Stability of contact discontinuity for the Boltzmann equation

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and Navier-Stokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation corresponding to the contact discontinuities for the Euler equations, we succeed in obtaining the time asymptotic stability of this wave pattern with a convergence rate. In our analysis, an intrinsic dissipative mechanism associated with this profile is found and used for closing the energy estimates.     

Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier-Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is of higher order than the heat-conductivity κ. Without loss of generality, we set μ≡0. Here we have no need to restrict the strength of the contact discontinuity to be small.     

Contact discontinuity with general perturbations for gas motions

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the Navier-Stokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the Navier-Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate , it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of . Thus, this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the L^∞ norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.     

Solutions of the Cauchy problem for the 2-D Euler system with fan-shaped structure

We study the Cauchy problem for the isentropic 2-D Euler system with initial data having discontinuity on a smooth curve. A local existence of a solution is established, which consists of shock wave, rarefaction wave and contact discontinuity.     

Global Stability of Multi-wave Configurations for the Compressible Non-isentropic Euler System

This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈ (1, 3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, onsisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.     

Exterior Problem for the Three-dimensional Euler Equations

A priori estimates for the exterior initial boundary value problems of the Euler equations are considered. The existence and uniqueness of a local solution is proved.     

The Viscosity Splitting Solutions of the Navier-Stokes Equations

The solutions of the initial boundary value problems or the Navier-Stokes equations are constructed by means of a viscosity splitting scheme. Convergence results are proved.     

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...     

Pointwise Stability of Contact Discontinuity for Viscous Conservation Laws with General Perturbations

The large time asymptotic behavior towards viscous contact waves for a class of systems of viscous conservation laws is studied in this paper for general initial perturbations. The high order deviation of the viscous solutions from the leading order ansatz is estimated pointwisely via the approximate Green function approach. The structural constraint on the left eigenvector belonging to the principal linearly degenerate family used in [13 Liu , T. , Xin , Z. ( 1988 ). Pointwise decay to contact discontinuities for systems of viscous conservation laws . Asian J. Math. 1 : 34 – 84 . [Google Scholar]] is removed so that our results hold, in particular, for the one-dimensional compressible Navier–Stokes equations of gas dynamics in both Lagrangian and Eulerian coordinates.     

Conditions on the Pressure for Vanishing Velocity in the Incompressible Fluid Flows in ℝ N

In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in ? ^N .     

An Eulerian-Lagrangian approach for incompressible fluids: Local theory

We study a formulation of the incompressible Euler equations in terms of the inverse Lagrangian map. In this formulation the equations become a first order advective nonlinear system of partial differential equations.

     

Solutions to the 2D Euler equations with velocity unbounded at infinity

We prove existence of solutions to the two-dimensional Euler equations with vorticity bounded and with velocity locally bounded but growing at infinity at a rate slower than a power of the logarithmic function. We place no integrability conditions on the initial vorticity. This result improves upon a result of Serfati which gives existence of a solution to the two-dimensional Euler equations with bounded velocity and vorticity.     

Notes on the Incompressible Euler and Related Equations on $\BR^N$\

The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on R^N and also presents Liouville type theorems for the incompressible and compressible fluid equations.     

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.     

Backward Euler Scheme,Singular Hölder Norms,and Maximal Regularity for Parabolic Difference Equations

We show finite difference analogues of maximal regularity results for discretizations of abstract linear parabolic problems. The involved spaces are discrete versions of spaces of Hölder continuous functions, which can be singular in 0. The main tools are real interpolation and Da Prato–Grisvard's theory of the sum of linear operators.