Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations

TL;DR

This paper proves the large-time stability and asymptotic convergence of viscous shock profiles for the outflow problem in barotropic Navier-Stokes equations, using the $a$-contraction method in Eulerian coordinates.

Contribution

It extends stability analysis of viscous shocks to the outflow problem in Eulerian coordinates, handling free boundary issues and proving decay of the shift speed.

Findings

Viscous shock profiles are asymptotically stable under small perturbations.

The time-dependent shift in shock position decays to zero over time.

The shock profile retains its shape asymptotically in the Eulerian framework.

Abstract

We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of $a$-contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free boundary value problem due to the absence of a boundary condition for the fluid density, we consider the problem in the Eulerian coordinate instead. Although the $a$-contraction method is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulty arising from a free boundary.