Asymptotic limits of the isentropic compressible viscous magnetohydrodynamic equations with Navier-slip boundary conditions

In this paper, two kinds of asymptotic limits to the isentropic compressible viscous magnetohydrodynamic equations in a three-dimensional bounded domain Ω with Navier-slip boundary conditions are discussed. One is the incompressible limit with ill-prepared initial data, and the other is the combined inviscid and incompressible limit with well-prepared initial data. In the first case, we show that the weak solutions of the compressible viscous magnetohydrodynamic equations converge weakly to the weak solutions of the incompressible viscous magnetohydrodynamic equations provided the index of the fluid friction coefficient α 1 ≥ 1 as the Mach number goes to 0. Moreover, the convergence of the velocity in L 2 ( 0 , ∞ ; L 2 ( Ω ) ) is indeed strong under some geometrical assumptions on the domain Ω and α 1 ≥ 1 2 . In the second case, we show that the weak solution of the compressible viscous magnetohydrodynamic equations converges to the local strong solution of the ideal incompressible magnetohydrodynamic equations. Furthermore, the convergence rates are also obtained.