Preface CHAPTER 1 The Physics Background of Landau-Lifshitz-Bloch Equation 1.1 Landau-Lifshitz Equation 1.2 Landau-Lifshitz-Bloch Equation 1.3 Landau-Lifshitz-Bloch Equation with Temperature Effect CHAPTER 2 Smooth Solutions of the Landau-Lifshitz-Bloch Equation 2.1 Existence of Smooth Solutions in Two Dimensions 2.2 Existence of Smooth Solutions for Small Initial Values in Three Dimensions 2.3 Uniqueness of Smooth Solutions CHAPTER 3 The Initial-Boundary Value Problem of Landau-Lifshitz Equation>The Initial-Boundary Value Problem of Landau-Lifshitz Equation 3.1 Landau-Lifshitz-Bloch-Maxwell Equation 3.1.1 Approximate Solutions and a Priori Estimates 3.2 The Existence of Generalized Solutions 3.3 Regularity and Global Smooth Solutions 3.3.1 In the Case of d = 2 3.3.2 In the Case of d = 3 CHAPTER 4 Landau-Lifshitz-Bloch-Maxwell Equations with Temperature Effect 4.1 The System with Temperature Effect 4.2 The Existence of Global Weak Solution 4.3 The Existence of Global Smooth Solution 4.4 The Uniqueness of Global Smooth Solution CHAPTER 5 The Periodic Initial Value Problem for the High-Dimensional GeneralizedLandau-Lifshitz-Bloch-Maxwell Equations 5.1 The Periodic Initial Value Problem for the Landau-Lifshitz-Bloch-Maxwell Equations 5.2 The Approximate Solution to the Periodic Initial Value Problem 5.3 The Estimation of the Approximate Solution 5.4 Existence of Global Weak Solutions 5.5 The Solution to the Initial Value Problem of the High-Dimensional Generalized Landau-Lifshitz-Bolch Equation 5.5.1 The Approximate Solutions are Uniformly Bounded and Convergent 5.5.2 The Global Weak Solution for an Infinitely Long Cylinder 5.5.3 Uniqueness of Smooth Solutions CHAPTER 6 Weak and Strong Solutions to Landau-Lifshitz-Bloch-Maxwell Equations 6.1 Physical Background 6.2 Solutions to the Viscosity Problem 6.2.1 Global Solutions to the ODE (6.2.24)-(6.2.32) 6.2.2 Existence of Weak Solution for the Viscosity Problem 6.3 A Prior Estimates Uniform in ε and Existence of Global Weak Solutions 6.4 Global Smooth Solution for Problem (6.1.1)-(6.1.4) CHAPTER 7 Smooth Solutions of the Fractional Order Landau-Lifshitz-Bloch Equation 7.1 A Priori Estimates for Local Smooth Solutions 7.2 Proof of Uniqueness of Solutions CHAPTER 8 Well-Posedness and Ergodicity of Solutions for Stochastic Landau-Lifshitz-Bloch Equations 8.1 Smooth Solutions of Stochastic Landau-Lifshitz-Bloch Equation 8.1.1 A Priori Estimates of Solutions 8.1.2 The Uniqueness of the Path 8.2 Ergodicity of Stochastic Landau-Lifshitz-Bloch Equation 8.2.1 Relevant Background 8.2.2 The Main Results of This Section 8.3 The Existence of an Invariant Measurable Set 8.3.1 Energy Estimates
8.3.3 Higher Regularity 8.3.4 Invariant Measure 8.4 Ergodicity: The Uniqueness of the Invariant Measure Set 8.4.1 Asymptotic Strong Feller Property 8.4.2 The Compact Property of Invariant Measures 8.4.3 Proof of the Gradient Flow Equation CHAPTER 9 The Initial Value Problem of the Landau-Lifshitz-Bloch Equation Coupled with Spin Polarization Transport Equation 9.1 Landau-Lifshitz-Bloch Equation Coupled with Spin Polarization Transport Equation 9.2 Existence of Global Smooth Solutions 9.3 Uniqueness for the Global Smooth Solution Bibliography
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