Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

Global weak solutions to a three-dimensional compressible non-Newtonian fluid

Pages: 1703 – 1733

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n6.a11

Authors

Li Fang (Department of Mathematics and Center for Nonlinear Studies (CNS), Northwest University, Xi’an, China)

Zhenhua Guo (Department of Mathematics and CNS, Northwest University, Xi’an, China; and School of Mathematics and Information, Guangxi University, Nanning, China)

Abstract

The paper concerns on the existence of global weak solutions to a compressible non-Newtonian fluid with the power-law type. The main contribution of this paper is to handle the power-law structure with the exponent $r \gt \frac{12\gamma}{5\gamma-3}$ when the pressure is related to $\rho^\gamma$ with $\gamma \gt 1$. The exponent $r$ is forced by the convective term and the convergent argument of approximate solution. Inspired by the weak formulation of the momentum equation, the existence of global weak solutions is proved relying on the Faedo–Galerkin method, weak compactness techniques and the monotonicity method.

Keywords

compressible non-Newtonian fluids, global weak solution, the monotonicity method

2010 Mathematics Subject Classification

35Q35, 76A05

Received 19 October 2020

Received revised 2 January 2022

Accepted 21 January 2022

Published 14 September 2022