Global existence of strong solutions to the multi-dimensional inhomogeneous incompressible MHD equations

Abstract

This paper is concerned with the Cauchy problem of the multi-dimensional incompressible magnetohydrodynamic equations with inhomogeneous density and fractional dissipation. It is shown that when $\alpha +\beta =1+\frac{n}{2}$ satisfying $1\le \beta \le \alpha \le \min \{\frac{3\beta }{2},\frac{n}{2},1+\frac{n}{4}\}$ and $\frac{n}{4}<\alpha$ for $n\ge 3$ , then the inhomogeneous incompressible MHD equations has a unique global strong solution for the initial data in Sobolev space which do not need a small condition.

19 pages