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

2016/11/17 by Liu, Cheng-Jie, Xie, Feng, Yang, Tong · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.05815

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.

Cited by

• Gevrey well-posedness of the hydrostatic MHD-wave system

Related

• Local-in-time Well-posedness of Boundary Layer System for the Full Incompressible MHD Equations by Energy Methods
• Well-posedness in Sobolev spaces of the two-dimensional MHD Boundary layer equations without viscosity
• Global well-posedness of the MHD boundary layer equations in the Sobolev Space
• Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces
• Local-in-time well-posedness for Compressible MHD boundary layer