On Landau-Lifshitz Equations

The Landau-Lifshitz equation bears a fundamental and important role in the understanding of magnetization dynamics, just as the Navier-Stokes equation does in that of fluid dynamics. As a nonlinear partial differential equation, it is an interesting field researched in mathematical physics.In this thesis, the anisotropic Landau-Lifshitz equations with external magnetic fields are researched. For initial-boundary value problems,·the existence of weak solutions,·the influence of different types of external fields and anisotropy (easy axis, easy plane),·the partial regularity of solutions in the interior domain and that up to the boundary,·the existence of partially regular solutions, and·the modern geometric analysis to the energy concentration phenomenon are most presented. These mathematical discussion does us a favor to understand the influence of terms in the equation, to catch the similarity and difference among many kinds of ferromagnetism, and to explore common properties and characteristic properties between the Dirichlet problem and the Neumann problem of the Landau-Lifshitz equation.