The Zero Mach Number Limit of the Three-Dimensional CompressibleViscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of thethree-dimension- al compressible viscous magnetohydrodynamicequations. More precisely, based on the local existence of thethree-dimensional compressible viscous magnetohydrodynamicequations, first the convergence-stability principle is established.Then it is shown that, when the Mach number is sufficiently small,the periodic initial value problems of the equations have a uniquesmooth solution in the time interval, where the incompressibleviscous magnetohydrodynamic equations have a smooth solution. Whenthe latter has a global smooth solution, the maximal existence timefor the former tends to infinity as the Mach number goes to zero.Moreover, the authors prove the convergence of smooth solutions ofthe equations towards those of the incompressible viscousmagnetohydrodynamic equations with a sharp convergence rate.