$. \n\nWe have also verified our analytical predictions by numerical simulations. Computing numerically the large deviation function is far from trivial. Even for the diffusive case the large deviation function predicted for $P(Q,t)$ (both annealed and quenched) in Ref. \\cite{derrida-gers} was\nnever verified numerically. In this paper, we used a sophisticated importance sampling method to compute numerically this large deviation \nfunction in the diffusive case up to an impressive accuracy of order $10^{-200}$. We further used the same technique to compute the large deviation function in the RTP case. \n\nThe formalism developed in this paper can be easily generalized in different directions. For instance, one can compute the flux distribution exactly for the case where there are, initially, arbitrary densities $\\rho_{\\rm left}$ and $\\rho_{\\rm right}$ to the left and to the right of the origin respectively, both the diffusive and for the RTP cases. One could also generalise this result in higher dimensions, with step-like initial conditions, where for instance one region of the space is initially occupied by particles with uniform density. For the diffusive case, the flux distribution in the presence of hard-core repulsions between particles was studied in Ref. \\cite{derrida-gers-sep} (for the simple symmetric exclusion process). It would be interesting to see whether our formalism can be generalized to study the flux distribution for RTP's with hard core repulsions. \n\n\n\n\\begin{acknowledgments}\nWe acknowledge support from the project 5604-2 of the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}