vanishing of vacuum states and blow up phenomena of the compressible navier stokes equations the navier stokes systems for compressible fluids with density dependent viscosities are considered in the present paper these equations in particular include the ones which are rigorously derived recently as the saint venant system for the motion of shallow water from the navier stokes system for incompressible flows with a moving free surface 14 these compressible systems are degenerate when vacuum state appears we study initial boundary value problems for such systems for both bounded spatial domains or periodic domains the dynamics of weak solutions and vacuum states are investigated rigorously

on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

weak and strong solutions for the incompressible navier stokes equations with damping in this paper we show that the cauchy problem of the navier stokes equations with damping u 1u 0 has global weak solutions for any 1 global strong solution for any 7 2 and that the strong solution is unique for any 7 2 5

nonlinear stability of large amplitude viscous shock wave for general viscous gas abstract in the present paper it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic navier stokes equations in which the pressure could be general and includes law and the viscosity coefficient is a smooth function of density the strength of shock wave could be arbitrarily large the proof is given by introducing a new variable which can formulate the original system into a new one and the elementary energy method introduced in 21

projection methods for nonlinear boundary value problems where l d l c x y is a linear map n x y is nonlinear and x and y are banach spaces recently cesari and mckenna s proved that if the underlying spaces possess a schauder basis or more generally they have what the authors call property l it is theoretically possible to establish a finite dimensional alternative problem for 1 3 their paper unities previous results 12 131 that established conditions under which 1 3 could be reduced to finite 282 0022 0396 85 3 00

global existence of weak solutions to 3d compressible primitive equations with degenerate viscosity in this paper we investigate the compressible primitive equations cpes with density dependent viscosity for large initial data the cpe model can be derived from the 3d compressible and anisotropic navier stokes equations by hydrostatic approximation motivated by the work of vasseur and yu siam j math anal 48 1489 1511 2016 invent math 206 935 974 2016 in which the global existence of weak solutions to the compressible navier stokes equations with degenerate viscosity was obtained we construct approximate solutions and prove the global existence of weak solutions to the cpe in this paper in our proof we first present the vertical velocity as a function of density and horizontal velocity which plays a role in using the faedo galerkin method to obtain the global existence of the approximate solutions then we obtain the key estimates of lower bound of the density the bresch desjardins entropy on the approximate solutions finally we apply compactness arguments to obtain global existence of weak solutions by vanishing the parameters in our approximate system step by step

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation abstract a short time existence theorem is proven for the initial boundary value problem for a class of quasilinear symmetric hyperbolic systems containing a constant coefficient spatial operator multiplied by a large parameter solutions u are shown to remain bounded independently of for a time independent of provided that u 0 and u l 0 are bounded independently of and certain structural conditions are satisfied this result is applied to the quasigeostrophic approximation of the shallow water equations and the constant pressure approximation in combustion and solutions for these cases are shown to converge to solutions of limiting equations as

cauchy problem for viscous rotating shallow water equations abstract we consider the cauchy problem for a viscous compressible rotating shallow water system with a third order surface tension term involved derived recently in the modeling of motions for shallow water with free surface in a rotating sub domain marche 2007 19 the global existence of the solution in the space of besov type is shown for initial data close to a constant equilibrium state away from the vacuum unlike the previous analysis about the compressible fluid model without coriolis forces see for instance danchin 2000 10 haspot 2009 16 the rotating effect causes a coupling between two parts of hodge s decomposition of the velocity vector field and additional regularity is required in order to carry out the friedrichs regularization and compactness arguments

asymptotics of the 1d compressible navier stokes equations with density dependent viscosity abstract we are concerned with the time asymptotic behavior toward rarefaction waves for strong non vacuum solutions to the cauchy problem of the one dimensional compressible navier stokes equations with degenerate density dependent viscosity the case when the pressure p and the viscosity coefficient for some parameters r is considered for 0 max 1 if the initial data is assumed to be sufficiently regular without vacuum and mass concentrations we show that the cauchy problem of the one dimensional compressible navier stokes equations admits a unique global strong non vacuum solution which tends to the rarefaction waves as time goes to infinity here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large the proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform in time positive lower and upper bounds on the specific volume

