rotational degeneracy of hyperbolic systems of conservation laws the purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy in this situation hyperbolicity is necessarily non strict so that the elementary waves have interesting patterns the discussion is centered around a theorem on existence and uniqueness of solutions of the riemann problem we give a unified presentation of examples from continuum mechanics

super diffusivity in a shear flow model from perpetual homogenization this paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy t d t y t dt y 0 0 and d 2 is a 2 times 2 skew symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales 12 21 h x 1 with h x 1 n 0 n h n x 1 r n where h n are smooth functions of period 1 h n 0 0 n and r n grow exponentially fast with n we can show that y t has an anomalous fast behavior y t 2 t 1 with 0 and obtain quantitative estimates on the anomaly using and developing the tools of homogenization

convergence of the fractional step lax friedrichs scheme and godunov scheme for the isentropic system of gas dynamics a convergence theorem of the fractional step lax friedrichs scheme and godunov scheme for an inhomogeneous system of isentropic gas dynamics 1 5 3 is established by using the framework of compensated compactness meanwhile a corresponding existence theorem of global solutions with large data containing the vacuum is obtained

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

diffusive limit of a tagged particle in asymmetric simple exclusion processes invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d 3 the method of proof is by martingale techniques which rely on the fact that symmetric random walks are transient in high dimensions c 2000 john wiley sons inc

rotationally invariant hyperbolic waves we use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves the equation can include weak dissipative dispersive or diffractive effects we give applications to equations from magnetohydrodynamics elasticity and viscoelasticity

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

on the superdiffusive behavior of passive tracer with a gaussian drift in the present article we consider a motion of a passive tracer particle whose trajectory satisfies the ito stochastic differential equation d x t v t x t dt sqrt 2 kappa d w t where w is a brownian motion v is a stationary gaussian random field with incompressible realizations independent of w and 0 we prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field the result is shown both for steady time independent and time dependent and markovian velocity fields in addition we provide explicit upper and lower bounds for the hurst exponent of the trajectory all previous rigorous results concerned explicitely solvable shear flows cases

global entropy solutions to exothermically reacting compressible euler equations abstract the global existence of entropy solutions is established for the compressible euler equations for one dimensional or plane wave flow of an ideal gas which undergoes a one step exothermic chemical reaction under arrhenius type kinetics we assume that the reaction rate is bounded away from zero and the total variation of the initial data is bounded by a parameter that grows arbitrarily large as the equation of state converges to that of an isothermal gas the heat released by the reaction causes the spatial total variation of the solution to increase however the increase in total variation is proved to be bounded in t 0 as a result of the uniform and exponential decay of the reactant to zero as t approaches infinity

invariant distributions and scaling limits for some diffusions in time varying random environments we consider a family of one dimensional diffusions in dynamical wiener mediums which are random perturbations of the ornstein uhlenbeck diffusion process we prove quenched and annealed convergences in distribution and under weigh ted total variation norms we find two kind of stationary probability measures which are either the standard normal distribution or a quasi invariant measure depending on the environment and which is naturally connected to a random dynamical system we apply these results to the study of a model of time inhomogeneous brox s diffusions which generalizes the diffusion studied by brox ann probab 14 4 1206 1218 1986 and those investigated by gradinaru and offret ann inst henri poincare probab stat 2011 we point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

extended divergence measure fields and the euler equations for gas dynamics a class of extended vector fields called extended divergence measure fields is analyzed these fields include vector fields in l p and vector valued radon measures whose divergences are radon measures such extended vector fields naturally arise in the study of the behavior of entropy solutions of the euler equations for gas dynamics and other nonlinear systems of conservation laws a new notion of normal traces over lipschitz deformable surfaces is developed under which a generalized gauss green theorem is established even for these extended fields an explicit formula is obtained to calculate the normal traces over any lipschitz deformable surface suitable for applications by using the neighborhood information of the fields near the surface and the level set function of the lipschitz deformation surfaces as an application we prove the uniqueness and stability of riemann solutions that may contain vacuum in the class of entropy solutions of the euler equations for gas dynamics

random walk in markovian environment we prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on z d we assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is markovian with strong spatial and temporal mixing properties

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

decay of periodic entropy solutions to degenerate nonlinear parabolic equations abstract under a precise nonlinearity diffusivity condition we establish the decay of space periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

diffusion in turbulence we prove long time diffusive behavior homogenization for convection diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix simple shear flow examples show that this result is sharp for flows that have stationary stream matrices

multiscale homogenization with bounded ratios and anomalous slow diffusion we show that the effective diffusivity matrix d v n for the heat operator t 1 2 rv n r in a periodic potential v n d p nd0 uk x rk obtained as a superposition of hlder continuous periodic potentials uk of period t d vd r d z d d 2 n uk 0 d 0 decays exponentially fast with the number of scales when the scale ratios rkc1 rk are bounded above and below from this we de duce the anomalous slow behavior for a brownian motion in a potential obtained as a superposition of an innite number of scales dyt d d t rv 1 yt dt

