Existence theorem and blow-up criterion of strong solutions to the two-fluid MHD equation in ${\mathbb{R}}^3$

Abstract

We first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony's paraproduct decomposition, it is proved that strong solution $(u,b)$ can be extended after $t=T$ if either $u\in L_T^q({\dot{B}}_{p,\infty }^0)$ with $\frac{2}{q}+\frac{3}{p}\le 1$ and $b\in L_T^1({\dot{B}}_{\infty ,\infty }^0)$, or $(\omega ,J)\in L_T^q({\dot{B}}_{p,\infty }^0)$ with $\frac{2}{q}+\frac{3}{p}\le 2$, where $ω(t)=\na\times u $ denotes the vorticity of the velocity and $J=\na\times b$ the current density.

18 pages