existence and features of similarity solutions for non boussinesq gravity currents in this paper the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions the current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density unlike earlier investigators we assume that the ambient fluid exerts no significant resisting action on the current and therefore the flow depth is expected to drop to zero at the front in the absence of friction in this context the shallow water equations are highly appropriate for computing the mean velocity and flow depth of the current the boundary condition imposed at the front leads to technical mathematical difficulties indeed unlike in the boussinesq case no regular solution to the shallow water equations satisfies the downstream condition but when the flow is supercritical at the channel inlet it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution which represents the head behavior to better understand this result and make sure that the result is physically relevant we consider the navier stokes equations within the high reynolds number limit approximate similarity solutions can be worked out which support our earlier analysis on the shallow water equations while the flow body is self similar and weakly rotational the head is not self similar but tends toward a self similarity shape at long times it is characterized by a strong vorticity a straight free surface and a nonuniform velocity profile which becomes flatter and flatter with time

unified approach to classical speed limit and thermodynamic uncertainty relation the total entropy production quantifies the extent of irreversibility in thermodynamic systems which is nonnegative for any feasible dynamics when additional information such as the initial and final states or moments of an observable is available it is known that tighter lower bounds on the entropy production exist according to the classical speed limits and the thermodynamic uncertainty relations here we obtain a universal lower bound on the total entropy production in terms of probability distributions of an observable in the time forward and backward processes for a particular case we show that our universal relation reduces to a classical speed limit imposing a constraint on the speed of the system s evolution in terms of the hatano sasa entropy production notably the obtained classical speed limit is tighter than the previously reported bound by a constant factor moreover we demonstrate that a generalized thermodynamic uncertainty relation can be derived from another particular case of the universal relation our uncertainty relation holds for systems with time reversal symmetry breaking and recovers several existing bounds our approach provides a unified perspective on two closely related classes of inequality classical speed limits and thermodynamic uncertainty relations

on some third order nonlinear boundary value problems existence location and multiplicity results abstract we prove an ambrosetti prodi type result for the third order fully nonlinear equation u t f t u t u t u t s p t with f 0 1 r 3 r and p 0 1 r continuous functions s r under several two point separated boundary conditions from a nagumo type growth condition an a priori estimate on u is obtained an existence and location result will be proved by degree theory for s r such that there exist lower and upper solutions the location part can be used to prove the existence of positive solutions if a non negative lower solution is considered the existence nonexistence and multiplicity of solutions will be discussed as s varies

new dissipated energy and partial regularity for thin film equations abstract we study the thin film equation h t h n h x x x x and prove that the energy h p h x 2 d x dissipates for suitable weak solutions when n p lies in the whole domain e defined by laugesen which extends partially the corresponding results of laugesen chugunova and taranets and then improve some previous results of partial regularity by chou and the first author

a level set approach for simulations of flows with multiple moving contact lines with hysteresis the level set method of sussman et al j comput phys 148 1999 81 is extended such that flows with multiple moving contact lines can be simulated accounting for inertia a relation between contact line speed and contact angle slip and contact line hysteresis the convergence properties of the method are investigated with particular attention on the resolution of the contact line stress singularity results are compared with a lubrication theory for spreading droplets

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

reverse draining of a magnetic soap film analysis and simulation of a thin film equation with non uniform forcing we analyze and classify equilibrium solutions of the one dimensional thin film equation with no flux boundary conditions and in the presence of a spatially dependent external forcing we prove theorems that shed light on the nature of these equilibrium solutions guarantee their validity and describe how they depend on the properties of the external forcing we then apply these results to the reverse draining of a one dimensional magnetic soap film subject to an external non uniform magnetic field numerical simulations illustrate the convergence of the solutions towards equilibrium configurations we then present bifurcation diagrams for steady state solutions we find that multiple stable equilibrium solutions exist for fixed parameters and uncover a rich bifurcation structure to these solutions demonstrating the complexity hidden in a relatively simple looking evolution equation finally we provide a simulation describing how numerical solutions traverse the bifurcation diagram as the amplitude of the forcing is slowly increased and then decreased

phase plane analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution a nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall here we discuss the physical processes of a thin liquid film evolution the classical thin film equation is being derived and the series solution is obtained by employing the adomian decomposition method adm in this paper a new way for the phase planes analysis and traveling wave solution by adm is presented the proposed algorithm is implemented using mathematical software mathematica 6 0

