Local-in-time Well-posedness of Boundary Layer System for the Full Incompressible MHD Equations by Energy Methods

TL;DR

This paper establishes the local-in-time well-posedness of the boundary layer system for the full incompressible MHD equations in two dimensions using energy methods, without requiring monotonicity conditions.

Contribution

It proves the existence and uniqueness of solutions for the MHD boundary layer system under general conditions, extending previous results by removing the monotonicity assumption.

Findings

Proved local-in-time existence of solutions.

Established uniqueness of solutions.

Applied energy methods to boundary layer equations.

Abstract

In this paper, we investigate the well-posedness theory for the MHD boundary layer system in two-dimensional space. The boundary layer equations are governed by the Prandtl type equations that are derived from the full incompressible MHD system with non-slip boundary condition on the velocity, perfectly conducting condition on the magnetic field, and Dirichlet boundary condition on the temperature when the viscosity coefficient depends on the temperature. To derive the Prandtl type boundary layer system, we require all the hydrodynamic Reynolds numbers, magnetic Reynolds numbers and Nusselt numbers tend to infinity at the same rate. Under the assumption that the initial tangential magnetic field is not zero, one applies the energy methods to establish the local-in-time existence and uniqueness of solution for the MHD boundary layer equations without the necessity of monotonicity condition.