Landau-Lifshitz方程及其相关方程消失极限问题

Landau-Lifshitz equation is the basic mathematical model depicting the magnetization motion law, plays a very important role in the study of magnetism. LL equation is closely related with Schrödinger mapping, harmonic heat flow equations. LL equation is a strong degradation, strong coupled quasilinear equations, especially solution limited on the spherical surface, many classical methods are difficult to use, so that some mathematical problems are still open. In order to explore these issues, this project research how the some terms of LL effect wellposedness of the equation in a variety of space; relationship between the limit and Schrödinger mapping, harmonic heat flow, incompressible Navier-Stokes equation and other equations as some of the parameters of LL equation disappear; This project focuses on the problem of vanishing sampling limits for LL equation with Gilburg damping proposed by professor Zhouping Xin. Our research method is first construct manifold by using Lie group methods and then use the analysis method of harmonic energy estimation in manifold.

Landau-Lifshitz方程是刻画磁化运动规律的基本数学模型, 在研究磁学中起着非常重要的作用. LL方程与Schrödinger映射, 调和热流等方程密切相关. 由于LL方程是一个强退化, 强耦合的拟线性方程组, 尤其解限制在球面上, 很多经典方法难以应用以至于关于这个方程一些数学问题仍然是公开的. 为了探讨这些问题, 本项目研究LL方程的各项在各种空间中对方程解的适定性影响；研究LL方程解的一些参数消失极限与Schrödinger映射, 调和热流, 不可压Navier-Stokes方程等其它方程解之间关系；本项目重点研究辛周平教授提议我们研究的具有Gilburg阻尼LL方程消失极限问题. 我们的研究方法是首先利用李群方法构造所需流形, 然后在流形上用调和分析方法进行能量估计.

Landau-Lifshitz方程是刻画磁化运动规律的基本数学模型,在研究磁学中起着非常重要的作用.关于Navier-Stokes方程和Maxwell方程耦合方程组我们建立了时空整体正则解的存在性.关于三维粘性依赖于密度柱对称不可压MHD方程,我们构造了一个逼近三维柱对称模型的一维模型,证明了该一维模型全局光滑解的存在性,通过对该一维模型进行扰动,构造三维柱对称模型,并证明了由一维模型构造出来的三维模型的解是全局光滑的.讨论Euler方程的解在有限时间段内的稳定性及在时间趋于无穷时的不稳定性.关于具有各向异性情形Landau-Lifshitz方程,我们获得了两个挺有趣结果:就一种著名的高阶效应场情形构造了一个显式解;就三个方向互不相同各向异性场情形也构造了一个具体的例子.就液晶方程我们证明在相同初边值条件下吉尔伯特阻尼系数趋于0时液晶方程弱解趋于极限方程的弱解,在H^2中讨论吉尔伯特阻尼系数趋于0时液晶方程弱解趋于极限方程弱解的条件,并得出液晶方程的阻力趋于0时系统能量小于等于初始时刻能量.