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

higher order equations related to thin films blow up and global existence the influence of the initial data we are interested in the following class of equations ksdv ht hhxx 12 hx 2 xx hx 3h x hx 6 h t periodic on 1 1 t 0 h 0 h0 with r 0 0 when 0 the model was established by j r king j r king two generalizations of the thin film equation math comput modelling 34 2001 737 756 here we show that if the initial data h0 0 23 then any admissible weak local solution h is necessarily nonnegative moreover there is no global weak solution on r of ksdv 0 and the blow up time must occur before t0 2h 0h02 h 0 2 provided that h0 is nonconstant h 0 is the average of h0 over 1 1 on the other hand if h0 0 then we have a value c 56 such that if c for all t 0 there is a nonpositive global weak solution h on 0 t being in particular in l2 0 t hper2 1 1 and if c we show that there exists a weak solution if is greater than 1 2 moreover adapting the energy method used by bernis f bernis finite speed of propagation and continuity of the interface for thin viscous flows adv differential equations 1 3 1996 337 368 in the case 1 and 0 we can show that weak solutions have a finite speed of propagation when 0 0 the model was established by spencer davis and voorhees b j spencer s h davis p w voorhees morphological instability in epitaxially strained dislocation free solid films nonlinear evolution appl math technical report 9201 dept of engineering sci and appl math mccormick school of eng and appl sci northwestern university evanston il 60208 september 1992 if 0 h0 0 then we show that there exists a global solution h 0 provided that h0 belongs to a certain class of functions if h0 0 blow up should occur in this case we show this fact under the dirichlet boundary conditions for the weak solution of ksdv 00 the blow up time tmax 1 13 ln 1 1 h0 1 t assuming h0 1 1 limt tmax 0t h l1 1 1 6d

on the determination of the steady film profile for a non newtonian thin droplet abstract a shooting method is used to determine a solution to a third order ode modeling the steady profile of a non newtonian thin droplet we compare a direct approach to an iterative approach using a secant method we obtain a nonlinear relationship between the contact angle and the position of the contact line r from this nonlinear relationship we use curve fitting to obtain an empirical law of the form tan r f k where k is the power law coefficient and f is a nonlinear function of k

cubic splines method for a system of third order boundary value problems we use uniform cubic polynomial splines to develop a numerical method for computing approximations to the solution of a system of third order boundary value problem associated with odd order obstacle problems such problems arise in physical oceanography and can be studied in the framework of variational inequality theory the convergence analysis of the new method is studied and an upper bound for the error is derived a numerical example is given to illustrate the efficiency of the new method

traveling wave solutions to thin film equations thin films can be effectively described by the lubrication approximation in which the equation of motion is mathit h mathit t mathit h mathit n mathit h mathit x mathit x mathit x mathit x 0 here h is a necessarily positive quantity which represents the height or thickness of the film different values of n especially 1 2 and 3 correspond to different physical situations this equation permits solutions in the form of traveling disturbances with a fixed form if u is the propagation velocity the resulting equation for the disturbance is mathit uh mathit x mathit h mathit n mathit h mathit x mathit x mathit x mathit x here quantitative and qualitative solutions to the equation are presented the study has been limited to the intervals in x where the solutions are positive it is found that transitions between different qualitative behaviors occur at n 3 2 3 2 and 1 2 for example if u is not zero solitonlike solutions defined on a finite interval are only possible for n3 more specific results can be obtained in the case in which the velocity is zero solitons occur for n2 for n 1 the region 3 2n is characterized by the presence of advancing front solutions with support on ensuremath infty t for ng1 2 single minimum solutions diverging at ifmmode pm else textpm fi ensuremath infty are possible the generic solution present for all positive values of n is a receding front which diverges at finite x for n0

a two dimensional similarity solution for capillary driven flows this work describes a two dimensional similarity solution of a viscous capillary flow in a thin film the solution describes two confronted contact lines which are being gradually welded by a point that advances at constant velocity along the common contact line the liquid has a partial wetting contact angle with respect to the substrate it is found that the capillary number is proportional to the cube of the contact angle the proportionality factor being found with high accuracy an important result is that the curvature diverges at the welding point thus showing that a singular point may exist in a capillary driven flow also the solution is immune to the contact line paradox since the energy dissipation rate is bounded

