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MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory
Author(s) -
Liu ChengJie,
Xie Feng,
Yang Tong
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21763
Subject(s) - magnetohydrodynamics , mathematics , sobolev space , prandtl number , uniqueness , boundary value problem , mathematical analysis , monotonic function , boundary layer , robin boundary condition , nonlinear system , boundary layer thickness , no slip condition , free boundary problem , mixed boundary condition , boundary (topology) , magnetic field , physics , mechanics , heat transfer , quantum mechanics
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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