Study On The Solutions Of Landau-Lifshitz Equations

Landau-Lifshitz equation describes the phenomenon of dynamic magnetization about magnetic material. Just as the Navier-Stokes equation does in that of fluid dynamics, it bears a fundamental role in the understanding of non-equilibrium magnetism. For this equation many physicists and mathematicians have made a mass of outstanding contributions, but the existence of global smooth solution is still an important open problem.In this thesis, the solutions of the Landau-Lifshitz equations with external magnetic fields were researched and some interesting results were also given. This dissertation consists of the following four aspects.First of all, we studied the piecewise smooth solutions of the Landau-Lifshitz equation with discontinuous external magnetic field. When the external magnetic field is constant, to the two Landau-Lifshitz equations with or without external magnetic field ,we prove that their solutions can be converted reciprocally by the two theorems, moreover, we also gave two practical application examples for the two theorems; we have also studied the asymptotic behavior between the solutions of the above two equations and proved a necessary and sufficient condition that the solutions of the Landau-Lifshitz equation with external magnetic field converge to the solutions of the Landau-Lifshitz equation without external magnetic field .Secondly, when the external magnetic field changes with time or space, we got the theorem which denotes the relations between the solution of Landau-Lifshitz equation and its external magnetic field. Employing this relationship, we gave their global smooth solution and the energy explosive solution for the two Landau-Lifshitz equations with external magnetic field which changes with time or space.Thirdly, because the Landau-Lifshitz equation without external magnetic and the dissipation item bears an important role in the study of other types of Landau-Lifishitz equation, and we specifically found some explicit or implicit column symmetric solutions for the Landau-Lifshitz equation, further we depicted a number of images for the solutions through the Maple software.Finally, we studied the initial boundary value problem for the Landau-Lifshitz equation with discontinuous external magnetic field, and focused on studying the existence of weak solutions of the Landau-Lifshitz equation with discontinuous external magnetic field which is bounded or unbounded.These studies is helpful to in-depth understanding of Landau-Lifshitz equation, in particular, benefits the understanding the relationship between the solutions of the two Landau-Lifshitz equations with or without external magnetic field.