dynamical behaviors for 1d compressible navier stokes equations with density dependent viscosity abstract the dynamical behaviors of vacuum states for one dimensional compressible navier stokes equations with density dependent viscosity coefficient are considered it is first shown that a unique strong solution to the free boundary value problem exists globally in time the free boundary expands outwards at an algebraic rate in time and the density is strictly positive in any finite time but decays pointwise to zero time asymptotically then it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum the solution is piecewise regular and contains a piece of continuous vacuum before the time t 0 which is compressed at an algebraic rate and vanishes at the time t meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially

on one dimensional compressible navier stokes equations with degenerate viscosity and constant state at far fields in this paper we are concerned with the cauchy problem for one dimensional compressible isentropic navier stokes equations with density dependent viscosity 0 and pressure p 1 we will establish the global existence and asymptotic behavior of weak solutions for any 0 and 1 under the assumption that the density function keeps a constant state at far fields in particular in the case that 0 12 we obtain the large time behavior of the strong solution obtained by mellet and vasseur when the solution has a lower bound no vacuum

global well posedness and long time dynamics for a higher order quasi geostrophic type equation abstract in this paper we study a higher order viscous quasi geostrophic type equation this equation was derived in 11 as the limit dynamics of a singularly perturbed navier stokes korteweg system with coriolis force when the mach rossby and weber numbers go to zero at the same rate the scope of the present paper is twofold first of all we investigate well posedness of such a model on the whole space r 2 we prove that it is well posed in h s for any s 3 globally in time interestingly enough we show that this equation owns two levels of energy estimates for which one gets existence and uniqueness of weak solutions with different regularities namely h 3 and h 4 regularities this fact can be viewed as a remainder of the so called bd entropy structure of the original system in the second part of the paper we investigate the long time behavior of these solutions we show that they converge to the solution of the corresponding linear parabolic type equation with same initial datum and external force our proof is based on dispersive estimates both for the solutions to the linear and non linear problems

normal numbers and finite automata we give an elementary and direct proof of the following theorem a real number is normal to a given integer base if and only if its expansion in that base is incompressible by lossless finite state compressors these are finite automata augmented with an output transition function such that the automata input output behaviour is injective they are also known as injective finite state transducers as a corollary we obtain v n agafonov s theorem on the preservation of normality on subsequences selected by finite automata

effective velocity in compressible navier stokes equations with third order derivatives abstract a formulation of certain barotropic compressible navier stokes equations with third order derivatives as a viscous euler system is proposed by using an effective velocity variable the equations model for instance viscous korteweg or quantum navier stokes flows the formulation in the new variable allows for the derivation of an entropy identity which is known as the bd bresch desjardins entropy equation as a consequence of this estimate a new global in time existence result for the one dimensional quantum navier stokes equations with strictly positive particle densities is proved

finite state dimension classical hausdorff dimension sometimes called fractal dimension was recently effectivized using gales betting strategies that generalize martingales thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary infinite sequences in this paper we use gales computed by multi account finite state gamblers to develop the finite state dimensions of sets of binary sequences and individual binary sequences the theorem of eggleston quart j math oxford ser 20 1949 31 36 relating hausdorff dimension to entropy is shown to hold for finite state dimension both in the space of all sequences and in the space of all rational sequences binary expansions of rational numbers every rational sequence has finite state dimension 0 but every rational number in 0 1 is the finite state dimension of a sequence in the low level complexity class ac0 our main theorem shows that the finite state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information lossless finite state compressors

on the uniqueness of strong solution to the incompressible navier stokes equations with damping abstract in this paper we show that the cauchy problem of the incompressible navier stokes equations with damping u 1 u 0 has global strong solution for any 3 and the strong solution is unique when 3 5 this improves earlier results

existence of global weak solutions for 3d degenerate compressible navier stokes equations in this paper we prove the existence of global weak solutions for 3d compressible navier stokes equations with degenerate viscosity the method is based on the bresch and desjardins commun math phys 238 211 223 2003 entropy conservation the main contribution of this paper is to derive the mellet and vasseur commun partial differ equ 32 431 452 2007 type inequality for weak solutions even if it is not verified by the first level of approximation this provides existence of global solutions in time for the compressible barotropic navier stokes equations the result holds for any gamma 1 in two dimensional space and for 1 gamma 3 in three dimensional space in both case with large initial data possibly vanishing on the vacuum this solves an open problem proposed by lions mathematical topics in fluid mechanics vol 2 compressible models 1998

