Weak and Strong Solutions of Equations of Compressible Magnetohydrodynamics

Abstract

A plasma is a mixture of ions, electrons and neutral particles. At the macroscopic level, beyond the kinetic description necessary in rarefied situations, dynamics of dense plasmas is governed by the interaction between the fluid components and the electromagnetic fields. When dealing with collisional plasmas [64] [67], it is often appropriate to use one-fluid approximations which are much simpler than a complete kinetic theory and relevant for a number of applications (astrophysics [20] [89] [100] [104], plasma physics [25] [27], electrometallurgy [45] etc...).Moreover a further approximation is obtained when one studies the interaction between electromagnetic fields and a electric-conducting fluid with electrical effects negligeable compared to magnetic effects. This simplified scheme is called magnetohydrodynamics (MHD). It unifies classical fluid dynamics and magnetism of continuum media in a coupled way: electromagnetic fields induce currents in the moving fluid which produce forces which in turn modify the electromagnetic fields. If one considers a compressible heat-conducting fluid, the equations governing the system will be the compressible Navier-Stokes-Fourier system coupled to the set of Maxwell's equations through suitable momenta and energy sources.

Fluid–rigid body interaction in a compressible electrically conducting fluid

“…Moreover, consider some external forces 𝑔, 𝐽 ∈ 𝐿 ∞ ((0, 𝑇) × Ω) and some initial data 0 ≤ 𝜌 0 ∈ 𝐿 𝛾 (Ω), (𝜌𝑢) 0 ∈ 𝐿 1 (Ω), 𝐵 0 ∈ 𝐿 2 (Ω) such that…”

“…the 𝛿-kernel of the initial domain 𝑆 𝑖 0 of the 𝑖-th body, where 𝛿 > 0 is chosen sufficiently small such that for all 𝑖 = 1, … , 𝑚 the 𝛿-neighborhood (𝑂 𝑖 ) 𝛿 = {𝑥 ∈ ℝ 3 ∶ dist(𝑥, (𝑂 𝑖 ) 𝛿 ) < 𝛿} of 𝑂 𝑖 coincides with 𝑆 𝑖 0 . Such 𝛿 > 0 exists due to the 𝐶 2…”

“…Let Ω, 𝑆 1 0 , … , 𝑆 𝑚 0 ⊂ ℝ 3 , 𝑚 ∈ ℕ, be bounded domains of class 𝐶 2 ⋂ 𝐶 0,1 . Let further {𝑢 𝑘 } 𝑘∈ℕ be a sequence of vector fields bounded in 𝐿 2 (0, 𝑇; 𝐻 1,2 (Ω)) uniformly with respect to 𝑘 and let each 𝑢 𝑘 be compatible with the system {𝑆 𝑖 0 , 𝑋 𝑖 𝑘 } 𝑚 𝑖=1 , where 𝑋 𝑖 𝑘 (𝑡) ∶ ℝ 3 → ℝ 3 , 𝑡 ∈ [0, 𝑇], 𝑖 = 1, … , 𝑚 denotes an isometry. Then, there exist isometries 𝑋 𝑖 (𝑡) ∶ ℝ 3 → ℝ 3 such that, for a suitable subsequence, it holds that 𝑋 𝑖 𝑘 → 𝑋 𝑖 in 𝐶 ( [0, 𝑇]; 𝐶 loc ( ℝ 3 ))…”

“…In order to classify our result within the wide range of works in MHD and FSI, we give a brief summary of the related literature. The existence of weak solutions to the MHD problem for a compressible fluid but without any rigid bodies involved is, for example, shown in [42] and [2]. The fluids considered in the latter one of these articles are electrically as well as thermally conducting and moreover the existence of strong solutions is addressed therein.…”

Global existence of a diffusion limit with damping for the compressible radiative Euler system coupled to an electromagnetic field

“…In the present work we couple the previous model to an electromagnetic field through the so called magnetohydrodynamic approximation (MHD) [3].…”