非齐次不可压Navier-Stokes和MHD方程的整体适定性研究

发布时间：2019-04-10 06:59

【摘要】：本文主要研究两类重要的流体力学方程组:非齐次不可压缩Navier-Stokes方程组和非齐次不可压缩磁流体力学(MHD)方程组.Navier-Stokes方程组是一类描述粘性不可压缩流体动量守恒的运动方程组.磁流体力学(MHD)方程组主要描述磁场与导电流体(等离子体,液态金属等)之间的相互影响.本文主要内容有一下几个方面:第1章主要介绍了 Navier-Stokes方程组和MHD方程组的物理背景以及研究研究现状,并简单叙述了本文的主要工作结果.第2章列出了后面章节需要用到的一些预备知识,包括一些重要的不等式以及Littlewood-Paley 分解理论等.第3章研究了三维非齐次不可压缩的Navier-Stokes方程组的整体适定性.通过利用加权的Chemin-Lerner型Besov空间范数技巧,方程的代数结构以及Gagliardo-Nirenberg不等式,分别对速度场的水平分量和竖直分量进行能量估计,我们可以得到多项式小性初值条件下Navier-Stokes方程组的整体适定性.这实质改进了 Paicu和Zhang (J. Funct. Anal.2012)中指数型小性初值的条件结论.第4章给出了三维非齐次不可压缩的MHD方程组也有第3章中Navier-Stokes方程组类似的结果,并且我们说明了初始速度场和磁场的任意一个分量可以很大.第5章证明了三维非齐次不可压缩的MHD方程在无密度小性且带有高度震荡的初始速度场和磁场的条件下的整体适定性.第6章证明了带有随时间变化的背景磁场的二维无磁场耗散的MHD方程,在速度可随时间线性增长的特解附近的整体稳定性.这也是考虑带有随时间变化的背景磁场问题的首次尝试.
[Abstract]:In this paper, we mainly study two kinds of important hydrodynamic equations: inhomogeneous incompressible Navier-Stokes equations and inhomogeneous incompressible (MHD) equations in magnetohydrodynamic mechanics. Navier-Stokes equations are a kind of equations describing viscous incompressible flow. Equations of motion for the conservation of body momentum. The (MHD) equations of magnetohydrodynamics describe the interaction between magnetic field and conductive fluid (plasma, liquid metal, etc.). The main contents of this paper are as follows: in chapter 1, the physical background and research status of Navier-Stokes equations and MHD equations are introduced, and the main results of this paper are briefly described. Chapter 2 lists some preparatory knowledge needed in the later chapters, including some important inequalities and Littlewood-Paley decomposition theory. In chapter 3, the global well-posedness of three-dimensional inhomogeneous incompressible Navier-Stokes equations is studied. By using the weighted Chemin- Lerner type Besov space norm technique, the algebraic structure of the equation and the Gagliardo-Nirenberg inequality, the energy of the horizontal component and the vertical component of the velocity field are estimated respectively. We can obtain the global well-posedness of the Navier-Stokes equations under the condition of polynomial small initial value. This essentially improves Paicu and Zhang (J. Funct.. The conditional conclusion of the initial value of exponential miniaturization in Anal.2012. In chapter 4, we show that the inhomogeneous incompressible three-dimensional MHD equations are similar to those of the Navier-Stokes equations in Chapter 3, and we also show that any component of the initial velocity field and magnetic field can be very large. In chapter 5, we prove the global well-posedness of the three-dimensional inhomogeneous incompressible MHD equation under the condition of initial velocity field and magnetic field without density and high oscillation. In chapter 6, we prove the global stability of the two-dimensional non-magnetic dissipative MHD equation with a time-dependent background magnetic field near the special solution where the velocity can increase linearly with time. This is also the first attempt to consider a background magnetic field with time-varying variations.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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