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

TL;DR

This paper establishes the local well-posedness of MHD boundary layer equations in Sobolev spaces without requiring the monotonicity condition, highlighting the stabilizing role of magnetic fields.

Contribution

It proves local existence and uniqueness of solutions for MHD boundary layers without the classical monotonicity assumption, expanding the mathematical understanding of MHD boundary layer stability.

Findings

Well-posedness established without monotonicity condition

Magnetic field stabilizes the boundary layer

Local-in-time existence and uniqueness proven

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.