A note on the zero Mach number limit of compressible Euler equations

This note presents a short and elementary justification of the classical zero Mach number limit for isentropic compressible Euler equations with prepared initial data. We also show the existence of smooth compressible flows, with the Mach number sufficiently small, on the (finite) time interval where the incompressible Euler equations have smooth solutions.

     

The low Mach number limit in Besov spaces for compressible magnetohydrodynamics equations

In this paper, we develop a continuation principle for general hyperbolic singular limit problems in more general Besov spaces, which covers the cases of usual Sobolev spaces with higher regularity in and the critical Besov space. As an application, we give a simple justification for the low Mach number limit of compressible magnetohydrodynamics equations. More precisely, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval where the incompressible magnetohydrodynamics equations have smooth solutions, and the definite convergence orders are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

^** Email: dlevy{at}math.stanford.edu We derive a second-order, semi-discrete central-upwind schemefor the incompressible 2D Euler equations in the vorticity formulation.The reconstructed velocity field preserves an exact discreteincompressibility relation. We state a local maximum principlefor a fully discrete version of the scheme and prove it usinga convexity argument. We then show how similar convexity argumentscan be used to prove that the scheme maps certain Orlicz spacesinto themselves. The consequences of this result on the convergenceof the scheme are discussed. Numerical simulations support theexpected properties of the scheme.     

Global well-posedness of the non-isentropic full compressible magnetohydrodynamic equations

In this paper, we are concerned with Cauchy problem for the multi-dimensional (N ≥ 3) non-isentropic full compressible magnetohydrodynamic equations. We prove the existence and uniqueness of a global strong solution to the system for the initial data close to a stable equilibrium state in critical Besov spaces. Our method is mainly based on the uniform estimates in Besov spaces for the proper linearized system with convective terms.     

Convergence of compressible Euler-Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler-Poisson equations in plasma physics in the torus T^d. For well prepared initial data the convergence of solutions of compressible Euler-Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl-div decomposition of the gradient.     

Euler方程的解在Besov空间上的存在性和惟一性

利用交换子及不可压缩流的性质,得到了Eu ler方程在Besov空间的端点情形时存在性和惟一性.     

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.     

A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations

Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations.In this survey paper,we explain how these methods may be implemented so as to study the compresible Navier-Stokes equations in the whole space.We shall investigate both the initial value problem in critical Besov spaces and the low Mach number asymptotics.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

Remarks on the nonlinear instability of incompressible Euler equations

In this paper, we consider the nonlinear instability of incompressible Euler equations. If a steady density is non-monotonic, then the smooth steady state is a nonlinear instability. First, we use variational method to find a dominant eigenvalue which is important in the construction of approximate solutions, then by energy technique and analytic method, we obtain the dynamical instability result.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.     

Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler-Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.     

带密度的不可压Euler方程在临界Besov空间中的适定性

本文证明了带密度的不可压Euler方程在临界Besov空间中的局部适定性,并且只用涡度场给出了强解的一个爆破准则.另外,本文关于带密度的不可压磁流体方程得到了类似结果.     

Precise regularity results for the Euler equations

It has already been proved, under various assumptions, that no singularity can appear in an initially regular perfect fluid flow, if the L^∞ norm of the velocity's curl does not blow up. Here that result is proved for flows in smooth bounded domains of (d?2) when the regularity is expressed in terms of Besov (or Triebel-Lizorkin) spaces.     

Convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations

We establish the convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations in this paper. The convergence is rigorously proved on the time interval where the smooth solution to the incompressible Euler equations exists. The proof relies on the compactness argument and the so-called relative-entropy method.     

Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions

The compressible Navier-Stokes-Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.