Dynamics of Partial Differential Equations

Volume 16 (2019)

Number 3

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

Pages: 225 – 251

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n3.a1

Authors

Zhengzheng Chen (School of Mathematical Sciences, Anhui University, Hefei, China)

Yeping Li (Department of Mathematics, East China University of Science and Technology, Shanghai, China)

Mengdi Sheng (School of Mathematical Sciences, Anhui University, Hefei, China)

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.

Keywords

compressible Navier–Stokes–Korteweg system, viscous shock profiles, asymptotic stability, boundary effect

2010 Mathematics Subject Classification

Primary 35B40, 35Q35. Secondary 35L65.

Received 1 December 2018

Published 30 August 2019