on uniqueness of weak solutions for the thin film equation abstract in any number of space variables we study the cauchy problem related to the fourth order degenerate diffusion equation s h h h 0 this equation derived from a lubrication approximation also models the surface tension dominated flow of a thin viscous film in the hele shaw cell our focus is on uniqueness of weak solutions in the complete wetting regime when a zero contact angle between liquid and solid is imposed in this case we transform the problem by zooming into the free boundary and look at small lipschitz perturbations of a quadratic stationary solution in the perturbational setting the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide optimal conditions on the initial data under which a unique solution exists in fact this solution can be used to define a class of functions in which the original initial value problem has a unique weak solution moreover we show that the interface between empty and wetted regions is an analytic hypersurface in time and space

exact polynomial solutions to some nonlinear diffusion equations abstract a class of non group invariant exact solutions to a number of nonlinear diffusion equations is constructed expressed in the appropriate variables these solutions are polynomials in the spatial variable the method is mainly applicable to equations which may be written in an appropriate quadratically nonlinear form a number of equivalence transformations between various nonlinear diffusion equations are also derived

finite time generalization of the thermodynamic uncertainty relation for fluctuating currents in nonequilibrium steady states the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving we show that this relation holds not only for the long time limit of fluctuations as described by large deviation theory but also for fluctuations on arbitrary finite time scales this generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments for which infinite time scales are not accessible importantly often this finite time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long time limit we illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

time rescaling and asymptotic behavior of some fourth order degenerate diffusion equations in this paper we investigate the large times behavior of weak solutions to the fourth order degenerate parabolic equation ut u nuxxx x modeling the evolution of thin films in particular for all n 0 we prove exponential decay of u x t towards its mean value 1 u dx in l1 norm for long times and we give the explicit n dependent rate of decay the result is based on classical entropy estimates and on detailed lower bounds for the entropy production

instabilities in gravity driven flow of thin fluid films this paper presents theoretical computational and experimental aspects of the instability development in the flow of thin fluid films the theoretical part involves basic fluid me chanics and presents derivation of the thin film equation using lubrication approximation a simplified version of this equation is then analyzed analytically using linear stability analysis and also numerically the results are then compared directly to experiments the experimental part outlines the setup as well as data acquisition and analysis this immediate comparison to experiments is very useful for gaining better insight into the interpretation of various theoretical and computational results

fluctuation theorem uncertainty relation the fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations such as the second law of thermodynamics and the jarzynski equality recently the thermodynamic uncertainty relation was discovered which states that the fluctuation of observables is lower bounded by the entropy production in the present letter we derive a thermodynamic uncertainty relation from the fluctuation theorem we refer to the obtained relation as the fluctuation theorem uncertainty relation and it is valid for arbitrary dynamics stochastic as well as deterministic and for arbitrary antisymmetric observables for which a fluctuation theorem holds we apply the fluctuation theorem uncertainty relation to an overdamped langevin dynamics for an antisymmetric observable we demonstrate that the antisymmetric observable satisfies the fluctuation theorem uncertainty relation but does not satisfy the relation reported for current type observables in continuous time markov chains moreover we show that the fluctuation theorem uncertainty relation can handle systems controlled by time symmetric external protocols in which the lower bound is given by the work exerted on the systems

droplet spreading intermediate scaling law by pde methods we consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity the standard lubrication approximation leads to an evolution equation for the film height h that is ill posed when the spreading is limited by the no slip boundary condition at the liquid solid interface due to a singularity at the moving contact line the most common relaxation of the no slip boundary condition removes this singularity but introduces a new physical length scale the slippage length b it is believed that this microscopic length scale only enters logarithmically in the effective that is macroscopic spreading behavior in this paper we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b more precisely we prove a scaling law for the diameter of the apparent that is macroscopic support of the droplet in time this is an intermediate scaling law it takes an initial layer to forget the initial droplet shape whereas after a long time the droplet is so thin that its spreading is governed by the physics on the scale b our proof works by deriving suitable estimates for physically relevant integral quantities the free energy the length of the apparent support and their respective rates of change as opposed to matched asymptotic methods this pde approach closely mimics a simple heuristic argument based on the gradient flow structure 2002 john wiley sons inc

exact wkb analysis of a schrodinger equation with a merging triplet of two simple poles and one simple turning point ii its relevance to the mathieu equation and the legendre equation abstract we develop the exact wkb analysis of an m2p1t merging two simple poles and one simple turning point schrodinger equation in part ii using a wkb theoretic transformation to the algebraic mathieu equation constructed in part i we calculate the alien derivative of its borel transformed wkb solutions at each fixed singular point relevant to the simple poles through the analysis of borel transformed wkb solutions of the legendre equations in the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic mathieu equation and the legendre equation

