This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.

Large time asymptotics of the solutions to non-symmetric Fokker-Planck type equations are studied by extending the entropy method to this case. We present a modified Bakry-Emery criterion that yields covergence of the solution to the steady state in relative entropy with an explicit exponential rate. In parallel it also implies a logarithmic Sobolev inequality w.r.t. the steady state measure. Explicit examples illustrate that skew-symmetric perturbations in the Fokker Planck operator can "help" to improve the constant in such a logarithmic Sobolev inequality.

The low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data is rigorously justified in the whole space R 3 . First, the uniform-in-Mach-number estimates of the solutions in a Sobolev space are established on a finite time interval independent of the Mach number. Then the low Mach number limit is proved by combining these uniform estimate with a theorem due to Métiver and Schochet [Arch. Ration. Mech. Anal. 158 (2001), 61-90] for the Euler equations that gives the local energy decay of the acoustic wave equations.

This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients and general initial data in the whole space R d (d = 2 or 3). It is rigorously showed that, as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient simultaneously go to zero, the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the ideal incompressible magnetohydrodynamic equations as long as the latter exists.

The low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, in which the effect of thermal conduction is taken into account, is rigorously justified in the framework of classical solutions with small density and temperature variations. Moreover, we show that for sufficiently small Mach number, the compressible magnetohydrodynamic equations admit a smooth solution on the time interval where the smooth solution of the incompressible magnetohydrodynamic equations exists. In addition, the low Mach number limit for the ideal magnetohydrodynamic equations with small entropy variation is also investigated. The convergence rates are obtained in both cases.