On the slightly compressible MHD system in the half-plane

Abstract

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 problemconverges to the incompressible one, which has a global solution intime. Hence, it is natural to expect that, for Mach numbersufficiently small, the compressible solution exists on any arbitrarytime 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 remainderpart exists on any arbitrary time interval. Since this holds also forthe solution of the irrotational Euler problem, this yieldsthe existence of the smooth compressible solution for the MHD system.