MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory

Liu Cheng-jie Shanghai Jiao Tong University Xie Feng Shanghai Jiao Tong University Yang Tong City University of Hong Kong

Analysis of PDEs mathscidoc:1903.03001

Communications on Pure and Applied Mathematics, 72, (1), 0063–0121, 2019.1
We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl-type equations that are derived from the incompressible MHD system with non-slip 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 local-in time existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the well-posedness 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.
MHD, Prandtl boundary layer, Well-Posedness theory, Sobolev spaces
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@inproceedings{liu2019mhd,
  title={MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory},
  author={Liu Cheng-jie, Xie Feng, and Yang Tong},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190322085236088901220},
  booktitle={Communications on Pure and Applied Mathematics},
  volume={72},
  number={1},
  pages={0063–0121},
  year={2019},
}
Liu Cheng-jie, Xie Feng, and Yang Tong. MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory. 2019. Vol. 72. In Communications on Pure and Applied Mathematics. pp.0063–0121. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190322085236088901220.
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