Communications in Mathematical Sciences

Volume 22 (2024)

Number 2

Stability and decay rate of viscous contact wave to one-dimensional compressible Navier-Stokes equations

Pages: 315 – 331

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n2.a2

Authors

Xinxiang Bian (School of Information and Mathematics, Yangtze University, Hubei, China)

Lingling Xie (Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China; and School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing, China)

Abstract

This paper studies the large-time asymptotic stability and optimal time-decay rate of viscous contact wave to one-dimensional compressible Navier–Stokes equations. We prove that one-dimensional compressible Navier–Stokes equations are asymptotically stable for viscous contact wave with arbitrarily large strength, under large initial perturbations. The time optimal decay rate of viscous contact wave is also obtained under the small initial perturbations. In the proof, the Lagrange transform is used to cancel the convection terms, which are difficult to estimate due to the lower spatial derivatives compared with the diffusion terms.

Keywords

compressible Navier–Stokes equations, stability, decay rate, viscous contact wave

2010 Mathematics Subject Classification

35Q30, 76N10

Full Text (PDF format)

Received 15 September 2022

Received revised 15 January 2023

Accepted 19 June 2023

Published 1 February 2024