Contents
Preface
CHAPTER 1 The Physics Background of Landau–Lifshitz–Bloch Equation 1
1.1 Landau–Lifshitz Equation 1
1.2 Landau–Lifshitz–Bloch Equation 2
1.3 Landau–Lifshitz–Bloch Equation with Temperature Effect 4
CHAPTER 2 Smooth Solutions of the Landau–Lifshitz–Bloch Equation 7
2.1 Existence of Smooth Solutions in Two Dimensions 9
2.2 Existence of Smooth Solutions for Small Initial Values in Three Dimensions 14
2.3 Uniqueness of Smooth Solutions 16
CHAPTER 3 The Initial-Boundary Value Problem of Landau–Lifshitz Equation 19
3.1 Landau–Lifshitz–Bloch–Maxwell Equation 19
3.1.1 Approximate Solutions and a Priori Estimates 22
3.2 The Existence of Generalized Solutions 28
3.3 Regularity and Global Smooth Solutions 34
3.3.1 In the Case of d = 2 34
3.3.2 In the Case of d = 3 43
CHAPTER 4 Landau–Lifshitz–Bloch–Maxwell Equations with Temperature Effect 49
4.1 The System with Temperature Effect 49
4.2 The Existence of Global Weak Solution 51
4.3 The Existence of Global Smooth Solution 59
4.4 The Uniqueness of Global Smooth Solution 64
CHAPTER 5 The Periodic Initial Value Problem for the High-Dimensional Generalized Landau–Lifshitz–Bloch–Maxwell Equations 67
5.1 The Periodic Initial Value Problem for the Landau–Lifshitz–Bloch–Maxwell Equations 67
5.2 The Approximate Solution to the Periodic Initial Value Problem 68
5.3 The Estimation of the Approximate Solution 69
5.4 Existence of Global Weak Solutions 73
5.5 The Solution to the Initial Value Problem of the High-Dimensional Generalized Landau–Lifshitz–Bloch Equation 74
5.5.1 The Approximate Solutions are Uniformly Bounded and Convergent 75
5.5.2 The Global Weak Solution for an Infinitely Long Cylinder 77
5.5.3 Uniqueness of Smooth Solutions 78
CHAPTER 6 Weak and Strong Solutions to Landau–Lifshitz–Bloch–Maxwell Equations with Polarization 85
6.1 Physical Background 85
6.2 Solutions to the Viscosity Problem 89
6.2.1 Global Solutions to the ODE (6.2.24)–(6.2.32) 91
6.2.2 Existence of Weak Solution for the Viscosity Problem 99
6.3 A Priori Estimates Uniform in ε and Existence of Global Weak Solutions 102
6.4 Global Smooth Solution for Problem (6.1.1)–(6.1.4) 106
CHAPTER 7 Smooth Solutions of the Fractional Order Landau–Lifshitz–Bloch Equation 115
7.1 A Priori Estimates for Local Smooth Solutions 116
7.2 Proof of Uniqueness of Solutions 121
CHAPTER 8 Well-Posedness and Ergodicity of Solutions for Stochastic Landau–Lifshitz–Bloch Equations 123
8.1 Smooth Solutions of Stochastic Landau–Lifshitz–Bloch Equation 123
8.1.1 A Priori Estimates of Solutions 126
8.1.2 The Uniqueness of the Path 129
8.2 Ergodicity of Stochastic Landau–Lifshitz–Bloch Equation 131
8.2.1 Relevant Background 131
8.2.2 The Main Results of This Section 135
8.3 The Existence of an Invariant Measurable Set 138
8.3.1 Energy Estimates 138
8.3.2 The Pathwise Uniqueness 141
8.3.3 Higher Regularity 143
8.3.4 Invariant Measure 148
8.4 Ergodicity: The Uniqueness of the Invariant Measure Set 153
8.4.1 Asymptotic Strong Feller Property 153
8.4.2 The Compact Property of Invariant Measures 160
8.4.3 Proof of the Gradient Flow Equation 163
CHAPTER 9 The Initial Value Problem of the Landau–Lifshitz–Bloch Equation Coupled with Spin Polarization Transport Equation 167
9.1 Landau–Lifshitz–Bloch Equation Coupled with Spin Polarization Transport Equation 167
9.2 Existence of Global Smooth Solutions 169
9.3 Uniqueness for the Global Smooth Solution 180
Bibliography 183