Some Mathematical Problems In The Ferromagnetic Chain Equation (s)

This dissertation concerns the mathematical aspects for equations of ferromagnetic chains and the related models. Ferromagnetic chain equation was first proposed in 1935 by physicists L.D. Landau and E.M. Lifshitz when studying the dispersive theory for magnetic conductivity in magnetic materials. It is an important dynamical equation of magnetization and frequently appears in condensation physics. Very recently some refined models were proposed by physicists to depict the magnetic dynamics in ferrimagnetic materials and, spin polarization transport effects were taken into account in ferromagnetic materials as well. Random effects, caused by such as photons, conducting electrons and nuclear spins, were also introduced into the Landau-Lifshitz equation to account for fluctuations of the magnetic moment orientation, which leads to the highly nonlinear multiplicative stochastic Landau-Lifshitz equation. We prove rigorously in mathematics the existence and uniqueness of solutions in some sense for these models as well as their asymptotic behaviors. In particular, we obtain the existence and uniqueness for the first time for the stochastic Landau-Lifshitz equation in one dimension and the existence and uniqueness of small solutions for spatial dimension two and three. As far as we know, this is the first mathematical result for SLLE and is important for further studies.In Chapter 1, we briefly introduce the physical background, historic results and the main results in our dissertation.In Chapter 2, we consider the existence of weak solutions for the dynamical equation in ferrimagnetic materials by penalty methods and Galerkin's approximation. By fixed point theorem and some a priori estimates the existence and uniqueness for smooth solutions are obtained and more importantly we establish the relationship between these equations and the classical wave maps.In Chapter 3, we focus on the spin-polarized transport equation in dimension 2, for which we get the existence and uniqueness of smooth solutions by inverse function theorem and some a priori estimates. In Chapter 4, nonexistence of vortex solutions for Landau-Lifshitz equation without Gilbert damping term is obtained.In Chapter 5, we are concerned with the multiplicative stochastic Landau-Lifshitz equation. It is pointed out that the Stratonovich stochastic integral should be utilized to get the proper thermal consistency, based on which the smooth solutions are obtained via the difference method and the It(o|^) formula. Also some blow up phenomena are discussed in this chapter.