Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier--Stokes Equations

Abstract

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 (VR, UR, SR)(t,x). If the initial data (v0 , u0 ,s0 )(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 (VR, UR, SR)(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.