global well posedness of the cauchy problem of two dimensional compressible navier stokes equations in weighted spaces abstract in this paper we study the global well posedness of classical solution to 2d cauchy problem of the compressible navier stokes equations with large initial data and vacuum it is proved that if the shear viscosity is a positive constant and the bulk viscosity is the power function of the density that is with 3 then the 2d cauchy problem of the compressible navier stokes equations on the whole space r 2 admits a unique global classical solution u which may contain vacuums in an open set of r 2 note that the initial data can be arbitrarily large to contain vacuum states various weighted estimates of the density and velocity are obtained in this paper and these self contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow

entropy rates and finite state dimension the effective fractal dimensions at the polynomial space level and above can all be equivalently defined as the c entropy rate where c is the class of languages corresponding to the level of effectivization for example pspace dimension is equivalent to the pspace entropy rate at lower levels of complexity the equivalence proofs break down in the polynomial time case the p entropy rate is a lower bound on the p dimension equality seems unlikely but separating the p entropy rate from p dimension would require proving p np we show that at the finite state level the opposite of the polynomial time case happens the reg entropy rate is an upper bound on the finite state dimension we also use the finite state genericity of ambos spies and busse automatic forcing and genericity on the diagonalization strength of finit automata in proc fourth int conf on discrete mathematics and theoretical computer science 2003 springer berlin pp 97 108 to separate finite state dimension from the reg entropy rate however we point out that a block entropy rate characterization of finite state dimension follows from the work of ziv and lempel compression of individual sequences via variable rate coding ieee trans inform theory 24 1978 530 536 on finite state compressibility and the compressibility characterization of finite state dimension by dai et al finite state dimension theoret comput sci 310 1 3 2004 1 33 as applications of the reg entropy rate upper bound and the block entropy rate characterization we prove that every regular language has finite state dimension 0 and that normality is equivalent to finite state dimension 1

existence of global strong solutions in critical spaces for barotropic viscous fluids this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n 2 we address the question of the global existence of strong solutions for initial data close to a constant state having critical besov regularity first this article shows the recent results of charve and danchin arch ration mech anal 198 1 233 271 2010 and chen et al commun pure appl math 63 1173 1224 2010 with a new proof our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity which allows us to cancel out the coupling between the density and the velocity as in haspot well posedness in critical spaces for barotropic viscous fluids 2009 second we improve the results of charve and danchin 2010 and chen et al 2010 by adding as in charve and danchin 2010 some regularity on the initial data in low frequencies in this case we obtain global strong solutions for a class of large initial data which rely on the results of hoff arch rational mech anal 139 303 354 1997 hoff commun pure appl math 55 11 1365 1407 2002 and hoff j math fluid mech 7 3 315 338 2005 and those of charve and danchin 2010 and chen et al 2010 we conclude by generalizing these results for general viscosity coefficients

global existence of weak solutions to the compressible quantum navier stokes equations with degenerate viscosity we study the compressible quantum navier stokes qns equations with degenerate viscosity in the three dimensional periodic domains on the one hand we consider qns with additional damping terms motivated by the recent works of li and xin e print arxiv 1504 06826 and p antonelli and s spirito arch ration mech anal 225 1161 1199 2017 we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the bd entropy estimate using this system we obtain the global existence of weak solutions to the compressible qns equations with damping terms for large initial data moreover we obtain some new a priori estimates which can avoid using the assumption that the gradient of the velocity is a well defined function which was indeed used directly in the work of vasseur and yu siam j math anal 48 1489 1511 2016 invent math 206 935 974 2016 on the other hand in the absence of damping terms we also prove the global existence of weak solutions to the compressible qns equations without the lower bound assumption on the dispersive coefficient which improves the previous result of p antonelli and s spirito arch ration mech anal 225 1161 1199 2017

ekman layers of rotating fluids the case of general initial data in this paper we describe the weak solutions of the navier stokes equations with a large coriolis term as the rossby and the ekman numbers go to zero in a special domain with various boundary conditions this work extends the results in the case of well prepared initial data in 19 30 2000 john wiley sons inc

