Contents -- Preface -- 1 Spin Waves and Equations of Ferromagnetic Spin Chain -- 1.1 Physics Background for the Equations of Ferromagnetic Spin Chain -- 1.1.1 Motion Equations for Magnetization -- 1.1.2 Landau-Lifshitz Equations -- 1.2 A Simple Derivation of Landau{Lifshitz Equation -- 1.2.1 Magnetically Ordered Crystals -- 1.2.2 The Wave Function and Spin Operator for the System of Two Electrons -- 1.2.3 Multi-electron Wave Function and Spin Operator -- 1.3 Equations for the Antiferromagnets -- 1.3.1 Antiferromagnetic Moments and Magnetic Energy -- 1.3.2 Equations for the Antiferromagnets -- 1.4 Spin Waves in Ferromagnets -- 1.4.1 Equilibrium State of Ferromagnets -- 1.4.2 Spin Waves in Ferromagnets -- 1.4.3 Damping of Spin Waves -- 1.5 Spin Waves in Antiferromagnets -- 1.5.1 Equilibrium States of Antiferromagnets -- 1.5.2 Spin Waves in Antiferromagnets -- 1.5.3 Electromagnetic Waves in Magnetically Ordered Crystals -- 1.6 Bibliography Comments -- 2 Integrability of Heisenberg Chain -- 2.1 Spin Waves and Solitary Waves -- 2.1.1 Spin Waves -- 2.1.2 Solitary Waves -- 2.1.3 Approximate Solutions -- 2.1.4 Equivalence to Nonlinear Schr odinger Equations -- 2.2 Geometric Representation for the Landau{Lifshitz Equations -- 2.2.1 Establishing the Natural Coordinate System -- 2.2.2 Geometric Representation of Landau-Lifshitz Equation -- 2.3 Inhomogeneous Heisenberg Chain -- 2.3.1 Inhomogeneous Ferromagnetic Equations -- 2.3.2 Inhomogeneous Heisenberg Chain -- 2.4 Spherical (Cylindrical) Symmetric Heisenberg Equations of Ferromagnetic Spin Chain -- 2.4.1 Radial Symmetric Equations -- 2.4.2 Painleve Property of the Radial Symmetric Nonlinear Schr odinger Equation -- 2.4.3 Normal Change for Radial Symmetric Equation and Matrix Form for the Radial Symmetric Heisenberg Equations.

2.4.4 B acklund Change for Radial Symmetric Nonlinear Schr odinger Equation and Solitary Solutions -- 2.5 Bibliography Comments -- 3 One-Dimensional Landau -Lifshitz Equations -- 3.1 Initial Boundary Value Problem of One-dimensional Ferromagnetic Spin Chain Equations -- 3.1.1 Initial Boundary Value Problem of Ferromagnetic Spin Chain Equations -- 3.1.2 Theory on Quasilinear Parabolic Equations -- 3.1.3 Approximate Solution to the Initial Boundary Problem of the System of Ferromagnetic Spin Chain -- 3.1.4 Weak Solution to the Ferromagnetic Spin Chain Equation -- 3.2 Nonlinear Initial-boundary Value Problem for the System of Ferromagnetic Spin Chain -- 3.2.1 Nonlinear Initial-boundary Value Problem for the System of Ferromagnetic Spin Chain -- 3.2.2 Discrete Solution of Nonlinear Ordinary Di erential Systems -- 3.2.3 Global Weak Solution for the Spin System -- 3.2.4 Global Weak Solution to the Equations of Ferromagnetic Spin Chain -- 3.2.5 Mixed Boundary Value Problem -- 3.2.6 The Mixed Boundary Problem of Equations of Ferromagnetic Spin Chain -- 3.3 Smooth Solution for the Ferromagnetic Spin Chain Systems -- 3.3.1 Smooth Solution to the Nonlinear Systems with Periodic Initial Boundary Conditions -- 3.3.2 Smooth Solution to the Equations of Ferromagnetic Spin Chain -- 3.4 Smooth Solution for the 1D Inhomogeneous Heisenberg Chain Equations -- 3.4.1 Inhomogeneous Heisenberg Chain Equations -- 3.4.2 " > 0: Local Smooth Solution -- 3.4.3 " > 0: Global Solution and Uniform Estimates -- 3.4.4 " = 0: Global Solution and Uniqueness -- 3.5 Measure-Valued Solution to the Strongly Degenerate Compressible Heisenberg Chain Equations -- 3.5.1 Compressible Heisenberg Chain Model and Compressible Heisenberg Chain System -- 3.5.2 Measure-Valued Solution to the Strongly Degenerate Equations -- 3.6 Bibliography Comments -- 4 Landau-Lifshitz Equations and Harmonic Maps.

