Well-posedness in Sobolev spaces of the two-dimensional MHD Boundary layer equations without viscosity

TL;DR

This paper proves the well-posedness of the 2D MHD boundary layer equations without viscosity in Sobolev spaces, extending previous results to a viscosity-free setting using pseudo-differential calculus.

Contribution

It establishes existence and uniqueness results for the viscosity-free MHD boundary layer system, complementing prior work that required viscosity or resistivity.

Findings

Well-posedness in Sobolev spaces for the system without viscosity

Use of pseudo-differential calculus to handle boundary integrals

Overcoming difficulties due to lack of diffusion for velocity

Abstract

We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.