Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density

Abstract

In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in ${\mathbb R}^3$. We establish the global well-posedness of the MHD system if the initial data satisfies $(ρ_0, u_0, H_0)∈ L^{∞}({\mathbb R}^3)× H^s({\mathbb R}^3)× H^s({\mathbb R}^3)$ with $\frac{1}{2} < s \le 1$ and $0 < \underline{ρ} \le ρ_0 \le \overline{ρ} < +∞,~~~~ \|(u_0, H_0)\|_{\dot{H}^{\frac 12}} \le c, $ for some small $c>0$ which only depends on $\underline{ρ}, \overline{ρ}$. As a byproduct, we also get the decay estimate of the solution.