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MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: WellPosedness Theory

ChengJie Liu Feng Xie Tong Yang

Analysis of PDEs mathscidoc:1912.43977

Communications on Pure and Applied Mathematics, 72, (1), 63-121, 2019.1
We study the wellposedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtltype equations that are derived from the incompressible MHD system with nonslip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the localitime existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the wellposedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for the MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. 2018 Wiley Periodicals, Inc.
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@inproceedings{chengjie2019mhd,
  title={MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: WellPosedness Theory},
  author={ChengJie Liu, Feng Xie, and Tong Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211021748780541},
  booktitle={Communications on Pure and Applied Mathematics},
  volume={72},
  number={1},
  pages={63-121},
  year={2019},
}
ChengJie Liu, Feng Xie, and Tong Yang. MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: WellPosedness Theory. 2019. Vol. 72. In Communications on Pure and Applied Mathematics. pp.63-121. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211021748780541.
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