asymptotic behavior of a compressible two phase model with well formation interaction abstract in this work we consider a compressible gas liquid model with a well reservoir interaction term that is relevant for coupled wellbore reservoir flow systems involved in e g drilling operations main focus is on deriving estimates that are independent of time under suitable conditions on the well reservoir interaction term we obtain such estimates which allow prediction of the long time behavior of the gas and liquid masses moreover we also obtain a quantification of the convergence rates as a function of time and gain some insight into the role played by the rate characterizing how fast the well reservoir interaction must die out the model is investigated in a free boundary setting where the initial mass is a mixture of both phases i e no single phase zone exists

a multidimensional model for the combustion of compressible fluids we consider a multidimensional model for the combustion of compressible reacting fluids the flow is governed by the navier stokes in eulerian coordinates and the chemical reaction is irreversible and is governed by the arrhenius kinetics the existence of globally defined weak solutions is established by using weak convergence methods compactness and interpolation arguments in the spirit of feireisl 16 and p l lions 24

well posedness in critical spaces for the system of compressible navier stokes in larger spaces this paper is dedicated to the study of viscous compressible barotropic fluids in dimension n geq2 we address the question of well posedness for it large data having critical besov regularity our result improves the analysis of r danchin in cite dw of chen et al in cite c2 and of the author in cite h1 h2 inasmuch as we may take initial density in b nn p 1 with 1 leq p infty our result relies on a new a priori estimate for the velocity where we introduce a new unknown called textit effective velocity to weaken one the coupling between the density and the velocity in particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density

global strong solutions to radial symmetric compressible navier stokes equations with free boundary in this paper we consider the two dimensional barotropic compressible navier stokes equations with stress free boundary condition imposed on the free surface as the viscosity coefficients satisfies 2 1 we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary in particular we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive and the free boundary propagates along the particle path and expand outwards at an algebraic rate

normality and two way automata we prove that two way transducers both deterministic and non deterministic cannot compress normal numbers to achieve this we first show that it is possible to generalize compressibility from one way transducers to two way transducers these results extend a known result normal infinite words are exactly those that cannot be compressed by lossless finite state transducers and more generally by bounded to one non deterministic finite state transducers we also argue that such a generalization cannot be extended to two way transducers with unbounded memory even in the simple form of a single counter

two level mixed finite element methods for the navier stokes equations with damping abstract in this paper the mixed finite element methods are considered for the navier stokes equations with damping optimal error estimates of the h 1 norm and l 2 norm for the velocity and the l 2 norm for the pressure are obtained moreover two kinds of two level methods based on the oseen correction and newton correction are presented finally two numerical examples are implemented to confirm the analysis

lagrange structure and dynamics for solutions to the spherically symmetric compressible navier stokes equations the compressible navier stokes system cns with density dependent viscosity coefficients is considered in multi dimension the prototype of the system is the viscous saint venat model for the motion of shallow water a spherically symmetric weak solution to the free boundary value problem for cns with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path which is continuous away from the symmetry center detailed regularity and lagrangian structure of this solution have been obtained in particular it is shown that the particle path is uniquely defined starting from any non vacuum region away from the symmetry center along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved starting from any non vacuum point at a later on time a particle path is also uniquely defined backward in time which either reaches at some initial non vacuum point or stops at a small middle time and connects continuously with vacuum in addition the free boundary is shown to expand outward at an algebraic rate in time and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up this finally leads to the formation of vacuum state almost everywhere as the time goes to infinity

on the dynamics of navier stokes equations for a shallow water model in this paper we study a free boundary problem of one dimensional compressible navier stokes equations with a density dependent viscosity which include in particular a shallow water model under some suitable assumptions on the initial data we obtain the global existence uniqueness and the large time behavior of weak solutions in particular it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations

phragmen lindelof alternative results for the shallow water equations for transient compressible viscous flow abstract in this paper the spatial behavior of the viscous shallow water equations in a two dimensional channel domain is studied a differential inequality for suitable energy associated with the solutions of the viscous shallow water equations in a semi infinite channel is derived from which we show the phragmen lindelof alternative results the main tool used is the energy method our results can be viewed as a version of saint venant s principle to the transient compressible viscous flow

