Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations

TL;DR

This paper studies the long-term behavior of solutions to the barotropic Navier-Stokes equations near viscous shock waves in a half-line, establishing convergence under boundary conditions and small initial perturbations, using a novel boundary-adapted method.

Contribution

It extends the $a$-contraction method with shifts to boundary value problems, removing previous restrictions and working directly on physical variables.

Findings

Solutions converge to viscous shock profiles over time.

The method applies to both impermeable wall and inflow boundary conditions.

No need for anti-derivative variables in $L^2$ space, unlike prior works.

Abstract

We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to $L^2$ space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption $\gamma\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems.