On the slightly compressible MHD system in the half-plane

In this paper we prove the existence of a smooth compressible solution for the MHD system in the half-plane. It is well-known that, as the Mach number goes to zero, the compressible MHD problem converges to the incompressible one, which has a global solution in time. Hence, it is natural to expect that, for Mach number sufficiently small, the compressible solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. In order to obtain the existence result, we decompose the solution as the sum of the solution of the irrotational Euler problem, the solution of the incompressible MHD system and the solution of the remainder problem which describes the interaction between the first two components. We show that the solution of the remainder part exists on any arbitrary time interval. Since this holds also for the solution of the irrotational Euler problem, this yields the existence of the smooth compressible solution for the MHD system.