Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework

TL;DR

This paper proves the validity of Prandtl boundary layer expansion for inhomogeneous incompressible MHD equations with strong boundary layers, under specific conditions on viscosity, resistivity, and magnetic field, within a Sobolev framework.

Contribution

It establishes the uniform $L^ abla$ estimates and validates the Prandtl ansatz for inhomogeneous MHD with strong boundary layers, extending previous results to variable density and magnetic interactions.

Findings

Validation of Prandtl boundary layer expansion in Sobolev space.

Handling of strong boundary layers with unbounded curl of boundary layers.

Extension to inhomogeneous MHD with variable density and magnetic field interactions.

Abstract

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics (MHD) equations in the half plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform $L^\infty$ estimates of the error functions comes from the unboundedness of curl of the strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl ansatz in $L^\infty$ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in \cite{LXY192}, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.