Boundary Layer Problems for the Two-dimensional Inhomogeneous Incompressible Magnetohydrodynamics Equations

TL;DR

This paper establishes local-in-time well-posedness for boundary layer problems in two-dimensional inhomogeneous incompressible MHD equations, considering non-zero tangential magnetic fields and small density perturbations, using energy methods in weighted Sobolev spaces.

Contribution

It proves the existence and uniqueness of solutions for inhomogeneous MHD boundary layers under specific conditions, extending prior results to inhomogeneous cases.

Findings

Well-posedness of inhomogeneous MHD boundary layer equations established.

Results hold for small density perturbations and non-zero tangential magnetic fields.

Homogeneous case well-posedness obtained with large initial data.

Abstract

In this paper, we study the well-posedness of boundary layer problems for the inhomogeneous incompressible magnetohydrodynamics(MHD) equations, which are derived from the two dimensional density-dependent incompressible MHD equations.Under the assumption that initial tangential magnetic field is not zero and density is a small perturbation of the outer constant flow in supernorm,the local-in-time existence and uniqueness of inhomogeneous incompressible MHD boundary layer equation are established in weighted Conormal Sobolev spaces by energy method. As a byproduct, the local-in-time well-posedness of homogeneous incompressible MHD boundary layer equations with any large initial data can be obtained.