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# 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

#### 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

The work of Li Fang was supported in part by the National Natural Science Foundation of China (Grant no. 11501445). The work of Zhenhua Guo was supported in part by the National Natural Science Foundation of China (Grant no. 11931013, 11671319).

Received 19 October 2020

Received revised 2 January 2022

Accepted 21 January 2022

Published 14 September 2022