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 $1\leq \beta\leq \alpha\leq\min {\frac{3\beta}{2},\frac{n}{2},1+\frac{n}{4}}$ and $\frac{n}{4}<\alpha$ for $n\geq3$ , 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.