【作者】 李丹；

【导师】 杜力力；

【作者基本信息】 四川大学 ， 应用数学， 2023， 博士

【摘要】 不可压缩流体主要由Navier-Stokes方程刻画,它是一类描述粘性不可压缩流体动量守恒的运动方程组,如果再考虑磁场对流体的影响,则主要由磁流体动力学方程描述,它主要描述磁场与导电流体之间的相互影响。本文主要研究三维不可压缩磁本原方程组的数学理论,包含具有完全粘性以及部分粘性或部分磁扩散的情形,建立了其强解的整体适定性以及磁本原方程组与相关的磁流体力学方程组之间的严格对应关系。本文主要内容有以下几个方面:第一章主要介绍了Navier-Stokes方程组,本原方程,磁流体动力学方程组的物理背景以及数学理论的研究现状,并简单叙述了本文的主要工作结果。第二章列出了后面章节需要用到的一些预备知识,包括三维积分形式的Ladyzhenskaya不等式、经典的方程组形式的Gronwall不等式、对数形式的Sobolev嵌入不等式、Aubin-Lions紧性引理,Minkowski不等式。第三章研究了磁本原方程组在三维薄型区域上的整体适定性,我们假设初值属于H2空间且没有小性要求,证明其初边值问题强解的整体存在性与唯一性。第四章严格证明了三维不可压缩的磁流体动力学方程组的解强收敛于磁本原方程组的解。也就是说,对于水平方向的初始速度场和磁场属于H1空间时,证明了磁流体动力学方程组的整体Leray-Hopf弱解强收敛于磁本原方程组的整体强解。当水平方向的初始速度场和磁场属于H2空间时,磁流体动力学方程组的局部强解可以延拓为整体强解。进一步得到磁流体动力学方程组的整体强解收敛于磁本原方程组的整体强解。收敛速率都与小宽高比极限同阶。在第三章的基础上,我们在第五章研究了一类速度场和磁场具有水平耗散和水平磁扩散的磁本原方程组的整体适定性,我们假设初值属于H~2空间,证明了初边值问题强解的整体存在性与唯一性。此外,我们还证明了对任意初值(?)∈H~1∩L~∞,且(?)∈L~m,这里m∈(2,∞),我们利用对数形式的Sobolev嵌入不等式和对数形式的Gronwall不等式建立了该模型强解的全局适定性。结合第五章的结论,我们在第六章证明了当初值(?)∈H~1∩L~∞,且(?)∈L~m,这里m∈(2,∞),具有水平耗散和水平磁扩散的磁流体动力学方程组的整体Leray-Hopf弱解强收敛于具有水平耗散和磁扩散的磁本原方程组的整体强解。当初值(?)∈H~2时,我们证明了在有限时间里,具有水平耗散和水平磁扩散的磁流体动力学方程组的强解强收敛于水平耗散和水平磁扩散的磁本原方程组的强解。两种收敛速率都为(?)阶，这里β=min{α-2,2}.更多还原

【Abstract】 The basic series of equations in theoretical fluid mechanics is Navier-Stokes equation.It is a class of motion equations that describe the viscous incompressible fluids conservation of momentum.If the magnetic field is considered,the fluid mechanics equations are mainly characterized by the Magnetohydrodynamics(MHD)equations,which mainly described the influence between the magnetic fields and conductive fluids.This paper mainly studies the mathematical theory of three-dimensional incompressible hydrostatic MHD equations,including the cases with complete viscosity and partial viscosity or partial magnetic diffusion,and establishes the global well-posedness of its strong solutions and the strict correspondence between the hydrostatic MHD equations and the related MHD equations.The main structure and contents of this paper are as follows,In chapter 1,we mainly introduce the background and the research status of the mathematical theory of Navier-Stokes equations,Primitive equations,and MHD system,and describe briefly the main works of this article.In chapter 2,we list some preliminary knowledge that will be needed in the later chapters,including Ladyzhenskaya-type inequalities for some kinds of three-dimensional integrals,logarithmic Sobolev inequality,a system version of the classic Gronwall inequality and Aubin-Lions compactness lemma,Minkowsky inequality.In chapter 3,we establish the global well-posedness of the 3D hydrostatic MHD equations on a thin domain.We assume that the initial value belongs to H2 space without any small assumption,we prove the global existence and uniqueness of the strong solutions of the initial boundary value problem.In chapter 4,we provide a rigorous justification of the solutions of the incompressible three-dimensional scaled magnetohydrodynamics(SMHD)equations to the solutions of the hydrostatic magnetohydrodynamics(HMHD)equations.For the H1-initial data case,we prove that global Leray-Hopf weak solutions of the three-dimensional SMHD equation strongly converge to the global strong solutions of the HMHD equations.In the H2-initial data case,the strong solutions of the SMHD can be extended to be a global one.As a consequence,we observe that the global strong solutions of the SMHD strong converge to the global strong solutions of the HMHD equations.The convergence rate is of the same order as the aspect raatio parameter.On the basis of chapter 3,we study the detailed proof of the global well-posedness of the strong solutions of the HMHD equations with only horizontal viscosity and horizontal magnetic resistivity in chapter 5.There is no smallness assumption on H2-initial data.We prove the global existence and uniqueness to the strong solutions of the initial boundary value problem.Moreover,we also establish the global well-posedness of strong solutions for this system,with any initial data(?)∈H~1∩L~∞,such that(?)∈L~m,for some m∈(2,∞),by using the logarithmic type anisotropic Sobolev inequality and the logarithmic type Gronwall inequality.Combining with the conclusion in Chapter 5,we prove that the initial data(?)∈H~1∩L~∞,such that(?)∈L~m,for some m∈(2,∞),the global Leray-Hopf weak solutions of the MHD system with horizontal dissipation and magnetic diffusion strongly converges to the global strong solutions of the HMHD equations with horizontal dissipation and magnetic diffusion in Chapter 6.When the initial data(?)∈H2,given a finite time,we prove that the strong solutions of MHD equations with horizontal dissipation and horizontal magnetic diffusion strongly converge to the strong solutions of the HMHD equations with horizontal dissipation and horizontal magnetic diffusion.The converrgence rate is of order (?),where β = min{α-2,2}.更多还原