non negative global weak solutions for a degenerated parabolic system approximating the two phase stokes problem abstract we establish the existence of non negative global weak solutions for a strongly coupled degenerated parabolic system which was obtained as an approximation of the two phase stokes problem driven solely by capillary forces moreover the system under consideration may be viewed as a two phase generalization of the classical thin film equation

on the exact wkb analysis of higher order simple pole type operators abstract higher order simple pole type operators that is higher order linear ordinary differential operators with a large parameter whose coefficients have simple poles at the origin are discussed from the viewpoint of the exact wkb analysis making use of the technique of microdifferential operators we clarify the singularity structure of the borel transform of their wkb solutions

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

optimal lower bounds on asymptotic support propagation rates for the thin film equation abstract we derive lower bounds on asymptotic support propagation rates for strong solutions of the cauchy problem for the thin film equation the bounds coincide up to a constant factor with the previously known upper bounds and thus are sharp our results hold in case of at most three spatial dimensions and n 1 2 92 the result is established using weighted backward entropy inequalities with singular weight functions to yield a differential inequality combined with some entropy production estimates the optimal rate of propagation is obtained to the best of our knowledge these are the first lower bounds on asymptotic support propagation rates for higher order nonnegativity preserving parabolic equations

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

on a fourth order degenerate parabolic equation in higher space dimensions we are concerned with the existence of non negative solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions ut div u n u 0 in 0 t u u 0 on 0 t and u x 0 u0 x in

estimating entropy production by machine learning of short time fluctuating currents thermodynamic uncertainty relations turs are the inequalities which give lower bounds on the entropy production rate using only the mean and the variance of fluctuating currents since the turs do not refer to the full details of the stochastic dynamics it would be promising to apply the turs for estimating the entropy production rate from a limited set of trajectory data corresponding to the dynamics here we investigate a theoretical framework for estimation of the entropy production rate using the turs along with machine learning techniques without prior knowledge of the parameters of the stochastic dynamics specifically we derive a tur for the short time region and prove that it can provide the exact value not only a lower bound of the entropy production rate for langevin dynamics if the observed current is optimally chosen this formulation naturally includes a generalization of the turs with the partial entropy production of subsystems under autonomous interaction which reveals the hierarchical structure of the estimation we then construct estimators on the basis of the short time tur and machine learning techniques such as the gradient ascent by performing numerical experiments we demonstrate that our learning protocol performs well even in nonlinear langevin dynamics we also discuss the case of markov jump processes where the exact estimation is shown to be impossible in general our result provides a platform that can be applied to a broad class of stochastic dynamics out of equilibrium including biological systems

existence and uniqueness results for third order nonlinear differential systems abstract sufficient conditions are given for the existence of solutions to third order boundary value problems for nonlinear differential systems some appropriate conditions are given to guarantee that the nagumo condition is satisfied by constructing appropriate a lower solution upper solution pair a concept that is defined in this paper the uniqueness result of the problem is also established the emphasis here is that the differential systems has nonlinear dependence on all over order derivatives and the boundary conditions are nonlinear

entropy production estimation with optimal current entropy production characterizes the thermodynamic irreversibility and reflects the amount of heat dissipated into the environment and free energy lost in nonequilibrium systems according to the thermodynamic uncertainty relation we propose a deterministic method to estimate the entropy production from a single trajectory of system states we explicitly and approximately compute an optimal current that yields the tightest lower bound using predetermined basis currents notably the obtained tightest lower bound is intimately related to the multidimensional thermodynamic uncertainty relation by proving the saturation of the thermodynamic uncertainty relation in the short time limit the exact estimate of the entropy production can be obtained for overdamped langevin systems irrespective of the underlying dynamics for markov jump processes because the attainability of the thermodynamic uncertainty relation is not theoretically ensured the proposed method provides the tightest lower bound for the entropy production when entropy production is the optimal current a more accurate estimate can be further obtained using the integral fluctuation theorem we illustrate the proposed method using three systems a four state markov chain a periodically driven particle and a multiple bead spring model the estimated results in all examples empirically verify the effectiveness and efficiency of the proposed method

