Abstract
In this paper, we study the existence of weak solutions for the compressible magnetohydrodynamics flows driven by stochastic external forcing. Our method is based on solving the system for each fixed representative of the random variable and applying an abstract result on the measurability of multi-valued maps.
Highlights
1 Introduction and main results This paper considers the following compressible magnetohydrodynamic flows driven by a stochastic external force in the isentropic case [ – ]: dρ + divx(ρu) dt =, ( . )
P(ρ) is the pressure, the viscosity coefficients of the flow satisfy ν + λ > and ν > ; ν > is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field, the perturbation W is a random variable represented for a.a. ω by a bounded function, sufficiently regular with respect to the spatial variable x ∈
(ρu) =, if ρ =, the electric field does not appear in the MHD equations ( . )-( . ), from ( . ), the electric field E is induced by moving conductive flow in the magnetic field, having the following relationship with the magnetic field H and the velocity u: E = ν ∇ × H – u × H
Summary
1 Introduction and main results This paper considers the following compressible magnetohydrodynamic flows driven by a stochastic external force in the isentropic case [ – ]: dρ + divx(ρu) dt = , Feireisl et al [ ] gave results on the existence of weak solutions for the three dimensional stochastic compressible Navier-Stokes system. Inspired by the work of [ ], this paper is devoted to the study of the existence of weak solutions for three dimensional stochastic compressible magnetohydrodynamic flows.
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