Existence Of Some Ferromagnetic Chain Equation

In this dissertation, some ferromagnetic chain equations based on Landau-Lifshitz equation are considered from PDE-theory viewpoint. They have been applied to magnetics, electronics and materials. There are also many results on solitons, dynamic systems and numerical computations.In chapter 1, we briefly introduce the background in physics and developments of Landau-Lifshitz equation, in addition, the main work of this dissertation is described.In chapter 2, when the effective magnetic field consists of exchange energy and anisotropy energy, we consider the corresponding Landau-Lifshitz equation with Gilbert term. The existence of periodical solutions is proved by the difference method; For the problem with Neumann boundary condition, we applying the penalty method and an auxiliary function α.In chapter 3, a one-dimensional isotropic biquadratic Heisenberg spin chain equation is considered. Using the method that will be introduced in Theorem 1.2, we prove the existence of weak solutions.Chapter 4 considers a modified Landau-Lifshitz equation proposed by A. Visintin[4]. We also use the method in Theorem 1.2 to prove that there exists a smooth 2-dimensional radical symmetric solution for this problem.Chapter 5 focuses on the existence of solutions for the dynamic equation of ferrimagnets. Observing the relation between this equation and the wave equation, we prove the uniqueness of the classical solution. The existence of smooth solutions for the Cauchy problem is also presented.