Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
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–Bolch 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 Prior Estimates Uniform in 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
