Some Studies About The Landau-Lifshitz Equation

In the paper,we study Landau-Lifshitz equations about the magnetic materials.The most important equation is LL equation,and Landau-Lifshitz equations describe an evolution of spin field in continuous ferromagnetic.We study the plane wave solutions of the Schrodinger map equations with the especial method.The first chapter is introduction,mainly introduce the development history of Landau-Lifshitz equation in mathematics and physics.In the second chapter,we study that some stationary solutions of Schrodinger map equation.Specially,we construct a type of plane wave solution of Landau-Lifshitz equation with the model | u | = 1.In addition,we discover the law which when the spin vector u is moving along one direction with the model | x | from 0 tend to ?,the spin vector u approaches the south pole(0,0,-1)from the north pole(0,0,1).Landau-Lifshitz equations describe an evolution of spin field in continuous ferromagnetic.Therefore,it is very significant to study the problems about magnetization movement.Many people study a lot of problems and construct many solutions about the Landau-Lifshitz equation,but no one to study the linear plane wave solution.So,in the paper,we construct some stationary solutions of Schrodinger map equation which contains a style of plane wave solutions.In the third chapter,we study the global Schrodinger maps equation in dimensions d ?2:small data in the critical Sobolev spaces.In the fourth chapter,we review and summarize the research result and still not be solve problems.