Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations

Abstract

The full compressible magnetohydrodynamic equations can be derived formally from the complete electromagnetic fluid system in some sense as the dielectric constant tends to zero. This process is usually referred as magnetohydrodynamic approximation in physical books. In this paper we justify this singular limit rigorously in the framework of smooth solutions for well-prepared initial data.

Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system

“…In [12], we studied the magnetohydrodynamic approximation for the isentropic electromagnetic fluid system in three-dimensional period domain and obtained the isentropic compressible magnetohydrodynamic equations with explicit convergence rates. Recently, we extended the results in [12] to the complete magnetohydrodynamic fluid system and obtained the full compressible magnetohydrodynamic equations [13]. We remark that the viscosities (including the shear and buck viscosities and heat conductivity coefficient) play a crucial role in the proof process of [13] and the inviscid case is left as an open problem there.…”

“…Recently, we extended the results in [12] to the complete magnetohydrodynamic fluid system and obtained the full compressible magnetohydrodynamic equations [13]. We remark that the viscosities (including the shear and buck viscosities and heat conductivity coefficient) play a crucial role in the proof process of [13] and the inviscid case is left as an open problem there.…”

“…The purpose of this paper is to solve this problem and give a rigorous derivation of the compressible magnetohydrodynamic equations (1.16)-(1.19) from the non-isentropic compressible Euler-Maxwell system (1.10)-(1.15) as the dielectric constant ǫ tends to zero. As in [13], we consider the system (1.10)- (1.15) in a periodic domain of R 3 , i.e., the torus T 3 " pR{p2πZqq 3 .…”

Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain

“…In [8,9], Kawashima and Shizuta established the global existence of smooth solutions for small data [11] and studied its zero dielectric constant limit ǫ 2 → 0 in the whole space R 2 . Recently, Jiang and Li [6] studied the zero dielectric constant limit ǫ 2 → 0 to the system (1.1)-(1.5) and obtained the convergence of the system (1.1)- (1.5) to the full compressible magnetohydrodynamic equations in T 3 , see also [7] on the similar results to the invisid case of (1.1)- (1.5). In [10], Li and Mu study the low Mach number limit ǫ 1 → 0 to the system (1.1)-(1.5) and obtained the convergence of the system (1.1)-(1.5) to the incompressible Navier-Stokes-Maxwell system in the torus T 3 .…”

Weak and Strong Solutions of Equations of Compressible Magnetohydrodynamics

“…This approximation corresponds to a strongly non-relativistic regime and has been justified in the 2dimensional case by S. Kawashima and Y. Shizuta in [63], and more recently in 3-dimensional space by S. Jiang and F. Li (see [57] for isentropic flows and [58] for the general case). Accordingly, equation ( 3) can be written in the form…”