Regularity criterion for the 3D generalized Newtonian fluids

TL;DR

This paper establishes a regularity criterion for 3D generalized Newtonian fluids, showing conditions under which weak solutions become strong solutions based on the velocity gradient's membership in a specific critical function space.

Contribution

It introduces a new regularity criterion for 3D generalized Newtonian fluids involving the velocity gradient in a critical Besov space, extending previous results.

Findings

Weak solutions become strong under the new criterion.

The criterion applies for specific ranges of the parameter p.

The velocity gradient's space membership is crucial for regularity.

Abstract

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for $\frac{5}{3} <p<\frac{11}{5} $ provided that the gradient of velocity $\nabla \boldsymbol{u}$ belongs to the critical space $L^{\frac{2}{2-(3-p)a}}(0,T;\dot{B}^{-a}_{\infty,\infty}(\mathbb{R}^3))$, where $a\in(\frac{3}{2},\frac{2}{3-p})$ if $p\in(\frac{5}{3},2)$ and $a\in(\frac{1}{p},\frac{2}{3-p})$ if $p\in[2,\frac{11}{5})$.