Regularity criteria for weak solutions of the Magneto-micropolar equations

Abstract

<p style='text-indent:20px;'>In this paper, we show that a weak solution <inline-formula><tex-math id="M1">$ (\mathbf{u},\mathbf{w},\mathbf{b})(\cdot,t) $</tex-math></inline-formula> of the magneto-micropolar equations, defined in <inline-formula><tex-math id="M2">$ [0,T) $</tex-math></inline-formula>, which satisfies <inline-formula><tex-math id="M3">$ \nabla u_3, \nabla_{h} \mathbf{w}, \nabla_{h} \mathbf{b} $</tex-math></inline-formula> <inline-formula><tex-math id="M4">$ \in L^{\frac{32}{7}}(0,T; $</tex-math></inline-formula> <inline-formula><tex-math id="M5">$ L^2(\mathbb{R}^3)) $</tex-math></inline-formula> or <inline-formula><tex-math id="M6">$ \partial_3 u_3, \partial_3 \mathbf{w}, \partial_3 \mathbf{b} \in L^{\infty}(0,T;L^2(\mathbb{R}^3)) $</tex-math></inline-formula>, is regular in <inline-formula><tex-math id="M7">$ \mathbb{R}^3\times(0,T) $</tex-math></inline-formula> and can be extended as a <inline-formula><tex-math id="M8">$ C^\infty $</tex-math></inline-formula> solution beyond <inline-formula><tex-math id="M9">$ T $</tex-math></inline-formula>.</p>