on temporal asymptotics for the pth power viscous reactive gas abstract in this paper we investigate the long time behaviour of solutions to the system governing a heat conductive viscous reactive pth power gas confined between two parallel plates for initial boundary value problems with the end points held at a prescribed temperature or insulated we prove the global existence of physically relevant solutions and establish their rate of convergence to equilibria for generic initial data the estimates for different boundary conditions are presented in a unified manner

global existence of weak solutions for the multicomponent reaction flows abstract a multidimensional model is introduced for the dynamic reversible reaction of compressible multicomponent gaseous mixture the purpose of this work is to develop a complete existence result in the case where the pressure depends only on the mixture composition and the barotropic correction without any additional assumption the existence of globally weak solutions of the navier stokes for multicomponent reaction flows is established by using weak convergence methods in the spirit of mellet vasseur 14 and li xin 12

superclose and superconvergence of finite element discretizations for the stokes equations with damping a class of conforming mixed finite element methods fems are applied to the stokes equations with damping with the help of the prior estimates for the exact and approximated solutions and the appropriate choice of the parameters a n and r appeared in the problem the superclose and superconvergence results for the velocity in h 1 norm and the pressure in l 2 norm are derived to our best knowledge this is the first superconvergence analysis of conforming mixed fems for this kind nonlinear problem

global weak solutions to 3d compressible navier stokes poisson equations with density dependent viscosity abstract in this paper we study the global existence of weak solutions to the compressible navier stokes poisson n s p equations with density dependent viscosity and non monotone pressure in a three dimensional torus our approach is based on the faedo galerkin method and the compactness arguments motivated by vasseur yu 29 and 30 we construct the approximate solutions and the key estimates ling in the elementary energy estimates b d entropy and mellet vasseur type inequality for the weak solutions here we need the conditions that the adiabatic constant satisfies 4 3 3 for 1 or 1 3 for 1 where is a sign constant of poisson equation which determines the physical meaning of the n s p system

finite energy method for compressible fluids the navier stokes korteweg model this is the first of a series of papers devoted to the initial value problem for the one dimensional euler system of compressible fluids and augmented versions containing higher order terms in the present paper we are interested in dispersive shock waves and analyze the zero viscosity capillarity limit associated with the navier stokes korteweg system specifically we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the euler system our method of proof combines energy and effective energy estimates a nonlinear sobolev inequality high integrability properties for the mass density and for the velocity and compactness properties based on entropies 2015 wiley periodicals inc

analysis of an incompressible navier stokes maxwell stefan system the maxwell stefan equations for the molar fluxes supplemented by the incompressible navier stokes equations governing the fluid velocity dynamics are analyzed in bounded domains with no flux boundary conditions the system models the dynamics of a multicomponent gaseous mixture under isothermal conditions the global in time existence of bounded weak solutions to the strongly coupled model and their exponential decay to the homogeneous steady state are proved the mathematical difficulties are due to the singular maxwell stefan diffusion matrix the cross diffusion terms and the different molar masses of the fluid components the key idea of the proof is the use of a new entropy functional and entropy variables which allows for a proof of positive lower and upper bounds of the mass densities without the use of a maximum principle

on the compactness of the reduced gravity two and a half layer equations abstract we consider the reduced gravity two and a half model in oceanic fluid dynamics and prove the stability of weak solutions in periodic domain t 2 the proof is based on the uniform a priori estimates and the method of weak convergence and the limit is carried out with the help of a new higher regularity estimate of the velocity which is derived by constructing a special test function

asymptotic behavior of weak solutions to the damped navier stokes equations abstract whether or not weak solutions of classical navier stokes equations decay to zero in l 2 as time tends to infinity was posed by leray in his pioneering paper 12 13 and has been well solved see 15 for instance this paper examines the three dimensional navier stokes equations with damping term u 1 u and proves that the weak solutions decay to zero in l 2 as time tends to infinity for any 1 by developing local in space estimate for weak solutions

global existence for the dirichlet problem for the viscous shallow water equations abstract a global existence and uniqueness theorem of strong solutions for the initial boundary value problem with dirichlet boundary conditions is established for small forcing and small initial data an exponentialc0decay rate is also established and the solution is shown to be classical fort 0 the positivity of the fluid heighthfort 0 is also proved

