Regularity of weak solutions to magneto-micropolar fluid equations. (English) Zbl 1240.35421

Yuan, Baoquan

Acta Math. Sci., Ser. B, Engl. Ed. 30, No. 5, 1469-1480 (2010).

Summary: In this article, we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in \(\mathbb R^3\). We obtain the classical blow-up criteria for smooth solutions \((u,\omega, b)\), i.e., \(u\in L^q(0,T;L^p(\mathbb R^3))\) for \(\frac 2q+\frac 3p\leq 1\) with \(3<p\leq \infty, u\in C([0,T); L^3(\mathbb R^3))\) or \(\nabla u\in L^q(0,T;L^p)\) for \(\frac 32<p\leq \infty\) satisfying \(\frac 2q+\frac 3p\leq 2\). Moreover, our results indicate that the regularity of weak solutions is dominated by the velocity \(u\) of the fluid. In the end-point case \(p=\infty\), the blow-up criteria can be extended to more general spaces \(\nabla u\in L^1(0,T; \dot{B}^0_{\infty,\infty}(\mathbb R^3))\).

MSC:

35Q35	PDEs in connection with fluid mechanics
35B65	Smoothness and regularity of solutions to PDEs
35D30	Weak solutions to PDEs
35B44	Blow-up in context of PDEs
76W05	Magnetohydrodynamics and electrohydrodynamics

Keywords:

magneto-micropolar fluid equations; regularity; blow-up