J. Math. Fluid Mech. 20 (2018), 359–378
2017 Springer International Publishing
Journal of Mathematical
Convergence of the Full Compressible Navier–Stokes–Maxwell System to the
Incompressible Magnetohydrodynamic Equations in a Bounded Domain II: Global
Jishan Fan, Fucai Li and Gen Nakamura
Abstract. In this paper we continue our study on the establishment of uniform estimates of strong solutions with respect to
the Mach number and the dielectric constant to the full compressible Navier–Stokes–Maxwell system in a bounded domain
Ω ⊂ R
. In Fan et al. (Kinet Relat Models 9:443–453, 2016), the uniform estimates have been obtained for large initial
data in a short time interval. Here we shall show that the uniform estimates exist globally if the initial data are small.
Based on these uniform estimates, we obtain the convergence of the full compressible Navier–Stokes–Maxwell system to the
incompressible magnetohydrodynamic equations for well-prepared initial data.
Mathematics Subject Classiﬁcation. 76W05, 35Q60, 35B25.
Keywords. Full compressible Navier–Stokes–Maxwell system, Zero Mach number limit, Zero dielectric constant limit, In-
compressible magnetohydrodynamic equations, Bounded domain.
In this paper we shall continue our study on the singular limit of the following full compressible Navier–
Stokes–Maxwell system in a bounded domain Ω ⊂ R
ρ +div(ρu)=0, (1.1)
∇p − μΔu − (λ + μ)∇div u =(E + u × b) × b, (1.2)
(ρe)+div(ρue)+pdiv u− div (κ∇T )
+ λ(div u)
+(E + u × b)
E − rot b + E + u × b =0, (1.4)
b +rotE =0, div b =0, (1.5)
where the unknowns ρ, u, p, e,T ,E,andb stand for the density, velocity, pressure, internal energy, tem-
perature, electric ﬁeld, and magnetic ﬁeld, respectively. The physical constants μ and λ are the shear
viscosity and bulk viscosity of the ﬂow and satisfy μ>0andλ +
μ ≥ 0. κ>0 is the heat conduc-
> 0 is the (scaled) Mach number, and
> 0 is the (scaled) dielectric constant.
(∇u + ∇u
), where ∇u
denotes the transpose of the matrix ∇u.
In [8,9], Kawashima and Shizuta established the global existence of smooth solutions to the system
(1.1)–(1.5) for small data and studied its zero dielectric constant limit
→ 0 in the whole plane R
Recently, Jiang and Li  studied the zero dielectric constant limit
→ 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
.In, they obtained similar results to the inviscid case of (1.1)–(1.5). In , Li and Mu