The Study On Blowup Solutions Of Ferromagnetic Equation And Related Map Equations

In this dissertation, we study the blowup problem of ferromagnetic model for magnetically ordered materials and related models. We will study the blowup solution of ferromagnetic model and related models by using the comparison prin-ciple, modulation method, exact treatment of the solution, asymptotic expansion method of the perturbation, classical energy methods and some important inequal-ities.In Chapter1, first we briefly introduce the history of ferromagnetic models. Then we show the models and its research progress. Finally the structure of this dissertation and the main research contents are introduced.In Chapter2, two different systems which can be regard as the extreme cases of Landau-Lifshitz-Gilbert equation (LLG) are considered. When the gyromagnetic term of LLG vanishes away, we consider some special solutions for a modified harmonic map heat flow which map from (2+1)-dimensional space-time into the2-sphere. The existence of regular initial data leading to blow up in finite time is established. When the Gilbert term is omitted, a blowup solution is obtained by constructing a blowup solution of Landau-Lifshitz equation.In Chapter3, we study the class of equivariant solution of the Harmonic map equation (extreme case of Landau-Lifshitz-Gilbert). We define the Frenet basis, renormalized variables and the linear Hamiltonian. We construct the approximate solution and its localized version. We describe our bootstrap assumptions, set up the orthogonality conditions, define the nonlinear energies for the remainder radi- ation term and derive the modulation equations and the mixed energy/Morawetz type identity. We conclude the proof of the main result of this section establishing a finite time blowup and the accompanying asymptotics.In Chapter4, we construct the exact solution of2or3dimensional space-time Landau-Lifshitz equation (LL equation) raised in the ferromagnetic materials. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and non-symmetric coordinates. The types of solutions cover the finite time blow-up solution, vortex solution and periodic solution. In the end, we sketch some solutions and their spatial curvatures.In Chapter5, we present some exact solutions of multi-dimensional space-time (in-)homogeneous LL equations under the radially symmetric coordinates.3different non-blowup solutions of LL equations are presented. We also construct some blowup solutions of LL equations. These two different kinds of solutions all admit the infinity energy in the initial time. All the solutions for LL equations are on the cone. Furthermore, two exact blow up solutions on S2are obtained for the different inhomogeneous LL equations. The question of whether a solution of the inhomogeneous, isotropic LL equations can develop a finite time singularity on S2from smooth initial data is not clear. Our result show that the blowup phenomenon can really happen on this initial data.In Chapter6, two exact blowup solutions of (2+1)-dimensional space-time inhomogeneous isotropic Landau-Lifshitz equation (IILL for short) are constructed under suitable transformations. These blowup examples show that some solutions of the2-dimensional IILL on S2can develop a finite time singularity from smooth initial data with finite energy in finite (or infinite) spacial domain. Some properties about one kind of these solutions are illustrated by the graphs.In Chapter7, exact solutions for the multidimensional Schrodinger map equa-tion (SM for short) on hyperbolic2-space H2cone are obtained. Consequently we show the non-traveling wave solution on H2is a finite energy solution on the finite spacial domain. The question of whether a solution of SM can develop a finite time singularity on H2with smooth initial data is not clear. Our result show that blowup can really happen on this initial data. In addition, some exact global smooth solutions are constructed.In Chapter8, we consider the2+1dimensional space-time isotropic anti-ferromagnetic equation (IAF for short) to the2-sphere for equivariant data of homotopy number N≥1. Using the method of matched asymptotic expansions, we present an analysis of the asymptotic behavior of singularities arising in this special class of solutions. Specifically, a sharp description of the corresponding blowup rate and the stability are investigated in settings with certain symmetries. We also find out the blow up behavior of two different IAFs are very different. In the end, the blowup results are verified by numerical experiments.