Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

Zero dissipation limit problem of 1-D Navier–Stokes equations

Pages: 1305 – 1329



Shixiang Ma (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Danli Wang (Academy of Mathematics and System Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China; and School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing, China)


The zero dissipation limit problem of the Navier–Stokes equations with zero viscosity in the case of the superposition of two rarefaction waves and a contact discontinuity is considered in this paper. It is proved that when the heat conductivity coefficient tends to zero, there exists a unique global solution of the compressible Navier–Stokes equations which converges uniformly to the Riemann solution of the corresponding Euler equations away from the initial time and the contact discontinuity. In addition, the uniform convergence rate in terms of the heat conductivity coefficient is obtained. This result is proved by a combination of the energy method from [F.M. Huang, Y. Wang, and T. Yang, Kinet. Relat. Models, 3:685–728, 2010] and [S.X. Ma, J. Math. Anal. Appl., 387:1033–1043, 2012].


Navier–Stokes equations, composite wave, inviscid limit

2010 Mathematics Subject Classification

35L65, 35Q30, 76N10, 76N15

This paper’s subject classification codes were emended on 1 June 2022.

The first author is partially supported by the National Natural Science Foundation of China (Nos. 11771155, 11771297), and by the Natural Science Foundation of Guangdong Province (No. 2021A1515010249).

Received 23 February 2021

Accepted 30 November 2021

Published 26 May 2022