on superdiffusive behavior of a passive tracer in a random flow abstract in this note we consider a passive tracer model describing particle dispersion in a turbulent flow the trajectory of the particle is given by the solution of an ordinary differential equation x t f x t x 0 x 0 where f x is a divergence free random vector field that is spatially homogeneous and isotropic we show that trajectories of the tracer display superdiffusive behavior when the random velocity f x decorrelates at large distances but does it not rapidly but rather at some moderate rate the main tools used in the proofs are variational principles and tauberian type theorems

on homogenization of time dependent random flows we study a diffusion with a random time dependent drift we prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition this result generalizes the results of 13 7

asymptotic decay of besicovitch almost periodic entropy solutions to anisotropic degenerate parabolic hyperbolic equations abstract we prove the well posedness and the asymptotic decay to the mean value of besicovitch almost periodic solutions to nonlinear anisotropic degenerate parabolic hyperbolic equations

entropy solutions in l for the euler equations in nonlinear elastodynamics and related equations a compactness framework is established for approximate solutions to the euler equations in one dimensional nonlinear elastodynamics by identifying new properties of the lax entropies especially the higher order terms in the lax entropy expansions and by developing ways to employ these new properties in the method of compensated compactness then this framework is applied to establish the existence compactness and decay of entropy solutions in l for the euler equations in nonlinear elastodynamics with a more general stress strain relation than those for the previous existence results this compactness framework is further applied to solving the euler equations of conservation laws of mass momentum and energy for a class of thermoelastic media and the equations of motion of viscoelastic media with memory

asymptotic stability of riemann solutions for a class of multidimensional systems of conservation laws with viscosity we prove the asymptotic stability of two state nonplanar riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added the class considered here is those systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes in particular we obtain the uniqueness of the self similar l entropy solution of the two state nonplanar riemann problem the asymptotic stability to which the main result refers is in the sense of the convergence as t in lloc1 of the space of directions x t that is the solution u t x of the perturbed problem satisfies u t t r as t in lloc1 n where r is the self similar entropy solution of the corresponding two state nonplanar riemann problem

large time behavior and homogenization of solutions of two dimensional conservation laws we study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data under a very weak nonlinearity condition we prove that the solutions converge to constants as time goes to infinity even in one space dimension our results improve the earlier ones since we only require the fluxes to be nonlinear in a neighborhood of the mean value of the initial data we then use these results to study the homogenization problem for scalar conservation laws with oscillatory initial data

on the smallest singular value of multivariate vandermonde matrices with clustered nodes abstract we prove lower bounds for the smallest singular value of rectangular multivariate vandermonde matrices with nodes on the complex unit circle the nodes are off the grid groups of nodes cluster and the studied minimal singular value is bounded below by the product of inverted distances of a node to all other nodes in the specific cluster by using known and new upper bounds for the smallest singular value this completely settles the univariate case and pairs of nodes in the multivariate case both including reasonable sharp constants for larger clusters we show that the smallest singular value depends also on the geometric configuration within a cluster

compressible euler equations with general pressure law we study the hyperbolic system of euler equations for an isentropic compressible fluid governed by a general pressure law the existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new euler poisson darboux equation which is highly singular when the density of the fluid vanishes new properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy flux kernel are found

homogenization of a diffusion process in a divergence free random field on etudie le probleme d homogeneisation pour un processus de diffusion dans un champ de derive aleatoire independant de la divergence dans le cas particulier d un petit champ gaussien on derive un developpement asymptotique de la matrice de diffusion efficace en termes de sa densite spectrale

validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws nonlinear geometric optics with various frequencies for entropy so lutions only in l 1 of multidimensional scalar conservation laws is analyzed a new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling compactness homogenization and l 1 stability new multidimensional features are recognized especially including nonlinear propagations of oscillations with high frequencies the validity of nonlinear geometric optics for entropy solutions in l 1 of multidimensional scalar con servation laws is justified

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

asymptotic behavior of the solutions to the damped compressible euler equations with vacuum abstract the asymptotic behavior of solutions of the damped compressible euler equations is conjectured to obey to the famous porous media equations pmes the previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic in present paper we prove that the l entropy weak solution with vacuum obtained by the compensated compactness theory converges strongly in l loc p 1 p space to the unique similarity solution of the related pme as time goes to infinity

asymptotic stability of riemann solutions in bgk approximations to certain multidimensional systems of conservation laws abstract we prove the asymptotic stability of nonplanar two states riemann solutions in bgk approximations of a class of multidimensional systems of conservation laws the latter consists of systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes the asymptotic stability to which the main result refers is in the sense of the convergence as t in l loc 1 of the space of directions x t that is the solution z t x of the perturbed cauchy problem for the corresponding bgk system satisfies x z t t d r as t in l loc 1 r n where r is the self similar entropy solution of the two states nonplanar riemann problem for the system of conservation laws

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

zero relaxation and dissipation limits for hyperbolic conservation laws we are interested in hyperbolic systems of conservation laws with relaxation and dissipation particularly the zero relaxation limit such a limit is of interest in several physical situations including gas flow near thermo equilibrium kinetic theory with small mean free path and viscoelasticity with vanishing memory in this article we study hyperbolic systems of two conservation laws with relaxation for the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit for the marginally stable case where the equilibrium speed may equal one of the frozen speeds we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation our analysis includes the construction of suitable entropy pairs to derive energy estimates we need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems the theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit 1993 john wiley sons inc

