MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well-Posedness Theory |
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| Authors: | Cheng-Jie Liu Feng Xie Tong Yang |
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| Institution: | 1. Institute of Natural Sciences, Shanghai Jiao Tong University, 200240 Shanghai, P.R., China;2. School of Mathematical Sciences and LSC-MOE, Shanghai Jiao Tong University, 200240 Shanghai, P.R., China;3. Department of Mathematics, City University of Hong Kong, Tat Chee Avenue Kowloon, Hong Kong |
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| Abstract: | 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-i-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. © 2018 Wiley Periodicals, Inc. |
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