Stability of large-amplitude viscous shock under periodic perturbation for 1-d isentropic Navier-Stokes equations

TL;DR

This paper proves the nonlinear stability of large-amplitude viscous shocks in the 1D isentropic Navier-Stokes equations under space-periodic perturbations, showing solutions tend to a shifted shock despite oscillations at far fields.

Contribution

It introduces a novel approach to demonstrate stability by constructing an ansatz that captures oscillations, ensuring the difference remains in H^2 space over time.

Findings

Viscous shock solutions are stable under periodic perturbations.

Solutions tend to a shifted viscous shock with oscillations.

Stability holds for large-amplitude shocks with small initial perturbations.

Abstract

The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier-Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N-S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearly stable even though the perturbation oscillates at the far fields. The key point is to construct a suitable ansatz $ (\tilde{v},\tilde{u}) $, which carries the same oscillations of the solution $ (v,u) $ at the far fields, so that the difference $ (v-\tilde{v},u-\tilde{u}) $ belongs to the $ H^2(R) $ space for all $ t\geq 0. $