Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data

Abstract

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.

Incompressible Limit of the Nonisentropic Ideal Magnetohydrodynamic Equations

“…Recently, the authors [18] justified rigourously the low Mach number limit of classical solutions to the ideal or full compressible non-isentropic MHD equations with small entropy or temperature variations. When the heat conductivity and large temperature variations are present, the low Mach number limit for the full compressible non-isentropic MHD equations was shown in [19]. We emphasize here that the arguments in [19] are completely different from the present paper (at least in the derivation of the uniform estimates), and depend essentially on the positivity of fluid viscosities, magnetic diffusivity, and heat conductivity coefficients.…”

“…In this situation we can not write the total energy equation as a transport equation of entropy, which plays a key role in the proof. Instead, it is more convenient to use the temperature as an unknown in the total energy equation when the heat conductivity is positive, see [19] for more details.…”

“…Finally, applying the identity 19) we expect to get, as ǫ → 0, the following incompressible ideal non-isentropic MHD equations…”

Asymptotic Limits of the Full Compressible Magnetohydrodynamic Equations

“…Recently, the low Mach number limit for the full compressible MHD equations with small entropy or temperature variation was justified rigourously in [13]. When the heat conductivity and large temperature variations are present, the low Mach number limit for the full compressible nonisentropic MHD equations was shown in [15] for local smooth solutions. In [14], Jiang, Ju, and Li studied the incompressible limit of smooth solutions to the compressible nonisentropic ideal MHD equations with general initial data in the whole space R 3 when the initial data belong to H s (R 3 ) with s > 0 being an even integer.…”

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.…”