Logarithmically Improved Regularity Criterion for the 3D Micropolar Fluid Equations

Hui Zhang

doi:10.1155/2014/386269

Abstract

We study the regularity of weak solutions to the incompressible micropolar fluid equations. We obtain an improved regularity criterion in terms of vorticity of velocity in Besov space. It is proved that if the vorticity field satisfies ∫0T∇×uB˙∞,∞0/1+log1+∇×uB˙∞,∞0dt<∞ then the strong solution can be smoothly extended after time T.

Highlights

This paper focuses on the incompressible micropolar fluid equations in R3∂tu + (u ⋅ ∇) u − Δu + ∇p − ∇ × w = 0,∂tw − Δw − ∇ (∇ ⋅ w) + 2w + u ⋅ ∇w − ∇ × u = 0, (1)∇ ⋅ u = 0, u (x, 0) = u0 (x), w (x, 0) = w0 (x), where u(x, t) is the velocity field, w(x, t) is the microrotational velocity field, and p = p(x, t) is the scalar pressure field, while (u0, w0) are the given initial data with ∇ ⋅ u0 = 0 in the sense of distribution.Micropolar fluid system was firstly developed by Eringen [1, 2]
It is a type of fluids which exhibits microrotational effects and microrotational inertia and can be viewed as a non-Newtonian fluid
It can describe many phenomena that appear in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes system and that is important to the scientists working with the hydrodynamic-fluid problems and phenomena

Summary

Introduction

Micropolar fluid system was firstly developed by Eringen [1, 2]. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [3]. The purpose of this paper is to study the regularity of weak solutions to the micropolar fluid system (1). By means of the Littlewood-Paley decomposition methods and function decomposition technique, Dong and Zhang [7, 8] recently prove the regularity of weak solutions under the velocity condition and the pressure condition in Besov spaces. Yuan proved [9] some classical regularity criteria of weak solutions to the Navier-Stokes equation which holds for the micropolar fluid equations. Motivated by the ideas of [11,12,13,14], this paper is to establish logarithmically improved regularity criterion in terms of the vorticity

International Journal of Analysis

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