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

decay of almost periodic solutions of conservation laws we consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables which are almost periodic in the space variables in a generalized sense introduced by stepanoff and wiener which extends the original one of h bohr we prove that if u x t is such a solution whose inclusion intervals at time t with respect to 0 satisfy l epsiv t t 0 as t and such that the scaling sequence u t x t u t x t t is pre compact as t in l loc 1 d 1 then u x t decays to its mean value which is independent of t as t the decay considered here is in l 1 loc of the variable x t which implies as we show that as t where m x denotes taking the mean value with respect to x in many cases we show that if the initial data are almost periodic in the generalized sense then so also are the solutions we also show in these cases how to reduce the condition on the growth of the inclusion intervals l t with t as t for fixed 0 to a condition on the growth of l 0 with as 0 which amounts to imposing restrictions only on the initial data we show with a simple example the existence of almost periodic non periodic functions whose inclusion intervals satisfy any prescribed growth condition as 0 the applications given here include inviscid and viscous scalar conservation laws in several space variables some inviscid systems in chromatography and isentropic gas dynamics as well as many viscous 2 2 systems such as those of nonlinear elasticity and eulerian isentropic gas dynamics with artificial viscosity among others in the case of the inviscid scalar equations and chromatography systems the class of initial data for which decay results are proved includes in particular the l generalized limit periodic functions our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

large time behavior of entropy solutions of conservation laws abstract we are concerned with the large time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws we present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions these approaches allow the solutions of arbitrarily large oscillation without a priori assumption on the ways from which the solutions come the relation between the large time behavior of entropy solutions and the uniqueness of riemann solutions leads to an extensive study of the uniqueness problem we use a direct method to show the large time behavior of large l solutions for a class of m m systems including a model in multicomponent chromatography we employ the uniqueness of riemann solutions and the convergence of self similar scaling sequence of solutions to show the asymptotic behavior of large bv solutions for the 3 3 system of euler equations in thermoelasticity these results indicate that the riemann solution is the unique attractor of large discontinuous entropy solutions whose initial data are l l 1 or bv l 1 perturbation of the riemann data for these systems these approaches also work for proving the large time behavior of approximate solutions to hyperbolic conservation laws

periodic solutions of the degasperis procesi equation well posedness and asymptotics abstract we prove the well posedness of periodic entropy discontinuous solutions for the degasperis procesi equation partly motivated by the bounded periodic solutions found by vakhnenko and parkes 21 we study the long time asymptotic behavior of periodic entropy solutions

stability results for scattered data interpolation by trigonometric polynomials a fast and reliable algorithm for the optimal interpolation of scattered data on the torus mathbb t d by multivariate trigonometric polynomials is presented the algorithm is based on a variant of the conjugate gradient method in combination with the fast fourier transforms for nonequispaced nodes the main result is that under mild assumptions the total complexity for solving the interpolation problem at m arbitrary nodes is of order cal o m log m this result is obtained by the use of localized trigonometric kernels where the localization is chosen in accordance with the spatial dimension d numerical examples show the efficiency of the new algorithm

convection diffusion equation with space time ergodic random flow we prove the homogenization of convection diffusion in a time dependent ergodic incompressible random flow which has a bounded stream matrix and a constant mean drift we also prove two variational formulas for the effective diffusivity as a consequence we obtain both upper and lower bounds on the effective diffusivity

an invariance principle for diffusion in turbulence we prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials the result generalizes to multiple particles motion driven by a common velocity field and independent molecular brownian motions

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

a liouville theorem for stationary and ergodic ensembles of parabolic systems a first order liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity furthermore the paper establishes almost surely an intrinsic large scale mathcal c 1 alpha regularity estimate for caloric functions

a martingale approach to homogenization of unbounded random flows we study the asymptotic behavior of brownian motion in steady unbounded incompressible random flows we prove an invariance principle for almost all realizations of random flows the key compactness result is obtained by moser s iterative scheme in pde theory

spatially periodic solutions in relativistic isentropic gas dynamics we consider the initial value problem with periodic initial data for the euler equations in relativistic isentropic gas dynamics for ideal polytropic gases which obey a constitutive equation relating pressure p and density p 2 with 1 0 1 depending on the initial bounds the solution decays in l loc 1 to its mean value as t

the vanishing viscosity method in one dimensional thermoelasticity the vanishing viscosity method is applied to the system of conservation laws of mass momentum and energy for a special class of one dimensional thermoelastic media that do not conduct heat two types of vanishing viscosity are considered newtonian and artificial in both cases accompanied by vanishing heat conductivity it is shown that in either case one can pass to the zero viscosity limit by the method of compensated compactness provided that velocity and pressure are uniformly bounded oscillations in the entropy field may propagate along the linearly degenerate characteristic field but do not affect the compactness of the velocity field or the pressure field a priori bounds on velocity and pressure are established albeit only for the case of artificial viscosity