global stability of weak solutions for a multilayer saint venant model with interactions between the layers in this paper we investigate the existence of global weak solutions for the multilayer model introduced by audusse et al which is related to incompressible free surface flows we prove the global stability of weak solutions over the torus we observe that this model admits the so called bd entropy and a gain of integrability on the velocity in the spirit of the work of mellet and vasseur it allows us to obtain enough compactness estimates in order to show the stability of global weak solutions

global solutions to a model for exothermically reacting compressible flows with large discontinuous initial data we prove the global existence of solutions of the navier stokes equations describing the dynamic combustion of a compressible exothermically reacting fluid and we study the large time behavior of solutions giving necessary and sufficient conditions for complete combustion in certain cases the adiabatic constants and specific heats of the burned product and unburned reactant fluids may differ and the initial data may be large and discontinuous

two grid mfeas for the incompressible stokes type variational inequality with damping abstract in this paper we study two grid mixed finite element algorithms mfeas for the stationary incompressible stokes equations with damping and nonlinear slip boundary conditions we set up the weak formulation and obtain the well posedness then employing the mixed finite element method some optimal error estimates between the exact solution and the approximation solution are derived two kinds of two grid mfeas based on the stokes correction and newton type correction are established some error estimates for two grid mfeas are obtained several numerical results are presented to confirm our theoretical findings

pressureless euler with nonlocal interactions as a singular limit of degenerate navier stokes system abstract we show that weak solutions of degenerate navier stokes equations converge to the strong solutions of the pressureless euler system with linear drag term newtonian repulsion and quadratic confinement the proof is based on the relative entropy method using the artificial velocity formulation for the one dimensional navier stokes system

highly rotating viscous compressible fluids in presence of capillarity effects we study here a singular limit problem for a navier stokes korteweg system with coriolis force in the domain mathbb r 2 times 0 1 and for general ill prepared initial data taking the mach and the rossby numbers proportional to a small parameter varepsilon rightarrow 0 we perform the incompressible and high rotation limits simultaneously moreover we consider both the constant and vanishing capillarity regimes in this last case the limit problem is identified as a 2 d incompressible navier stokes equation in the variables orthogonal to the rotation axis if the capillarity is constant instead the limit equation slightly changes keeping however a similar structure due to the presence of an additional surface tension term in the vanishing capillarity regime various rates at which the capillarity coefficient goes to 0 are considered in general this produces an anisotropic scaling in the system the proof of the results is based on suitable applications of the rage theorem combined with microlocal symmetrization arguments

asymptotic behavior of compressible navier stokes equations with density dependent viscosity and vacuum in this paper we study the one dimensional navier stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density precisely when the viscosity coefficient is proportional to with 0 where is the density we give the asymptotic behavior and the decay rate of the density function x t furthermore the behavior of the density function x t near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied the analysis is based on some new mathematical techniques and some new useful estimates this fills a final gap on studying navier stokes equations with the viscosity coefficient dependent on the density

chemically reacting mixtures in terms of degenerated parabolic setting the paper analyzes basic mathematical questions for a model of chemically reacting mixtures we derive a model of several finite component compressible gas taking rigorously into account the thermodynamical regime mathematical description of the model leads to a degenerate parabolic equation with hyperbolic deviation the thermodynamics implies that the diffusion terms are non symmetric not positively defined and cross diffusion effects must be strongly marked the mathematical goal is to establish the existence of weak solutions globally in time for arbitrary number of reacting species a key point is an entropy like estimate showing possible renormalization of the system

mac finite difference scheme for stokes equations with damping on non uniform grids abstract in this paper we consider the numerical methods for stationary stokes equations with damping the mark and cell mac method has been applied to discretize the problem on non uniform grids we establish the lbb condition and the stability for the mac scheme then we obtain the second order super convergence in l2 norm for both velocity and pressure on non uniform grids we also obtain the second order convergence for some terms of h1 norm of the velocity and the other terms of h1 norm are second order convergence on uniform grids numerical experiments using the mac scheme show agreement of the numerical results with theoretical analysis

