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

TL;DR

This paper proves the global existence of strong solutions for multi-dimensional inhomogeneous incompressible MHD equations with fractional dissipation under specific conditions, without requiring small initial data.

Contribution

It establishes the first global existence result for strong solutions to inhomogeneous MHD equations with fractional dissipation in multiple dimensions without smallness assumptions.

Findings

Global strong solutions exist under specified fractional dissipation conditions.

No small initial data condition is needed for the existence of solutions.

Results apply to multi-dimensional cases with inhomogeneous density.

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.