Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

TL;DR

This paper extends the stability analysis of viscous shock waves in compressible Navier-Stokes equations to large initial perturbations with significant density oscillations, using elementary energy methods.

Contribution

It demonstrates that stability results hold for large initial density oscillations, broadening the applicability of previous stability analyses.

Findings

Viscous shock wave stability persists under large density oscillations.

Elementary energy methods effectively establish uniform bounds on density.

Stability results are extended beyond small perturbations.

Abstract

This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, F. M. Huang, A. Matsumura and X. D. Shi showed that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and the key point is to deduce the desired uniform positive lower and upper bounds on the density.