Weak solutions for the Stokes system for compressible non-Newtonian fluids with unbounded divergence

TL;DR

This paper establishes the existence of weak solutions for a compressible non-Newtonian fluid model at low Reynolds numbers, overcoming challenges posed by unbounded divergence of velocity using advanced mathematical techniques.

Contribution

It introduces a novel approach combining regularity theory and logarithmic inequalities to handle unbounded divergence in weak solutions for the Stokes system.

Findings

Existence of weak solutions under relaxed divergence conditions

Extension of methods to non-Newtonian fluid models

Application of BMO function inequalities in PDE analysis

Abstract

We investigate the existence of weak solutions to a certain system of partial differential equations, modelling the behaviour of a compressible non-Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack of the $L^\infty$ estimate on the divergence of the velocity field. The result was obtained by combining the regularity theory for singular operators with a certain logarithmic integral inequality for $BMO$ functions, which allowed us to adjust the method from (Feireisl et al., 2015) to more relaxed conditions on the velocity.