具有外磁场的Landau-Lifshitz方程  

The Landau-Lifshitz Equation with External Magnetic Field

作　　者：陈旭 苏金 黎泽 CHEN Xu;SU Jin;LI Ze(School of Mathematics and Statistics,Ningbo University,Ningbo 315211,China)

机构地区：[1]宁波大学数学与统计学院,浙江宁波315211

出　　处：《应用数学》2023年第2期319-326,共8页Mathematica Applicata

基　　金：国家自然科学基金(1200010237);浙江省自然科学基金(LY22A010005);2022年浙江省大学生科技创新活动计划暨新苗人才计划(2022R405B073)。

摘　　要：本文研究具有外磁场的Landau-Lifshitz方程的全局动力学,具体包括局部适定性、全局适定性、周期解的存在性.首先,利用移动标架法将Landau-Lifshitz方程转化为半线性Schrodinger方程,从而利用色散方程的技巧得到任意维的局部适定性和一维的全局适定性.其次,通过使用Landau-Lifshitz方程的对称性将周期解的存在性转化为常微分方程,从而证明了具有非零常值外磁场的Landau-Lifshitz方程具有非平凡的周期解,同时对于时间相关的外磁场,我们构造了若干具有典型动力学意义的特解.In this paper,we study the global dynamics of Landau-Lifshitz equations with external magnetic fields.Specifically speaking,the results include local well-posedness,global well-posedness,the existence of periodic solutions and the long-time behavior of solutions.We transform the Landau-Lifshitz equation into a semilinear Schrodinger equation by using moving frame method,and then use the techniques of dispersive equations to obtain the local well-posedness of any dimension and the global well-posedness of one dimension.In addition,we use the symmetry of the Landau-Lifshitz equation to transform the existence of periodic solutions into ordinary differential equations,and thus prove that the Landau-Lifshitz equation with nonzero constant external magnetic field has a nontrivial periodic solution.Moreover,for time dependent magnetic fields,we construct several special solutions with typical dynamical behaviors.

关 键 词：LANDAU-LIFSHITZ方程 适定性 特解 

分 类 号：O175.24[理学—数学]