Asymptotic stability of viscous shock profiles for the 1D compressible Navier-Stokes-Korteweg system with boundary effect

TL;DR

This paper proves the asymptotic stability of viscous shock profiles for the 1D compressible Navier-Stokes-Korteweg system with boundary effects, showing solutions tend to a shifted shock profile under small initial perturbations.

Contribution

It establishes the global existence and stability of viscous shock profiles for the system with boundary effects, a new result in this context.

Findings

Solutions tend to a shifted viscous shock profile over time.

Global strong solutions exist under small initial perturbations.

Boundary effects are effectively handled in the analysis.

Abstract

This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. The analysis is based on the elementary $L^2$-energy method and the key point is to deal with the boundary estimates.