Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

The existence and limit behavior of the shock layer for 1D stationary compressible non-Newtonian fluids

Pages: 239 – 253

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a11

Authors

Zhenhua Guo (School of Mathematics and Information, Guangxi University, Nanning, China)

Yifan Su (School of Mathematics and Center for Nonlinear Studies (CNS), Northwest University, Xi’an, China)

Jinjing Liu (Department of Mathematics, Yunnan University, Kunming, China)

Abstract

In this paper, we first define the shock layer to a class of stationary compressible non-Newtonian fluids in one dimension. Then the existence and uniqueness of the shock layer are established. In addition, the limit behavior of the shock layer is analyzed. It is shown that, as the viscosity coefficient and the heat conductivity coefficient vanish, the shock layer to the non-Newtonian fluids tends to a shock wave of the corresponding Euler equations. It is also shown that, as the viscosity coefficient tends to zero, the shock layer goes to a non-viscous shock layer to the non-Newtonian fluids, while as heat-conductivity coefficient tends to zero, the shock layer converges to a thermally non-conducting shock layer to the non-Newtonian fluids.

Keywords

non-Newtonian fluids, Navier–Stokes equations, Euler equations, shock layer, shock wave

2010 Mathematics Subject Classification

35L67, 74J40, 76A05

Received 27 August 2019

Accepted 7 May 2022

Published 27 December 2022