Existence theorem and blow-up criterion of the strong solutions to the Magneto-micropolar fluid equations

TL;DR

This paper establishes the existence of strong solutions and a blow-up criterion for magneto-micropolar fluid equations in three dimensions, using Littlewood-Paley decomposition to relate blow-up to vorticity behavior.

Contribution

It proves the existence of strong solutions in $H^s$ spaces and introduces a Beale-Kato-Majda type blow-up criterion based solely on vorticity.

Findings

Existence of strong solutions for initial data in $H^s( ^3)$ with $s>3/2$.

Development of a vorticity-based blow-up criterion.

Application of Littlewood-Paley decomposition in analysis.

Abstract

In this paper we study the magneto-micropolar fluid equations in $\R^3$, prove the existence of the strong solution with initial data in $H^s(\R^3)$ for $s> {3/2}$, and set up its blow-up criterion. The tool we mainly use is Littlewood-Paley decomposition, by which we obtain a Beale-Kato-Majda type blow-up criterion for smooth solution $(u,\omega,b)$ which relies on the vorticity of velocity $\nabla\times u$ only.