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Regularity criterion for the 3D generalized Newtonian fluids

Qiao Liu, Xincheng Shi

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 $p\in[2,\frac{11}{5})$.

Regularity criterion for the 3D generalized Newtonian fluids

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 provided that the gradient of velocity belongs to the critical space , where if and if .
Paper Structure (5 sections, 4 theorems, 66 equations)

This paper contains 5 sections, 4 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Auxiliary results
  5. Proof of Theorem \ref{['theorem 1.1']}

Key Result

Theorem 1.1

Let $\frac{5}{3}<p<\frac{11}{5}$ and $\boldsymbol{u}_0\in W^{1,2}_\sigma(\mathbb{R}^3)\cap W^{1,p}(\mathbb{R}^3)$. Assume that $\boldsymbol{u}$ is a weak solution to 1.1 such that the following inclusion holds where $a$ satisfies then the solution $\boldsymbol{u}$ is a strong solution of 1.1 on $\mathbb{R}^3\times(0,T]$.

Theorems & Definitions (9)

  • Definition 1.1: Weak solution
  • Definition 1.2: Strong solution
  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.1
  • proof : Proof of Lemma 2.3