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

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 $H^3\left(\Omega \right)$ of the Sobolev space $H^3\left(\Omega \right)$, obtained by addition of suitable boundary conditions on one portion of the boundary, such that for initial data in $H^3\left(\Omega \right)$ there exists a solution in the same space $H^3\left(\Omega \right)$, 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”.