thermodynamic uncertainty relation for arbitrary initial states the thermodynamic uncertainty relation tur describes a trade off relation between nonequilibrium currents and entropy production and serves as a fundamental principle of nonequilibrium thermodynamics however currently known turs presuppose either specific initial states or an infinite time average which severely limits the range of applicability here we derive a finite time tur valid for arbitrary initial states from the cram er rao inequality we find that the variance of an accumulated current is bounded from below by the instantaneous current at the final time which suggests that the boundary is constrained by the bulk we apply our results to feedback controlled processes and successfully explain a recent experiment which reports a violation of a modified tur with feedback control we also derive a tur that is linear in the total entropy production and valid for discrete time markov chains with nonsteady initial states the obtained bound exponentially improves the existing bounds in a discrete time regime

analysis of adomian decomposition applied to a third order ordinary differential equation from thin film flow abstract the adomian decomposition method is applied to the third order ordinary differential equation y y k obtained by considering a travelling wave solution admitted by a generalized thin film equation or for thin film flow down a vertical wall the adomian decomposition method leads to a power series approximation to the solution of y y k we show that the domain of convergence of the adomian decomposition solution is dependent on k we truncate the adomian decomposition solution at a suitable order to ensure that the truncated solution satisfies the contact line condition y 0 at some point x x we then determine the contact angle where tan d y d x at x x and plot the variation of contact angle with increasing k

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

sensory capacity an information theoretical measure of the performance of a sensor for a general sensory system following an external stochastic signal we introduce the sensory capacity this quantity characterizes the performance of a sensor sensory capacity is maximal if the instantaneous state of the sensor has as much information about a signal as the whole time series of the sensor we show that adding a memory to the sensor increases the sensory capacity this increase quantifies the improvement of the sensor with the addition of the memory our results are obtained with the framework of stochastic thermodynamics of bipartite systems which allows for the definition of an efficiency that relates the rate with which the sensor learns about the signal with the energy dissipated by the sensor which is given by the thermodynamic entropy production we demonstrate a general trade off between sensory capacity and efficiency if the sensory capacity is equal to its maximum 1 then the efficiency must be less than 1 2 as a physical realization of a sensor we consider a two component cellular network estimating a fluctuating external ligand concentration as signal this model leads to coupled linear langevin equations that allow us to obtain explicit analytical results

thin film equations with partial wetting energy existence of weak solutions abstract the capillarity driven evolution with slip of a thin liquid film over a dry surface is considered in the regime of partial wetting the focus is on the simplest model case of a constant non zero dynamic contact angle in the lubrication approximation for the analytical treatment of the corresponding free boundary problem a new strategy is proposed based on the introduction of an ad hoc class of disjoining pressures which tend to concentrate at triple junctions a first investigation of this approach yields the existence of weak solutions which satisfy the dissipation relation as an inequality and which are different from those with zero contact angle a heuristic argument is also presented in order to clarify the connection between contact angle and dissipation relation it shows that moving droplets which satisfy the dissipation relation as an equality are forced to have the prescribed contact angle

numerical solutions of third order system of boundary value problems we develop a two stage numerical method for computing approximate solutions of a system of third order boundary value problems associated with odd order obstacle problems such a problem arise in physical oceanography and can be studied in the framework of variational inequality theory we show that the present method is of order two and gives numerical results which are better than the other available results numerical example is presented to illustrate the applicability of the new method our results can be viewed as an important and significant refinement of the previously known results

non linear higher order boundary value problems describing thin viscous flows near edges abstract two boundary value problems for non linear higher order ordinary differential equations are analyzed which have been recently proposed in the modeling of steady and quasi steady thin viscous flows over a bounded solid substrate the first problem concerns steady states and consists of a third order ode for the height of the liquid the ode contains an unknown parameter the flux and the boundary conditions relate near the edges of the substrate the height and its second derivative to the flux itself for this problem non existence and non uniqueness results are proved depending on the behavior as the flux approaches zero of the height function the function which relates the height to the flux near the edge out of which the liquid flows the second problem concerns quasi steady states and consists of a fourth order ode for the suitably scaled height of the liquid non linear boundary conditions relate the height to the flux near the edges of the substrate for this problem the existence of a solution is proved for a suitable class of height functions

