Huijiang Zhao

OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH POTENTIAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp - Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.

Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier-Stokes equations. Assume that the corresponding Riemann problem to the compressible Euler equations can be solved by rarefaction waves (V-R, U-R, S-R)(t, x). If the initial data (v(0), u(0), s(0))(x) to the nonisentropic compressible Navier-Stokes equations is a small perturbation of an approximate rarefaction wave constructed as in [ S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249-252], then we show that, for the general gas, the Cauchy problem admits a unique global smooth solution (v, u, s)(t, x) which tends to (V-R, U-R, S-R)(t, x) as t tends to infinity. A global stability result can also be established for the nonisentropic ideal polytropic gas, provided that the adiabatic exponent gamma is close to 1. Furthermore, we show that for the isentropic compressible Navier-Stokes equations, the corresponding global stability result holds, provided that the resulting compressible Euler equations are strictly hyperbolic and both characteristical fields are genuinely nonlinear. Here, global stability means that the initial perturbation can be large. Since we do not require the strength of the rarefaction waves to be small, these results give the nonlinear stability of strong rarefaction waves for the one-dimensional compressible Navier-Stokes equations.

Global Solutions to the One-Dimensional Compressible Navier--Stokes--Poisson Equations with Large Data

In this paper, we study the global solutions with large data away from vacuum to the Cauchy problem of the one-dimensional compressible Navier--Stokes--Poisson system with density-dependent viscosi...

GLOBAL SOLUTIONS TO THE BOLTZMANN EQUATION WITH EXTERNAL FORCES

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

One-dimensional Compressible Navier--Stokes Equations with Temperature Dependent Transport Coefficients and Large Data

This paper is concerned with the Cauchy problem of the one-dimensional compressible Navier--Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A result on the existence and uniqueness of a globally smooth nonvacuum solution is obtained provided that the

THE VLASOV–POISSON–BOLTZMANN SYSTEM FOR SOFT POTENTIALS

(\gamma-1)\cdot (H^3({\bf R})

Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation

-norm of the initial perturbation)

A new energy method for the Boltzmann equation

1

Global Stability of Strong Rarefaction Waves of the Jin–Xin Relaxation Model for the p-System

is the adiabatic gas constant. This is a Nishida--Smoller type global solvability result with large data.

One-dimensional compressible heat-conducting gas with temperature-dependent viscosity

An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov-Poisson-Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov-Poisson-Boltzmann system for angular cutoff soft potentials with