global existence of strong solution for viscous shallow water system with large initial data on the irrotational part abstract we are interested in studying the cauchy problem for the viscous shallow water system in dimension n 2 we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations more precisely our smallness assumption on the initial data is supercritical for the scaling of the equations it allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension n 2 to do this we introduce the notion of quasi solutions which consists in solving the pressureless viscous shallow water system we can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density this smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the quasi solutions indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations to finish we prove the existence of global strong solution with large initial data when n 2 provided that the mach number is sufficiently large

asymptotic limit for rotational quantum compressible navier stokes equations with multiple scales abstract in this paper we consider the degenerate quantum compressible navier stokes equations giving rise to a variety of mathematical problems in many areas we study the asymptotic limit for the rotational compressible navier stokes equations with quantum term and the ill prepared initial data

steady compressible navier stokes fourier system for monoatomic gas and its generalizations we consider steady compressible navier stokes fourier system for a gas with pressure p and internal energy e related by the constitutive law p i 1 e 1 we show that for any 3 there exists a variational entropy solution i e solution satisfying the weak formulation of balance of mass and momentum entropy inequality and global balance of total energy this result includes the model for monoatomic gas 5 if 5 these solutions also fulfill the weak formulation of the pointwise total energy balance

the linearized 2d inviscid shallow water equations in a rectangle boundary conditions and well posedness we consider the linearized 2d inviscid shallow water equations in a rectangle a set of boundary conditions is proposed which make these equations well posed several different cases occur depending on the relative values of the reference velocities u 0 v 0 and reference height phi 0 sub or super critical flow at each part of the boundary

existence of global weak solution for a reduced gravity two and a half layer model we investigate the existence of global weak solution to a reduced gravity two and a half layer model in one dimensional bounded spatial domain or periodic domain also we show that any possible vacuum state has to vanish within finite time then the weak solution becomes a unique strong one

on the steady compressible navier stokes fourier system we study the motion of the steady compressible heat conducting viscous fluid in a bounded three dimensional domain governed by the compressible navier stokes fourier system our main result is the existence of a weak solution to these equations for arbitrarily large data a key element of the proof is a special approximation of the original system guaranteeing pointwise uniform boundedness of the density as well as the positiveness of the temperature therefore the passage to the limit omits tedious technical tricks required by the standard theory basic estimates on the solutions are possible to obtain by a suitable choice of physically reasonable boundary conditions

a mixed fem for navier stokes type variational inequality with nonlinear damping term abstract in this work we consider a mixed finite element method fem for incompressible navier stokes equations with nonlinear damping term and friction boundary conditions after establishing the variational formulation we show the well posedness of this problem subsequently we focus our attention on the mixed fem and analyze the galerkin approximation to this system then some optimal error estimates are deduced in the end some iterative algorithms are presented and numerical results are given to verify the theoretical analysis

lp estimates for quantities advected by a compressible flow abstract we consider the evolution of a quantity advected by a compressible flow and subject to diffusion when this quantity is scalar it can be for instance the temperature of the flow or the concentration of some pollutants because of the diffusion term one expects the equations to have a regularizing effect however in their euler form the equations describe the evolution of the quantity multiplied by the density of the flow the parabolic structure is thus degenerate near vacuum when the density vanishes in this paper we show that we can nevertheless derive uniform l p bounds that do not depend on the density in particular the bounds do not degenerate near vacuum furthermore the result holds even when the density is only a measure we investigate both the scalar and the system case in the former case we obtain l bounds in the latter case the quantity being investigated could be the velocity field in compressible navier stokes type of equations and we derive uniform l p bounds for some p depending on the ratio between the two viscosity coefficients the main additional difficulty in that case being to deal with the second viscosity term involving the divergence of the velocity such estimates are to our knowledge new and interesting since they are uniform with respect to the density the proof relies mostly on a method introduced by de giorgi to obtain regularity results for elliptic equations with discontinuous diffusion coefficients

preservation of normality by transducers abstract we consider input deterministic finite state transducers with infinite inputs and infinite outputs and we consider the property of borel normality on infinite words when these transducers are given by a strongly connected set of states and when the input is a borel normal sequence the output is an infinite word such that every word has a frequency given by a weighted automaton over the rationals we prove that there is an algorithm that decides in cubic time whether an input deterministic transducer preserves normality

