Nonlinear Stability of Large Amplitude Viscous Shock Wave for General Viscous Gas

TL;DR

This paper proves the nonlinear stability of large amplitude viscous shock waves in isentropic Navier-Stokes equations with general pressure laws and density-dependent viscosity, using a novel variable transformation and energy methods.

Contribution

It introduces a new variable to reformulate the system and applies an elementary energy method to establish stability for large amplitude shocks with general pressure laws.

Findings

Large amplitude viscous shock waves are nonlinearly stable.

Stability holds for general pressure laws including γ-law.

The proof employs a new variable and energy method.

Abstract

In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes $\gamma$-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in Matsumura-Nishihara [On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2 (1985), no. 1, 17-25.].