Regularity Criterion Of The Weak Solution To The 3D Micropolar Fluid Flows

The mathematical theory of micropolar fluid dynamics began over 40 years ago when Eringe in 1966 published his pioneering works on the micropolar fluid motion equations. The micropolar fluid motion model accounts for micro-rotation effects and micro-rotation inertia in a fluid motion system.When the micro-rotation effects are neglected or w=0, the micropolar fluid flows re-duces to the incompressible Navier-Stokes equations From a viewpoint of model, therefore, Navier-Stokes flow is viewed as a simplified micropolar fluid.This paper is concerned with the regularity criteria for weak solutions to the 3D (three-dimensional) micropolar equations with respect to the velocity field and the pressure field in Besov spaces and Lorentz spaces.The article mainly includes the following several parts:In the chapter 1:We give the backgrounds of the 3D micropolar fluid flows, prelimi-naries and relative results of the micropolar equations.In the chapter 2:Applying the Fourier localization technique and the Bony's para-product decomposition, we study the regularity criteria for the three-dimensional microp-olar fluid equations in Besov spaces, we prove the regularity criteria of weak solutions to (1.1-1.2) with respect to the velocity field u∈L2/1+r (0,T; B∞,∞r(R3)), with-1<r<1.In the chapter 3:We study the regularity criteria of the weak solution to the three-dimensional micropolar fluid equations by applying the innovative function decomposition technique; whenπ∈Lq(0,T; Bp,∞r(R3)), with2/q+3/p=2+r,3/2+r<p<∞,-1<r≤1 we prove the regularity criteria of the three-dimensional micropolar fluid equations with respect to the pressure field in Besov spaces.In the chapter 4:We prove the regularity criteria of the weak solution to the three-dimensional micropolar fluid equations with respect to some partial derivatives of the velocity or the pressure field in Lorentz spaces. Firstly, we study the regularity criterion of weak solution to the system(1.1-1.2)when (?)3u∈Lp(0,T；Lq,∞(R3)),with 2/p+3/q 1,3<g≤∞；Secondly,we get the regularity criterion of the weak solution when (?)1u1,(?)2u2∈Lp(0,T；Lq,∞(R3)), with 2/p+3/q=2,3/2<q≤∞；Thirdly,we obtain the regularity criterion (?)3π∈Lp(0,T；Lq(R3)),with 2/p+3/q=4/7,12/7<q≤∞.