Nonlinear stability of rarefaction waves for the compressible Navier–Stokes equations with zero heat conductivity

Abstract

This paper is concerned with the time-asymptotic nonlinear stability of rarefaction waves to the Cauchy problem of the one-dimensional compressible Navier–Stokes equations with zero heat conductivity. Under the assumption that the unique global entropy solution to the resulting Riemann problem of the corresponding compressible Euler equations consists of rarefaction waves only, then if both the initial perturbation and the strengths of rarefaction waves are assumed to be suitably small, we show that its Cauchy problem admits a unique global solution which tends time-asymptotically toward the rarefaction waves. This result is proved by using the elementary energy method and the argument developed by Kawashima and Okada (1982).