δ-Viscosity Solution, Blow Up Solution And Global Solution Of Multidimensional Landau-Lifshitz Equations

For generalized Landau-Lifhsitz equations in multidimensions,Zhou Yulin and Guo Doling showed the global existence of weak solution in the case without Gilbert term in 1986. Nai-Heng Chang,J. Shatah,K. Uhlenbeck considered the initial value problem for the 2-dimensional cylindrical symmetric case In 1999,they proved that there exists one global smooth solution under the energy small initial condition. Recently,Ding Shijin and Guo Doling obtain the partial regularity of the weak solutions,whether it has global smooth solution is still an important open problem.To investigate the global existence of the smooth solution for Landau-Lifshitz equation in multidimensions (n > 2),we study the following four topics in this thesis:First,we prove the existence of the solutions of the static Landau-Lifshitz equation with multi-direct effective field and with Dirichlet boundary condition,and establish the stability of the solution of Landau-Lifshitz equation.Second,we introduce some new concepts such as J-viscosity solutions etc and provide some relative properties in order to study the existence and limiting behavior of the solution of the multidimensional Landau-Lifshitz equations. By virtue of these results we prove that there exists a smooth solution of the multidimensional Landau-Lifshitz equation with values in unit sphere. We show also that there are two disjoint open subsets such that the solution tends to (0,1,0) and (0,-1.0) on their arbitrary inner compact sets respectively,and to (0,0,1) somewhere in the interface which separates the two open subsets.Third,we present some exact global solutions with values in unit sphere for 2-dimensional Landau-Lifshitz equations with certain initial-boundary conditions,Next,we give a tuft of initial data which generate a group of exact global smooth solutions with values in the unit sphere. These solutions,as we shall prove later,form a continuum. This topic combined with the second one imply that we partly reply the open problem about the existence of global smooth solution for multidimensional Landau-Lifshitz equations with initial-boundary conditions.Fourth,We present some exact blow up solutions for n-dimensional Landau-Lifshitz equations with certain initial-boundary conditions.