Stability of Large Amplitude Viscous Shock Wave for 1-D Isentropic Navier-Stokes System in the Half Space

TL;DR

This paper proves the stability of large amplitude viscous shock waves in the 1-D isentropic Navier-Stokes system within a half space, extending previous results to shocks of arbitrary strength.

Contribution

It demonstrates the stability of large amplitude viscous shock waves in the half space, removing the smallness restriction on shock strength from prior work.

Findings

Viscous shock waves are stable for initial conditions far from the boundary.

Stability holds for shocks of arbitrarily large strength.

Results extend previous stability results to larger shocks.

Abstract

In this paper, the asymptotic-time behavior of solutions to an initial boundary value problem in the half space for 1-D isentropic Navier-Stokes system is investigated. It is shown that the viscous shock wave is stable for an impermeable wall problem where the velocity is zero on the boundary provided that the shock wave is initially far away from the boundary. Moreover, the strength of shock wave could be arbitrarily large. This work essentially improves the result of [A. Matsumura, M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146(1): 1-22, 1999], where the strength of shock wave is sufficiently small.