Two Regularity Criteria For The Weak Solutions To The 3D Incompressible Micropolar Fluid Equations

The mathematics theory of the micropolar fluid equation have been a new dynamics over the years since Eringen introduced the micropolar fluid equation in 1960s.In this paper,we study the following initial value problem of the incompressible micropolar fluid equations.#12 Hereu =(u1(x,t),u2(x,t),u3(x,t),P=P(x,t),? =(?1(x,t),?2(x,t),?3(x t)are respectively the unknown velocity field of the flow,the pressure of the flow,and the micro-rotational velocity,(u0,?0)is given initial data with div u0 =0 in the sense of distribution.This paper is concerned with the two regularity criteria of the 3D micropolar fluid equations:the first one is for div(u/|u|)with the condition,and the other is for a directional derivative of pressure such as(?)3P.More precisely,we prove that if div(u/|u|)? Lq(0,T;Lp(R3)),with2/q+3/p?1/2,6?p,??,4?q??,or(?)3P ? L?(0,T;L?(R3))wither 2/?+3/??1,3??<?,2<???,p a then the weak solution(u(x,t),?(x,t))of the incompressible micropolar fluid equations is regular in R3 ×(0,T).