两类可压磁流体力学方程的研究
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摘要
流体力学是力学的一个分支,它是研究流体(包括液体及气体)这样一个连续介质的宏观运动规律以及它与其他运动形态之间的相互作用.1822年,Navier建立了粘性流体的基本运动方程；1845年,Stokes又以更合理的基础导出了这个方程,并将其所涉及的宏观力学基本概念论证的令人信服.这组方程便是沿用至今的Navier-Stokes方程(简称N-S方程).而磁流体力学是研究导电流体在电磁场中的运动,他是可压的Navier-Stokes方程和磁场中Maxwell方程的耦合.
磁流体力学有一系列广泛的应用,与其有重要关联的就是许多工程问题,例如持续的等离子体约束受控热核聚变,核反应堆的液态金属冷却和金属的电磁铸件.除此之外我们可以发现在地球物理和天体物理学中也有应用,一个最重要的例子就是发电机问题,也就是地球磁场在他的液态金属核内的起源问题.
对于数学家和工程师而言,三维粘性可压热传导磁流体力学方程的弱解的紧性问题一直备受关注.为了更好的理解这个问题,人们提出许多数学模型.在这里我们选取两个有特点的模型并且得到比较好的结果来填补已有结果的空白.
本文致力于研究等熵的时间离散化可压磁流体力学方程和非等熵的稳态可压磁流体力学方程.具体地说,我们将研究如下两个模型：这里ρ是流体的密度,u是速度场,S是粘性应力张量,P是压力,f是外力,Eρe(ρ,θ)+1/2ρ|u|2,e是内能,q是热通量,H是磁场强度,(?)0是真空中的磁导率,σ是电导率.
体系(1]具有下面的边值条件：问题(2)需要赋予下面的边值条件：在第一部分中(第2章),我们主要研究边值问题(1)和(3)的弱解存在性问题,首先,在一定的假设条件下,我们用一种全新的方法证明了此模型中动能密度和密度的更高可积性.接下来,我们运用Leray-schauder不动点定理进一步证明了此模型的逼近体系的光滑解存在性.最后在取极限过程中我们运用分布函数理论来得到密度的强收敛性进而得到等熵的时间离散化可压磁流体力学方程的弱解存在性.我们有下面的结论：
定理0.0.1.如果γ>1,α>0,有M=ρu⑧u,并且存在κ>0使得对所有的Ω’(?)Ω,序列ρ(?)|u(?)|。|2和ρ。分别在L1+“((Ω’)和Lγ(1+k)(Ω’)上是有界的.
定理0.0.2.对任意的γ>1,α>0,F∈Cβ(Ω),非负函数f,g∈Cβ(Ω),问题(2.12)-(2.13)有一个解ρ∈∈C2+β(Ω),u∈∈C2+β(Ω),Ht∈C2+β(Ω).存在一个不依赖于ε常数c使得
这个结果结合定理0.5.1得到
定理0.0.3.如果γ>1和α>0,问题(2.1)-(2.2)至少存在一个弱解ρ∈Lγ((Ω),u∈Ⅱ01((Ω),H∈Ⅱ1((Ω)满足(2.3).
第二部分中(第3章),我们主要研究边值问题(2)和(4)的弱解存在性问题,首先,我们得到此模型中关于密度,速度,温度和磁场强度的先验估计.其次我们运用稳态流理论来得到逼近解的更高正则性.最后我们对逼近体系取极限,而在其过程中我们分别用二种不同的方法得到不同边界条件下密度的强收敛性进而得到等熵的稳态可压磁流体力学方程的弱解存在性.
定理0.0.4.令β∈[0,1),如果β=0那么区域(Ω∈C2不是径向对称的,取m=l+13γ-1/3γ-7,γ∈(7/3,3].令f∈L∞((Ω)且M>0.问题(3.1)-(3.2)存在一个弱解使得
我们关心的另一个问题将由下面的定理给出.
