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

  • * Corresponding author: Xiaoping Zhai

    * Corresponding author: Xiaoping Zhai
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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  • 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.

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


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