Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations

In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.     

Incompressible limit of solutions of multidimensional steady compressible Euler equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompressibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.     

GLOBAL ENTROPY SOLUTIONS TO AN INHOMOGENEOUS ISENTROPIC COMPRESSIBLE EULER SYSTEM

In this article, we develop a new technique to prove the global existene of entropy solutions to an inhomogeneous isentropic compressible Euler equations through the compensated compactness and vanishing viscosity method. In particular, the entropy solutions are uniformly bounded independent of time.     

Vanishing viscosity limit of short wave–long wave interactions in planar magnetohydrodynamics

We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schrödinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an aurora-type phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters, to a solution of the limit decoupled system involving the compressible Euler equations and a nonlinear Schrödinger equation. The vanishing viscosity limit serves to justify the SW–LW interactions in the limit equations as, in this setting, the SW–LW interactions cannot be defined in a straightforward way, due to the possible occurrence of vacuum.     

A note on barotropic compressible quantum Navier-Stokes equations

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations has been proved very recently, by Jüngel (2009) [1], if the viscosity constant is smaller than the scaled Plank constant. This paper extends the results to the case that the viscosity constant equals the scaled Plank constant. By using a new estimate on the square root of the solution, apparently not available in [1], the semiclassical limit for the viscous quantum Euler equations (which are equivalent to the barotropic compressible quantum Navier-Stokes equations) can be performed; then the results of this paper are obtained easily.     

The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray–Hopf weak solutions for the Navier–Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely determined and equals the shear flow solution of the Euler equations corresponding to this initial data. This simple example suggests that, also in more general situations, the vanishing viscosity limit of the Navier–Stokes equations could serve as a uniqueness criterion for weak solutions of the Euler equations.     

On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media

We study a model describing a compressible and miscible displacement in a porous medium. It consists of a coupled system of nonlinear parabolic partial differential equations. Using nonclassical estimates and renormalization tools, we prove the existence of relevant weak solutions for the problem. This is the first existence result obtained for a transport model containing both the coupling due to the compressibility assumption and the coupling due to the concentration dependent viscosity.     

EXISTENCE OF WEAK SOLUTIONS OF 2－D EULER EQUATIONS WITH INITIAL VORTICITY ω_0∈E（log~＋L）~α（α＞0）

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Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

Entropies and weak solutions of the compressible isentropic Euler equations

In this Note, we study the system of isentropic Euler equations for compressible fluids, with a general equation of state. We establish the existence of the fundamental kernel that generates the family of weak entropies, and study its singularities. The kernel is the solution of an equation of Euler-Poisson-Darboux type, and its partial derivative with respect to the density variable tends to a Dirac measure as the density approaches zero. We prove a new reduction theorem for the Young measures associated with the compressible Euler system. From these results, we deduce the existence, compactness, and asymptotic decay of measurable and bounded entropy solutions.     

ON THE WELL-POSEDNESS OF WEAK SOLUTIONS

This is to comment on the well-posedness of weak solutions for the initial value problem for partial differential equations. In recent decades, and particularly in recent years, there have been substantial progresses on construction by convex integration for the study of non-uniqueness of solutions for incompressible Euler equations, and even for compressible Euler equations. This prompts the question of whether it is possible to give a sense of well-posedness, which is narrower than the canonical Hadamard sense, so that the evolutionary equations are well-posed. We give a brief and partial review of the related results and offer some thoughts on this fundamental topic.     

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

We prove that any weak space-time \(L^2\) vanishing viscosity limit of a sequence of strong solutions of Navier–Stokes equations in a bounded domain of \({\mathbb R}^2\) satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that \(t-a.e.\) weak \(L^2\) inviscid limits of solutions of 3D Navier–Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.     

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.     

Compactness principles in nonlinear operator approximation theory

This paper is concerned with the approximate solution of nonlinear operator equations in abstract settings and with applications to integral and differential equations. A given operator with certain continuity and compactness or inverse compactness properties is a suitable limit of a sequence of operators with analogous properties which hold uniformly or asymptotically. Both fixed point equations and inhomogeneous equations are treated. Solutions of approximate problems converge to solutions of the given problem. This is an appropriate type of set convergence when solutions are not unique.     

DIFFUSIVE—DISPERSIVE TRAVELING WAVES AND KINETIC RELATIONS IV. COMPRESSIBLE EULER EQUATIONS

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumptions and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals' equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The authors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

Zero Dissipation Limit to Rarefaction Waves for the 1-D Compressible Navier-Stokes Equations

The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investi...     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

The vanishing viscosity limit for some symmetric flows

The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier–Stokes and the Euler solutions. Using properties of these correctors, we establish convergence of the Navier–Stokes solution to the Euler solution as viscosity vanishes with optimal rates of convergence. In addition, we investigate vorticity production on the boundary in the limit of vanishing viscosity. Our work significantly extends prior work in the literature.     

Existence of a Global Solution for a Scalar Conservation Law with a Source Term

We are concerned with a scalar conservation law with a source term. This equation is proposed to describe the qualitative behavior of waves for a general system in resonance with the source term by T.P. Liu. In addition to this, the scalar conservation law is used in various areas such as fluid dynamics, traffic problems etc.In the present paper, we prove the global existence and stability of entropy solutions to the Cauchy problem. The difficult point is to obtain the bounded estimate of solutions. To solve it, we introduce some functions as the lower and upper bounds. Therefore, our bounded estimate depends on the space variable. This idea comes from the generalized invariant region theory for the compressible Euler equation. The method is also applicable to other nonlinear problems involving similar difficulties. Finally, we use the vanishing viscosity method to construct approximate solutions and derive the convergence by the compensated compactness.