global existence of weak solution for quantum navier stokes poisson equations in this paper we consider the compressible quantum navier stokes poisson equations with a linear density dependent viscosity by the use of a singular pressure close to vacuum we prove the global in time existence of weak solutions in a three dimensional torus for large data in the sense of distribution

well posedness for the viscous rotating shallow water equations with friction terms we consider the cauchy problem for viscous rotating shallow water equations with friction terms the global existence of the solution in some hybrid spaces is shown for the initial data close to a constant equilibrium state away from the vacuum

an alternative method for boundary value problems with large nonlinearities where g r r is continuously differentiable and b and d are nonsingular 12 x n matrices historically lyapunov schmidt and the alternative method are able to reduce the problem to a finite dimensional one provided the norm of the nonlinear term is sufficiently small in the region of interest that is the weakly nonlinear case recently the finite dimensional reduction for ordinary differential equations with periodic boundary conditions was accomplished 161 in this paper a constructive technique is developed that allows us to reduce 1 1 to a finite dimensional problem

on the flow of chemically reacting gaseous mixture abstract we consider the cauchy problem for the system of equations governing flow of isothermal reactive mixture of compressible gases our main contribution is to prove sequential stability of weak solutions when the state equation essentially depends on the species concentration and the viscosity coefficients vanish on vacuum moreover under additional assumption on the cold component of the pressure in the regions of small density we prove the existence of weak solutions for arbitrary large initial data

analysis of an inviscid zero mach number system in endpoint besov spaces for finite energy initial data abstract the present paper is the continuation of work 18 devoted to the study of an inviscid zero mach number system in the framework of endpoint besov spaces of type b r s r d r 1 d 2 which can be embedded in the lipschitz class c 0 1 in particular the largest case b 1 1 and the case of holder spaces c 1 are taken into account the local in time well posedness of this system is proved under an additional finite energy hypothesis on the initial data the key to get this result is new a priori estimates for parabolic equations with variable coefficients in endpoint spaces b r s r d which are of independent interest in the special case of space dimension d 2 we are able to give a lower bound for the lifespan such that the solutions tend to be globally defined when the initial inhomogeneity is small there we will show refined a priori estimates in endpoint besov spaces for transport equations

viscosity effect on the degenerate lake equations abstract this paper concerns the effect of viscosity on the degenerate lake equations anelastic limit when the bottom topography vanishes on the shore we establish the existence and uniqueness of a global weak solutions for various choices of viscosity term in weighted sobolev spaces where the weight is assumed to be a power type weight our solution is constructed by adapting carefully the known works on the incompressible navier stokes equations in unweighted context finally we study in one case the vanishing viscosity limit when the solution of the inviscid lake equations is regular enough generalizing the results by martino et al 2001 to the degenerate case

an asymptotic limit of a navier stokes system with capillary effects a combined incompressible and vanishing capillarity limit in the barotropic compressible navier stokes equations for smooth solutions is proved the equations are considered on the two dimensional torus with well prepared initial data the momentum equation contains a rotational term originating from a coriolis force a general korteweg type tensor modeling capillary effects and a density dependent viscosity the limiting model is the viscous quasi geostrophic equation for the rotated velocity potential the proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms

on the isothermal compressible multi component mixture flow the local existence and maximal lp lq regularity of solutions abstract we consider the initial boundary value problem for the system of equations describing the flow of compressible isothermal mixture of arbitrary large number of components the system consists of the compressible navier stokes equations and a subsystem of diffusion equations for the species the subsystems are coupled by the form of the pressure and the strong cross diffusion effects in the diffusion fluxes of the species assuming the existence of solutions to the symmetrized and linearized equations proven in piasecki shibata and zatorska 2019 we derive the estimates for the nonlinear equations and prove the local in time existence and maximal l p l q regularity of solutions

stabilized low order finite elements for stokes equations with damping abstract in this paper we consider the stationary stokes equations with damping low order mixed finite element spaces are used to approximate the velocity and the pressure and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf sup condition the stability of this method is proved and the optimal order error estimates are derived by some nonlinear analysis techniques at last two numerical examples are implemented to test the stability and effectiveness of the proposed method