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

Fei Chen; Boling Guo; Xiaoping Zhai

doi:10.3934/krm.2019002

Article Contents

2019, Volume 12, Issue 1: 37-58. Doi: 10.3934/krm.2019002

This issue Previous Article Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks Next Article A stochastic algorithm without time discretization error for the Wigner equation

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
^* Corresponding author: Xiaoping Zhai

Received: February 28, 2017

Revised: January 31, 2018

Published: July 2018

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.

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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.

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Mathematics Subject Classification: Primary: 35Q35, 76D03; Secondary: 35B40.

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