A remark on the contact wave for the 1-D compressible Navier–Stokes equations

We revisit the classical work of Huang–Matsumura–Xin [F.M. Huang, A. Matsumura, and Z.P. Xin, Arch. Ration. Mech. Anal., 179:55–77, 2006] and Huang–Xin–Yang [F.M. Huang, Z.P. Xin, and T. Yang, Adv. Math., 219:1246–1297, 2008] for contact wave of the one-dimensional compressible Navier–Stokes equations. By using Huang–Matsumura–Xin–Yang’s approach and a detailed energy analysis, we prove the large-time asymptotic stability of a contact wave pattern with a better convergence rate for compressible Navier–Stokes equations under non-zero mass condition on the perturbation. This improves previous results of [F.M. Huang, A. Matsumura, and Z.P. Xin, Arch. Ration. Mech. Anal., 179:55–77, 2006] and [F.M. Huang, Z.P. Xin, and T. Yang, Adv. Math., 219:1246–1297, 2008].

Keywords

compressible Navier–Stokes equations, contact wave, time decay rate

2010 Mathematics Subject Classification

35B35, 35Q35, 76L05, 76N10

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This work was supported by the NNSFC Grant 11871229.

Received 8 July 2019

Accepted 14 September 2019

Published 1 April 2020