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March  2004, 3(1): 97-113. doi: 10.3934/cpaa.2004.3.97

On the slightly compressible MHD system in the half-plane

Paola Trebeschi 1,

1. 

Dipartimento di Matematica, Università di Brescia, Facoltà di Ingegneria, Via Valotti 9, 25133 Brescia, Italy

Received  February 2003 Revised  October 2003 Published  January 2004

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.
Keywords: Compressible MHD system, Euler equations, Mach number..
Mathematics Subject Classification: 35L50, 58J45, 46E35, 35Q35. .
Citation: Paola Trebeschi. On the slightly compressible MHD system in the half-plane. Communications on Pure and Applied Analysis, 2004, 3 (1) : 97-113. doi: 10.3934/cpaa.2004.3.97
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