On Research Of Compressible Magnetohydrodynamic Equations

Hydrodynamic mechanics is a branch of physics, which is the study of macroscopicmovement of continuous medium such as fluids (liquids, gases, and plasmas) and theinteraction between such a movement and other movement patterns.1822, Navier derivedthe fundamental equations of viscous fluids;1845, Stokes derived these equations throughthe more reasonable foundation. This is the set of famous equations, Navier-Stokesequations, which we are using today. It is the theoretical foundation of the fluid dynamics.Magnetohydrodynamics(MHD)is the theory of the macroscopic interaction of electricallyconducting fluids with magnetic fields. MHD flow is governed by the compressible Navier-Stokes equations and the Maxwell equations of the magnetic field.It has a very broad range of applications, it is of importance in connection with manyengineering problems, such as sustained plasma confinement for controlled thermonuclearfusion, liquid-metal cooling of nuclear reactions, and electromagnetic casting of metals.It also finds applications in geophysics and astronomy, where one prominent example isthe so-called dynamo problem, that is, the question of the origin of the Earth magnetic field in its liquid metal core.The question of the compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids has been one of the most concern problems for both engineers and mathematicians. Mathematicians have set up many models for understanding this question better. Here we selected the two character-istics models and got the better results to fill gaps in the existing results.This thesis is devoted to studies of weak solutions of isentropic time-discretized compressible MHD equations and non-isentropic steady compressible MHD equations. To be specific, we consider the following model: where p is the density of the fluid,!u is the velocity, S is the viscous part of the stress tensor, P is the pressure,f is the external force, E=pe(p,θ)+1/2ρ|u|2, e is internal energy, q is the heat flux, H is the magnetic field, μ0is permeability of vacuum, σ is the conductivity. system (1) with the following boundary conditions: Problem (2) with the following boundary conditions: The Part1,(containing Chapter2), we mainly concerned with the study of the existence of weak solutions for the boundary value problem(1)and(3), first, under some certain as-sumptions, we used a new method to prove a higher integrability of kinetic energy density and density of this model. Next, by using the Leray-schauder fixed point theorem we further proof the existence of the smooth solution for the approximation model. Finally, in the process of taking the limit, we get the strong convergence of density through us-ing distribution theory, then we acquire the existence of weak solutions for isentropic time-discretized compressible MHD equations.we have the following result:定理0.0.6.Ifγ＞1and α＞0,then M=ρu×u. Moreover, there exists κ＞0such that for all Ω’∈Ω, the sequences ρ∈|u∈|2and ρ∈are bounded in L1+K(Ω’) and Lγ(1+k)(Ω’), respectively.定理0.0.7. for all γ＞1, α＞0, F∈Cβ(Ω), and non-negative f,g∈Cβ(Ω), problem (2.12)-(2.13) has a solution ρ∈∈C2+β(Ω),u∈∈C2+β(Ω), H∈∈C2+β(Ω). There is a constant c independent of∈such thatThis result along with Theorem0.5.1implies,定理0.0.8.If γ＞1and α＞0, then problem (2.1)-(2.2) admits at least one weak solution ρ∈Lγ(Ω),u∈Ⅱ01(Ω),H∈Ⅱ1(Ω) satisfying (2.3).The Part2,(containing Chapter3), we mainly concerned with the study of the existence of weak solutions for the boundary value problem(2)and(4), firstly, we get the priori estimate of the density, velocity, temperature and magnetic field in the model. Sec-ondly, we get the higher regularity of approximation solution by using steady fluids issue, finally, taking the limits for the approximation system, we get the strong convergence of density under different boundary conditions through using two different methods, then we acquire the existence of weak solutions for non-isentropic steady compressible MHD equations. we have the following result:定理0.0.9.Letβ∈[0,1), the domainΩ∈C2be not axially symmetric if β=0and γ∈(7/3,3].Letf∈L∞(Ω)andM＞0.Then there exists a weak solution to problem (3.1)-(3.2) such that The same concerns also Theorem below, 定理0.0.10.Letβ=1,the domain Ω∈C2,and let m=l+1＞(3γ-1)/(3γ-7),γ＞7/3.Let f∈L∞(Ω)and M＞0.Then there exists a weak solution to problem (3.1)-(3.2) such that wheres(γ)=min{3(γ-1),2γ},r=min{2,(3m)/(m+1)}.