MHD boundary layers theory in Sobolev spaces without monotonicity. I. well-posedness theory

Cheng-Jie Liu; Feng Xie; Tong Yang

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arXiv · 1611.05815

MHD boundary layers theory in Sobolev spaces without monotonicity. I. well-posedness theory

Cheng-Jie Liu·Feng Xie·Tong Yang

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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-in-time existence, uniqueness of solution 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 MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics.

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BibTeX RIS

Cheng-Jie Liu, Feng Xie, Tong Yang. 2017-01-16. MHD boundary layers theory in Sobolev spaces without monotonicity. I. well-posedness theory. https://arxiv.org/abs/1611.05815

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