# American Institute of Mathematical Sciences

February  2019, 12(1): 37-58. doi: 10.3934/krm.2019002

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

 1 School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, China 2 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 3 School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong 518060, China

* Corresponding author: Xiaoping Zhai

Received  March 2017 Revised  February 2018 Published  July 2018

Fund Project: The first author is supported by the Postdoctoral Science Foundation of China grant 2017M620688, the second author is supported by NSFC grant 11731014, 11571254 and the third author is supported by NSFC grant 11601533

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.
Citation: Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002
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