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

tightening the uncertainty principle for stochastic currents we connect two recent advances in the stochastic analysis of nonequilibrium systems the loose uncertainty principle for the currents which states that statistical errors are bounded by thermodynamic dissipation and the analysis of thermodynamic consistency of the currents in the light of symmetries employing the large deviation techniques presented by gingrich et al phys rev lett 116 120601 2016 prltao0031 900710 1103 physrevlett 116 120601 and pietzonka barato and seifert phys rev e 93 052145 2016 2470 004510 1103 physreve 93 052145 we provide a short proof of the loose uncertainty principle and prove a tighter uncertainty relation for a class of thermodynamically consistent currents j our bound involves a measure of partial entropy production that we interpret as the least amount of entropy that a system sustaining current j can possibly produce at a given steady state we provide a complete mathematical discussion of quadratic bounds which allows one to determine which are optimal and finally we argue that the relationship for the fano factor of the entropy production rate var mean 2 is the most significant realization of the loose bound we base our analysis both on the formalism of diffusions and of markov jump processes in the light of schnakenberg s cycle analysis

exact wkb analysis of a schrodinger equation with a merging triplet of two simple poles and one simple turning point i its wkb theoretic transformation to the mathieu equation abstract we develop the exact wkb analysis of an m2p1t merging two simple poles and one simple turning point schrodinger equation our emphasis is put on the analysis of the singularity structure of its borel transformed wkb solutions near fixed singular points relevant to the two simple poles contained in the potential of the equation in part i we focus our attention on the construction and analytic properties of a wkb theoretic transformation that transforms an m2p1t equation to an algebraic mathieu equation that transformation plays an important role in part ii 7 when we discuss the singularity structure of borel transformed wkb solutions of an m2p1t equation

non polynomial spline approach to the solution of a system of third order boundary value problems we use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate the applicability and efficiency of the new method

universal bounds on current fluctuations for current fluctuations in nonequilibrium steady states of markovian processes we derive four different universal bounds valid beyond the gaussian regime different variants of these bounds apply to either the entropy change or any individual current e g the rate of substrate consumption in a chemical reaction or the electron current in an electronic device the bounds vary with respect to their degree of universality and tightness a universal parabolic bound on the generating function of an arbitrary current depends solely on the average entropy production a second stronger bound requires knowledge both of the thermodynamic forces that drive the system and of the topology of the network of states these two bounds are conjectures based on extensive numerics an exponential bound that depends only on the average entropy production and the average number of transitions per time is rigorously proved this bound has no obvious relation to the parabolic bound but it is typically tighter further away from equilibrium an asymptotic bound that depends on the specific transition rates and becomes tight for large fluctuations is also derived this bound allows for the prediction of the asymptotic growth of the generating function even though our results are restricted to networks with a finite number of states we show that the parabolic bound is also valid for three paradigmatic examples of driven diffusive systems for which the generating function can be calculated using the additivity principle our bounds provide a general class of constraints for nonequilibrium systems

small and waiting time behavior of the thin film equation we consider the small time behavior of interfaces of zero contact angle solutions to the thin film equation for a certain class of initial data through asymptotic analyses we deduce a wide variety of behavior for the free boundary point these are supported by extensive numerical simulations

asymmetric travelling waves for the thin film equation abstract we study the existence uniqueness and asymptotic behaviour of travelling wave solutions to the thin film equation with non zero contact angle at one interface we obtain explicit lower and upper bounds for these solutions moreover we analyse in an appendix the stability of a related class of stationary travelling wave solutions

the asymptotic solution of a class of third order boundary value problems arising in the theory of thin film flows this paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third order boundary value problems which govern the motion of thin liquid films subject to viscous capillary and gravitational forces precise conditions specifying where and when the third order derivative terms in the differential equations can be neglected are derived and improved estimates for the actual solutions in terms of solutions of the lower order models are constructed the paper also contains a technique for replacing a third order problem with an asymptotically equivalent second order one that may have wider applicability

nonnegativity preserving convergent schemes for the thin film equation we present numerical schemes for fourth order degenerate parabolic equations that arise e g in lubrication theory for time evolution of thin films of viscous fluids we prove convergence and nonnegativity results in arbitrary space dimensions a proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting hence the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix furthermore by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

upper bounds on waiting times for the thin film equation the case of weak slippage we derive upper bounds on the waiting time of solutions to the thin film equation in the regime of weak slippage n in 2 frac 32 11 in particular we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary for n in 2 frac 32 11 our estimates are sharp for n 2 they are sharp up to a logarithmic correction term note that the case n 2 corresponds with a grain of salt to the assumption of the navier slip condition at the fluid solid interface we also obtain results in the regime of strong slippage n in 1 2 however in this regime we expect them not to be optimal our method is based on weighted backward entropy estimates hardy s inequality and singular weight functions we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long

