Abstract
We are concerned with the Dirichlet problem of the three-dimensional steady viscous compressible magnetohydrodynamic (MHD) equations. It is proved that for any specific heat ratio $\gamma>1$ , the Dirichlet problem of the steady compressible MHD equations on a bounded domain $\Omega\subset\mathbb{R}^{3}$ admits a renormalized weak solution. Our method relies upon the weighted estimates of both pressure and kinetic energy for the approximate system, and the method of weak convergence developed by Lions and Feireisl.
Highlights
We consider the steady compressible magnetohydrodynamic equations in a bounded domain ⊂ R : div(ρu) =, ( . )–μ u – (λ + μ)∇ div u + ρ(u · ∇)u + ∇P(ρ) – (∇ × H) × H = ρf,–ν H – ∇ × (u × H) =, div H =, where ρ ≥, u = (u, u, u ), and P(ρ) = ργ with γ > being the specific heat ratio are the fluid density, velocity, and pressure, respectively
H = (H, H, H ) is the magnetic field, ν > is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field
The important progress in the spatial threedimensional case is due to the seminal work of Lions [ ], where he obtained the existence of renormalized weak solutions of the Navier-Stokes equations for γ
Summary
One important restrictions to the value of γ is due to the a priori estimates It is a natural and interesting problem to investigate the existence of weak solutions to the Dirichlet problem In the case of steady compressible MHD flows, the equations for the magnetic field are governed by an elliptic system and we cannot get the bound of H L and in turn H H in terms of the velocity coefficient u directly since here the coefficient u we consider is an arbitrary data. ) to get the existence of the weak solutions (ρδ, uδ, Hδ) to the following system: div(ρδuδ) = , –μ uδ – (λ + μ)∇ div uδ + ρδuδ · ∇uδ + ∇Pδ(ρ) – (∇ × Hδ) × Hδ = ρδf, –ν Hδ – ∇ × (uδ × Hδ) = , div Hδ = ,.
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