On an inhomogeneous boundary value problem for steady compressible magnetohydrodynamics flow
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
On an inhomogeneous boundary value problem for steady compressible magnetohydrodynamics flow Shengquan Liu and Ming Cheng
Abstract. In this paper, we are concerned with the three-dimensional stationary equations of compressible magnetohydrodynamic (MHD) isentropic flow in a bounded cylinder domain Ω = (0, 1) × Ω0 with an inhomogeneous boundary condition. We obtain the existence and uniqueness of a strong solution provided that the data are around a given constant flow. From our result, it reveals that the magnetic Reynolds number plays an important role in the stability of steady compressible MHD equations. Mathematics Subject Classification. 76W05, 35D35, 76N10. Keywords. Compressible magnetohydrodynamics equations, Strong solution, Inhomogeneous boundary condition.
1. Introduction In the present paper, we are devoted to investigating the following 3D stationary equations of compressible magnetohydrodynamic isentropic flow: ⎧ div(ρv) = 0 in Ω, ⎪ ⎪ ⎨ ρv · ∇v + ∇P (ρ) − 1 Δv − 1 ∇divv − S(∇ × B) × B = 0 in Ω, Re1 Re2 (1.1) 1 ∇ × (∇ × B) − ∇ × (v × B) = 0 in Ω, ⎪ ⎪ Rm ⎩ divB = 0 in Ω, where ρ, v, B denote the unknown density of the fluid, the velocity field and the magnetic field, respectively. The Reynolds number Re1 and second Reynolds number Re2 satisfy the physical restrictions Re1 > 0, 3Re1 ≥ Re2 ,
1 1 + > 0. Re1 Re2
The magnetic Reynolds number Rm and coupling parameter S are two positive constants. The pressure P is given by P (ρ) = Aργ with A > 0, γ ≥ 1. The domain Ω = (0, 1) × Ω0 is a bounded cylinder, where Ω0 ⊂ R2 is a given bounded domain with C 2 boundary. Therefore, the boundary of Ω is divided into three parts ∂Ω = Γin ∪ Γout ∪ Γ0 : Γin = Ω0 × {0}, Γout = Ω0 × {1}, Γ0 = ∂Ω0 × [0, 1]. Γin , Γout , Γ0 are called the inflow part, the outflow part, and the characteristic part, respectively. 0123456789().: V,-vol
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S. Liu and M. Cheng
ZAMP
Throughout this paper, we focus on the system (1.1) supplied with the following inhomogeneous boundary conditions: ⎧ on ∂Ω, ⎪v · n = d ⎪ ⎪ 2 ⎪ n · D(v) · τ + f v · τ = g , i = 1, 2 on ∂Ω, ⎨ Re1 i i i 1 (1.2) on ∂Ω, Rm ∇ × B − (v × B) × n = κ ⎪ ⎪ on ∂Ω, ⎪B · n = l ⎪ ⎩ on Γin , ρ = ρin where D(v) denotes the dilatation tensor as 1 D(v) = (vixj + vjxi ), i, j = 1, 2, 3, 2 n and τi (i = 1, 2) are the unit outer normal and tangent vectors to ∂Ω. f > 0 is a friction coefficient. We assume that the given functions satisfy g1 , g2 ∈ W 1−1/p, p (∂Ω), ρin ∈ W 1, p (Γin ) , d ∈ W 2−1/p, p (∂Ω) for some p ∈ (3, 6]. Since the singularity occurs at the junctions of Γin and Γout with Γ0 , the functional spaces W q,s (∂Ω) (s, q ∈ R) are algebraic sums of spaces defined on the boundary, that is, W q,s (∂Ω) = W q,s (Γin ) + W q,s (Γout ) + W q,s (Γ0 ) . Especially, d|Γ0 = 0 means that Γ0 is an impermeable wall. Here are some more words to describle the boundary conditions. Referring to [1], the conditions (1.2)1,2 on v are called slip conditions. It reveals that the velocity on the boundary is related with the tangent
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