Incompressible Limit of the Compressible Magnetohydrodynamic Equations with Vanishing Viscosity Coefficients

Abstract

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.

Inviscid incompressible limit for the strong stratified flow of a chemically reacting gaseous mixture

“…It should be pointed out that this kind of problem was first studied by Masmoudi [11] and then there are a lot of progressive works on this topic by Feireisl and Novotný [6] for the compressible Navier-StokesFourier system and by Jiang et al [8][9][10] for the compressible magnetohydrodynamic flows. Recently, Feireisl et al [5] have studied the inviscid incompressible limit of the weak solutions to the compressible Navier-Stokes equations of compressible flows with strong stratification using the relative entropy method.…”

Incompressible Limit of the Nonisentropic Ideal Magnetohydrodynamic Equations

“…In [21], Klainerman and Majda first studied the incompressible limit of the isentropic compressible ideal MHD equations in the spatially periodic case with well-prepared initial data. Recently, the incompressible limit of the isentropic viscous (including both viscosity and magnetic diffusivity) of compressible MHD equations with general data was studied in [12,16,17]. In [12], Hu and Wang obtained the convergence of weak solutions of the compressible viscous MHD equations in bounded, spatially periodic domains and the whole space, respectively.…”

“…In [16], the authors employed the modulated energy method to verify the limit of weak solutions of the compressible MHD equations in the torus to the strong solution of the incompressible viscous or partial viscous MHD equations (the shear viscosity coefficient is zero but the magnetic diffusion coefficient is a positive constant). In [17], the authors obtained the convergence of weak solutions of the viscous compressible MHD equations to the strong solution of the ideal incompressible MHD equations in the whole space by using the dispersion property of the wave equation if both shear viscosity and magnetic diffusion coefficients go to zero.…”

Singular Limit of the Rotational Compressible Magnetohydrodynamic Flows

“…For the case without rotational force, Masmoudi [2] proved the convergence of the weak solution of isentropic Navier-Stokes equations to the strong solution of the incompressible Euler equations in the 2-dimensional whole space R 2 and the space case by applying the related entropy method. Later, his result was extended to the isentropic compressible magnetohydrodynamic equations [3,4]. Feireisl and Novotný [5] studied the inviscid incompressible limit to the full Navier-Stokes-Fourier system in the whole space.…”