A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity

Yu-Zhu Wang,Liping Hu,Yin-Xia Wang

doi:10.1155/2011/128614

Abstract

We study the incompressible magneto-micropolar fluid equations with partial viscosity in . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

Highlights

In 2, the authors have proven that a weak solution to 1.1 has fractional time derivatives of any order less than 1/2 in the two-dimensional case
Rojas-Medar and Boldrini 5 proved the existence of weak solutions by the Galerkin method, and in 2D case, proved the uniqueness of the weak solutions
A Beale-Kato-Majda type blow-up criterion for smooth solution u, v, b to 1.1 that relies on the vorticity of velocity ∇ × u only is obtained by Yuan 7

Summary

Introduction

The incompressible magneto-micropolar fluid equations in Rn n 2, 3 take the following form:. The existences of weak and strong solutions for micropolar fluid equations were proved by Galdi and Rionero and Yamaguchi , respectively. Regularity criteria of weak solutions to the micropolar fluid equations are investigated in. In , the authors gave sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations. A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field in Morrey-Campanato spaces was established see. We obtain a Beale-Kato-Majda type blowup criterion of smooth solutions to the magneto-micropolar fluid equations 1.2.

Preliminaries

L2 exp