定理0.0.5.令β=1,区域Ω∈C2,取m=l+1＞(3γ-1)/(3γ-7),γ＞7/3令f∈L∞(Ω)且M>0.问题(3.1)-(3.2)存在一个弱解使得这里s(γ)＝min{3(γ-1),2γ},r=min{2,(3m)/(m+1)}.
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 that
This 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)}.
磁流体力学有一系列广泛的应用,与其有重要关联的就是许多工程问题,例如持续的等离子体约束受控热核聚变,核反应堆的液态金属冷却和金属的电磁铸件.除此之外我们可以发现在地球物理和天体物理学中也有应用,一个最重要的例子就是发电机问题,也就是地球磁场在他的液态金属核内的起源问题.
对于数学家和工程师而言,三维粘性可压热传导磁流体力学方程的弱解的紧性问题一直备受关注.为了更好的理解这个问题,人们提出许多数学模型.在这里我们选取两个有特点的模型并且得到比较好的结果来填补已有结果的空白.
本文致力于研究等熵的时间离散化可压磁流体力学方程和非等熵的稳态可压磁流体力学方程.具体地说,我们将研究如下两个模型：这里ρ是流体的密度,u是速度场,S是粘性应力张量,P是压力,f是外力,Eρe(ρ,θ)+1/2ρ|u|2,e是内能,q是热通量,H是磁场强度,(?)0是真空中的磁导率,σ是电导率.
体系(1]具有下面的边值条件：问题(2)需要赋予下面的边值条件：在第一部分中(第2章),我们主要研究边值问题(1)和(3)的弱解存在性问题,首先,在一定的假设条件下,我们用一种全新的方法证明了此模型中动能密度和密度的更高可积性.接下来,我们运用Leray-schauder不动点定理进一步证明了此模型的逼近体系的光滑解存在性.最后在取极限过程中我们运用分布函数理论来得到密度的强收敛性进而得到等熵的时间离散化可压磁流体力学方程的弱解存在性.我们有下面的结论：
定理0.0.1.如果γ>1,α>0,有M=ρu⑧u,并且存在κ>0使得对所有的Ω’(?)Ω,序列ρ(?)|u(?)|。|2和ρ。分别在L1+“((Ω’)和Lγ(1+k)(Ω’)上是有界的.
定理0.0.2.对任意的γ>1,α>0,F∈Cβ(Ω),非负函数f,g∈Cβ(Ω),问题(2.12)-(2.13)有一个解ρ∈∈C2+β(Ω),u∈∈C2+β(Ω),Ht∈C2+β(Ω).存在一个不依赖于ε常数c使得
这个结果结合定理0.5.1得到
定理0.0.3.如果γ>1和α>0,问题(2.1)-(2.2)至少存在一个弱解ρ∈Lγ((Ω),u∈Ⅱ01((Ω),H∈Ⅱ1((Ω)满足(2.3).
第二部分中(第3章),我们主要研究边值问题(2)和(4)的弱解存在性问题,首先,我们得到此模型中关于密度,速度,温度和磁场强度的先验估计.其次我们运用稳态流理论来得到逼近解的更高正则性.最后我们对逼近体系取极限,而在其过程中我们分别用二种不同的方法得到不同边界条件下密度的强收敛性进而得到等熵的稳态可压磁流体力学方程的弱解存在性.
定理0.0.4.令β∈[0,1),如果β=0那么区域(Ω∈C2不是径向对称的,取m=l+13γ-1/3γ-7,γ∈(7/3,3].令f∈L∞((Ω)且M>0.问题(3.1)-(3.2)存在一个弱解使得
我们关心的另一个问题将由下面的定理给出.
定理0.0.5.令β=1,区域Ω∈C2,取m=l+1＞(3γ-1)/(3γ-7),γ＞7/3令f∈L∞(Ω)且M>0.问题(3.1)-(3.2)存在一个弱解使得这里s(γ)＝min{3(γ-1),2γ},r=min{2,(3m)/(m+1)}.
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 that
This 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)}.
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