A note on Ideal Magneto-Hydrodynamics with perfectly conducting boundary conditions in the quarter space

TL;DR

This paper establishes local well-posedness for the ideal magneto-hydrodynamics equations in a quarter space with mixed boundary conditions, using a reflection technique to handle the non-uniformly characteristic boundary.

Contribution

It introduces a specific Sobolev space framework ensuring existence and regularity of solutions for MHD in a quarter space with mixed boundary conditions.

Findings

Existence of solutions in a specialized Sobolev space for initial data.

Persistence of full $H^3$-regularity over short time.

Use of reflection technique simplifies the proof due to geometry.

Abstract

We consider the initial-boundary value problem in the quarter space for the system of equations of ideal Magneto-Hydrodynamics for compressible fluids with perfectly conducting wall boundary conditions. On the two parts of the boundary the solution satisfies different boundary conditions, which make the problem an initial-boundary value problem with non-uniformly characteristic boundary. We identify a subspace ${{\mathcal H}}^3(\Omega)$ of the Sobolev space $H^3(\Omega)$, obtained by addition of suitable boundary conditions on one portion of the boundary, such that for initial data in ${{\mathcal H}}^3(\Omega)$ there exists a solution in the same space ${{\mathcal H}}^3(\Omega)$, for all times in a small time interval. This yields the well-posedness of the problem combined with a persistence property of full $H^3$-regularity, although in general we expect a loss of normal regularity near the boundary. Thanks to the special geometry of the quarter space the proof easily follows by the "reflection technique".