Communications in Mathematical Sciences

Volume 22 (2024)

Number 6

Asymptotic stability of nonlinear wave for an inflow problem to the compressible Navier-Stokes-Korteweg system

Pages: 1501 – 1528

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n6.a3

Authors

Yeping Li (School of Mathematics and Statistics, Nantong University, Nantong, China)

Yujie Qian (School of Mathematics and Statistics, Nantong University, Nantong, China)

Rong Yin (School of Mathematics and Statistics, Nantong University, Nantong, China )

Abstract

In this paper, we are concerned with the inflow problem on the half line $(0,+\infty)$ for a one-dimensional compressible Navier-Stokes-Korteweg system, which is used to model compressible viscous fluids with internal capillarity, i.e., the liquid-vapor mixtures with phase interfaces. We first investigate that the asymptotic profile is a nonlinear wave: the superposition wave of a rarefaction wave and a boundary layer solution under the proper condition of the far fields and boundary values. The asymptotic stability on the nonlinear wave is shown under some conditions that the initial data are a small perturbation of the rarefaction wave and the strength of the stationary wave is small enough. The proofs are given by an elementary energy method.

Keywords

compressible Navier–Stokes–Korteweg equation, inflow problem, rarefaction wave, boundary layer solution, asymptotic stability, energy method

2010 Mathematics Subject Classification

35B40, 76W05

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The authors’ research was supported in part by the National Science Foundation of China (Grant Nos. 12171258 and 12331007).

Received 29 November 2022

Accepted 28 December 2023

Published 18 July 2024