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

Abstract:

The expansion waves for the compressible Navier-Stokes equations have recently been shown to be nonlinear stable. The nonlinear stability results are called local stability or global stability depending on whether the $H^1-$norm of the initial perturbation is small or not. Up to now, local stability results have been well established. However, for global stability, only partial results have been obtained. The main purpose of this paper is to study the global stability of rarefaction waves for the compressible Navier-Stokes equations. For this purpose, we introduce a positive parameter $t_0$ in the construction of smooth approximations of the rarefaction wave solutions for the compressible Euler equations so that the quantity $\ell =\frac {t_0}{\delta }$ ($\delta$ denotes the strength of the rarefaction waves) is sufficiently large to control the growth induced by the nonlinearity of the system and the interaction of waves from different families. Then by using the energy method together with the continuation argument, we obtain some nonlinear stability results provided that the initial perturbation satisfies certain growth conditions as $\ell \to +\infty$. Notice that the assumption that the quantity $\ell$ can be chosen to be sufficiently large implies that either the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially.