Two dimensional Landau-Lifshitz equations in micromagnetism

Landau-Lifshitz equations are fundamental equations in the theory of ferromagnetism. They describe the dynamics of the magnetization field inside ferromagnetic material. They were first derived by L. Landau and E. Lifshitz. In the thesis we study the Landau-Lifshitz equations with a damping term. These equations form a system of quasilinear parabolic equations. In the first part of the thesis, we study the simplified Landau-Lifshitz equations in which only the term that comes from exchange energy appears. We prove the existence, uniqueness and partial regularity for the weak solution of the equations. Our proof is simpler and clearer than the proof given by other authors. We also give a detailed description of the behaviour of the solution near point singularities. In the second part of the thesis, we study the full Landau-Lifshitz equations. The nonlocal term that comes from the self-induced magnetic field makes the equations hard to deal with by the conventional methods for parabolic equations. However, our method in the first part of the thesis can be extended without difficulty to prove the complete result of existence, uniqueness and partial regularity for the weak solution of the full Landau-Lifshitz equations. In the final part of the thesis, we discuss some interesting open problems.