4.1 Weak Solution to Multidimensional Ferromagnetic Spin Chain Equations -- 4.1.1 Existence of Weak Solution to Multidimensional Ferromagnetic Spin Chain Equations -- 4.1.2 Weak Solution to Multidimensional System of Ferromagnetic Spin Chain -- 4.2 Landau-Lifshitz Equations on Riemannian Manifold and Harmonic Maps -- 4.2.1 Landau-Lifshitz Equations and Harmonic Maps -- 4.2.2 Local Smooth Solution of L-L Equation -- 4.2.3 Global Smooth Solution to L-L Equation -- 4.2.4 On the Landau{Lifshitz Equation on Riemannian Surface -- 4.3 Generalized L-L Systems and Harmonic Maps -- 4.3.1 Generalized Landau-Lifshitz Systems -- 4.3.2 The Global Weak Solution to the Generalized L-L Equations -- 4.4 Regularity of Weak Solutions to the Two-Dimensional Landau-Lifshitz Equations -- 4.4.1 Cauchy Problem to Two-Dimensional L-L Equation -- 4.4.2 Proof of Regularity of Solution to L-L Equation -- 4.5 Ginzburg-Landau Approximation to Landau{Lifshitz Equations -- 4.5.1 Estimates for Strong Parabolic System -- 4.5.2 L and L Bounds for 2a(x)(1 -- 4.5.3 Higher Order Interior Estimates -- 4.5.4 Boundary Estimates -- 4.5.5 Energy Estimates -- 4.5.6 Hausdor Measure Estimate for Singularity -- 4.5.7 Passing to the Limits -- 4.5.8 Chen-Struwe Solution -- 4.6 Smooth Solution and Decay Estimates to the L-L System with Small Initial Data in Higher Dimensions -- 4.6.1 Initial Value Problem to the L{L System in Higher Dimensions -- 4.6.2 Decay of Solution to the Higher-Dimensional L-L Equations -- 4.7 Radial Solution -- 4.7.1 Two-Dimensional Radial Symmetric Landau-Lifshitz Equation -- 4.7.2 A Priori Estimates -- 4.7.3 Existence of Local Solutions -- 4.8 Bibliography Comments -- 5 Landau-Lifshitz-Maxwell Equations -- 5.1 Global Weak Solution in Three Dimensions -- 5.1.1 The Periodic Initial Value Problem.

5.1.2 Approximate Solutions and the A Priori Estimates for the Periodic Initial Value Problem -- 5.1.3 Existence of Generalized Solution -- 5.1.4 Existence of Solution for the Initial Problem -- 5.2 Global Smooth Solution in One or Two Dimensions with Small Initial Data -- 5.2.1 The Problem -- 5.2.2 A Priori Estimates -- 5.2.3 Existence of Global Smooth Solution -- 5.3 Global Smooth Solution to One-Dimensional L-L-M with Large Data -- 5.3.1 Viscosity Vanishing Method -- 5.3.2 Global Existence of Smooth Solution -- 5.4 Global Weak Solution to L{L{M System on Riemannian Manifold -- 5.4.1 L-L-M System on Riemannian Manifold -- 5.4.2 Existence of Generalized Solution -- 5.5 Partial Regularity for Stationary Solutions to L{L{M Equations -- 5.5.1 Quasi-Static Maxwell Equations -- 5.5.2 Definition of Stationary Solution -- 5.5.3 Estimates for Local Energy -- 5.5.4 Energy Decay and Partial Regularity -- 5.6 Weak Solutions to Landau-Lifshitz-Maxwell Equations with Polarization -- 5.6.1 The Problem and Physics Background -- 5.6.2 Viscosity Approximation -- 5.6.3 Solutions to the Viscosity Problem -- 5.6.4 Existence of Weak Solution for the Viscosity Problem -- 5.6.5 A Priori Estimates Uniform in " -- 5.6.6 Global Existence of Weak Solutions -- 5.7 Bibliography Comments -- 6 Long Time Behavior of Solutions to the System of Ferromagnetic Spin Chain -- 6.1 Existence and Stability of Steady State Solutions -- 6.1.1 One-Dimensional Landau-Lifshitz Equations -- 6.1.2 Stability of the Steady State Solutions -- 6.2 Asymptotic Behavior of L-L Equations -- 6.2.1 Estimates for Energy -- 6.2.2 Bifurcation and Chaos -- 6.3 Approximate Inertial Manifold for One-Dimensional L-L Equations -- 6.3.1 One-Dimensional Landau-Lifshitz Equations -- 6.4 Attractor of Landau-Lifshitz Equations on Riemannian Manifold -- 6.4.1 Landau-Lifshitz Equations on Riemannian Manifold.

6.4.2 The A Priori Estimates -- 6.4.3 The Global Attractor -- 6.4.4 The Estimates of the Upper Bounds of Hausdor. and Fractal Dimensions for the Attractors -- 6.5 The Attractors for Landau-Lifshitz-Maxwell Equations -- 6.5.1 Periodic Initial Value Problem to L-L-M Systems -- 6.5.2 A Priori Estimates -- 6.5.3 Dimension Estimate of Attractor -- 6.6 Bibliography Comments -- Bibliography.