A new blow up criterion for the 3D magneto-micropolar fluid flows without magnetic diffusion

Abstract

Abstract This note obtains a new regularity criterion for the three-dimensional magneto-micropolar fluid flows in terms of one velocity component and the gradient field of the magnetic field. The authors prove that the weak solution ( u , ω , b ) $(u,\omega,b)$ to the magneto-micropolar fluid flows can be extended beyond time t = T $t=T$ , provided if u 3 ∈ L β ( 0 , T ; L α ( R 3 ) ) $u_{3}\in L^{\beta }(0,T;L^{\alpha }(R^{3}))$ with 2 β + 3 α ≤ 3 4 + 1 2 α , α > 10 3 $\frac{2}{\beta }+\frac{3}{\alpha }\leq \frac{3}{4}+\frac{1}{2\alpha },\alpha > \frac{10}{3}$ and ∇ b ∈ L 4 p 3 ( p − 2 ) ( 0 , T ; M ˙ p , q ( R 3 ) ) $\nabla b\in L^{\frac{4p}{3(p-2)}}(0,T;\dot{M}_{p,q}(R^{3}))$ with 1 < q ≤ p < ∞ $1< q\leq p<\infty $ and p ≥ 3 $p\geq 3$ .

Keywords

• Magneto-micropolar fluid flow
• Regularity criterion
• Weak solution
• Morrey spaces