Existence of steady solutions for a model for micropolar electrorheological fluid flows with not globally $\log$--Hölder continuous shear exponent

Abstract

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function $\hat{p}$ that is $\log$--Hölder continuous and an electric field $\mathbf{E}$ for that $|\mathbf{E}{|}^2$ is bounded and smooth. Note that these conditions do not imply that the variable shear exponent $p=\hat{p}\circ |\mathbf{E}{|}^2$ is globally $\log$--Hölder continuous.

24 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2112.06512