The Finite Difference Scheme And Global Regular Solution For Landau-Lifshitz Equation

This dissertation consists of two parts. In the first part we consider the convergence and stability of finite difference scheme for Landau-Lifshitz equation in one dimensions space and 2d radial symmetric Landau-Lifshitz equation. Then, in second part, we consider the existence and uniqueness of global regular solution for one-dimensional and 2-D radial symmetric Landau-Lifshitz equation.This dissertation consists of five chapters. In chapter 1, we briefly introduce the background in physics and developments for the Landau-Lifshitz equation. In which the main work of the dissertation is described. In chapter 2, we study finite difference scheme of the nonhomgeneous boundary value problem for the Landau-Lifshitz equation in one-dimensional space. We first derive a difference scheme, that we show to preserve an important property of the continuous model 梩he conservation of the norm of magnetization. We then prove that, under certain suitable condition, the scheme is conditional convergent and stable. In chapter 3, we extend the idea of chapter 1 to finite difference scheme for 2-D radial symmetric Landau-Lifshitz equation and obtain a theorem for the stability and convergence of finite difference scheme. In chapter 4, using a space discretization and employing the method of induction, we prove that there exists a unique global regular solution for homogeneous Neumann boundary value problem of one-dimensional Landau-Lifshitz equation. In chapter 5, we study the regularity and the limit behavior of solution for 2-D radial symmetric Landau-Lifshitz equation. Employing Gronwall inequality with parameter, we show that the global regular solution of Landau-Lifshitz equation converge, globally in time, to that of its limit equation as the parameter tends to zero.