inferring entropy production from short experiments we provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in nonequilibrium systems in the steady state from the measurements of arbitrary current fluctuations our results are built upon the finite time generalization of the thermodynamic uncertainty relation and require only very short time series data from experiments we illustrate our results with exact and numerical solutions for two colloidal heat engines

on nontrivial traveling waves in thin film flows including contact lines we discuss dynamics of thin liquid films spreading down an inclined plane the fronts of these films are known to be unstable with respect to formation of finger like and triangular patterns in this work we concentrate on one particular aspect of these flows and that is the existence of nonlinear traveling waves we find evidence for presence of these waves for all inclination angles less than 90 to understand better the relevant pattern formation mechanism we explore via numerical simulations the bifurcation structure of the stability diagram close to the critical wavenumber the recovered structure is consistent with the existence of a subcritical bifurcation we discuss the connection between the bifurcation diagram and the existence of nontrivial traveling waves

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

uncertainty relations in stochastic processes an information inequality approach the thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production in statistical inference theory information inequalities assert that it is infeasible for any estimator to achieve an error smaller than the prescribed bound inspired by the similarity between the thermodynamic uncertainty relation and the information inequalities we apply the latter to systems described by langevin equations and we derive the bound for the fluctuation of thermodynamic quantities when applying the cram er rao inequality the obtained inequality reduces to the fluctuation response inequality we find that the thermodynamic uncertainty relation is a particular case of the cram er rao inequality in which the fisher information is the total entropy production using the equality condition of the cram er rao inequality we find that the stochastic total entropy production is the only quantity that can attain equality in the thermodynamic uncertainty relation furthermore we apply the chapman robbins inequality and obtain a relation for the lower bound of the ratio between the variance and the sensitivity of systems in response to arbitrary perturbations

a family of fourth order q logarithmic equations we prove the existence of global in time weak nonnegative solutions to a family of nonlinear fourth order evolution equations parametrized by a real parameter q 0 1 which includes the well known thin film q 1 2 and the derrida lebowitz speer spohn dlss equation q 1 subject to periodic boundary conditions in one spatial dimension in contrast to the gradient flow approach in 25 our method relies on dissipation property of the corresponding entropy functionals tsallis entropies resulting in required a priori estimates and extends the existence result from 25 to a wider range of the family members namely to 0 q 1 2 generalized beckner type functional inequalities yield an exponential decay rate of relative entropies which in further implies the exponential stability in the l1 norm of the constant steady state finally we provide illustrative numerical examples supporting the analytical results

proof of the finite time thermodynamic uncertainty relation for steady state currents this paper presents the finite time version of the thermodynamic uncertainty relation for nonequilibrium steady states which sets a bound on the magnitude of current fluctuations the authors use a methodology that is similar to the proof for the long time limit but with large deviation theory applied to an ensemble of many copies

generic properties of stochastic entropy production we derive an ito stochastic differential equation for entropy production in nonequilibrium langevin processes introducing a random time transformation entropy production obeys a one dimensional drift diffusion equation independent of the underlying physical model this transformation allows us to identify generic properties of entropy production it also leads to an exact uncertainty equality relating the fano factor of entropy production and the fano factor of the random time which we also generalize to non steady state conditions

a positivity preserving numerical scheme for a nonlinear fourth order parabolic system noindent a positivity preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied the system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time the resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions which is proved by an exponential transformation of variables the stability of the scheme is shown and convergence is proved in one space dimension numerical results concerning the switching behavior of a resonant tunneling diode are presented

singular integral equations arising in draining and coating flows abstract we present a very general existence theorem for a nonlinear singular integral equation using the schauder tychonoff fixed point theorem our result automatically guarantees the existence of a positive solution to a problem arising in draining and coating flows

hybrid methods for direct integration of special third order ordinary differential equations abstract in this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations odes y f x y namely hybrid methods for solving third order odes directly hmtd using the theory of b series order of convergence of the hmtd methods is investigated the main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two step hybrid method a three stage explicit hmtd is constructed results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper

a numerical and asymptotic study of some third order ordinary differential equations relevant to draining and coating flows some draining or coating fluid flow problems in which surface tension forces are important can be described by third order ordinary differential equations accurate computations are provided here