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laboutique.edpsciences.fr-R002767 03 01 01 R002767 00 ED E107 00 01 01 The Ferromagnetic Chain Equations at High Temperatures Analysis and Applications of the Landau–Lifshitz–Bloch Equations 01 eng 08 196 03 22 2038900 17 01 P15 EDP Sciences & Science Press 01 01 P15 EDP Sciences & Science Press 11 00 20250915 02 01 002942 03 9782759838486 15 9782759838486 03 01 D1 EDP Sciences 45 03 laboutique.edpsciences.fr-002942 03 01 01 002942 03 9782759838486 15 9782759838486 10 EA E107 10 00 01 R002767 ED E107 1 10 01 02 Current Natural Sciences 01 01 The Ferromagnetic Chain Equations at High Temperatures Analysis and Applications of the Landau–Lifshitz–Bloch Equations 1 A01 01 A2205 Boling GUO GUO, Boling Boling GUO Guo Boling, an academician of Chinese Academy of Sciences, a renowned expert in applied mathematics and computational mathematics, research fellow of Institute of Applied Physics and Computational Mathematics, has made great achievements in qualitative and numerical study on the solutions, soliton solutions of nonlinear evolution equations, as well as infinite-dimensional dynamical systems. In addition to over 700 essays published on important journals home and abroad, he is the author of over ten books and 15 volumes of essays. He is the winner of the first prize and the third prize of National Natural Science Award, Guanghua Engineering Science and Technology Award, the Ho Leung Ho Lee (HLHL) Foundation Award, etc. 2 A01 01 A2383 Qiaoxin LI LI, Qiaoxin Qiaoxin LI 3 A01 01 A2384 Fangfang LI LI, Fangfang Fangfang LI 4 A01 01 A2385 Yitong PEI PEI, Yitong Yitong PEI 5 A01 01 A2386 Ying ZHANG ZHANG, Ying Ying ZHANG 6 A01 01 A2387 Lichen ZHAO ZHAO, Lichen Lichen ZHAO 01 eng 08 194 03 01 20 Mathematics;ferromagnetic chain equation;Landau–Lifshitz–Bloch;LLB;materials 10 2011 MAT007020 29 CLIL3052 01 510 05 03 00 This book presents a comprehensive exploration of the mathematical theory and recent advances related to the ferromagnetic chain equations at high temperatures, with a particular focus on the Landau–Lifshitz–Bloch (LLB) equation and its extensions. More specifically, the work addresses key topics, including the LLB equation, the coupled Maxwell–LLB system, temperature-dependent LLB models, nonlinear electron polarization systems, and stochastic and fractional LLB equations. It highlights a series of innovative results: ranging from the existence of weak and smooth solutions to the construction of global attractors and the analysis of periodic behaviors. By integrating rigorous mathematical derivations with underlying physical principles, it serves as a valuable reference for researchers in mathematics, physics, and materials science. 02 00 This book presents a comprehensive exploration of the mathematical theory and recent advances related to the ferromagnetic chain equations at high temperatures, with a particular focus on the Landau–Lifshitz–Bloch (LLB) equation and its extensions. 04 00 ContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IIICHAPTER 1The Physics Background of Landau–Lifshitz–Bloch Equation . . . . . . . . . . . . 11.1 Landau–Lifshitz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Landau–Lifshitz–Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Landau–Lifshitz–Bloch Equation with Temperature Effect . . . . . . . . . 4CHAPTER 2Smooth Solutions of the Landau–Lifshitz–Bloch Equation. . . . . . . . . . . . . . . 72.1 Existence of Smooth Solutions in Two Dimensions . . . . . . . . . . . . . . . 92.2 Existence of Smooth Solutions for Small Initial Values in Three Dimensions . . . .. . 142.3 Uniqueness of Smooth Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16CHAPTER 3The Initial-Boundary Value Problem of Landau–Lifshitz Equation . . . . . . . . 193.1 Landau–Lifshitz–Bloch–Maxwell Equation . . . . . . . . . . . . . . . . . . . . . 193.1.1 Approximate Solutions and a Priori Estimates . . . . . . . . . . . . . 223.2 The Existence of Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Regularity and Global Smooth Solutions . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 In the Case of d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 In the Case of d = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43CHAPTER 4Landau–Lifshitz–Bloch–Maxwell Equations with Temperature Effect . . . . . . 494.1 The System with Temperature Effect . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 The Existence of Global Weak Solution. . . . . . . . . . . . . . . . . . . . . . . . 514.3 The Existence of Global Smooth Solution . . . . . . . . . . . . . . . . . . . . . . 594.4 The Uniqueness of Global Smooth Solution . . . . . . . . . . . . . . . . . . . . . 64CHAPTER 5The Periodic Initial Value Problem for the High-Dimensional Generalized Landau–Lifshitz–Bloch–Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1 The Periodic Initial Value Problem for the Landau–Lifshitz–Bloch–Maxwell Equations . . .. 675.2 The Approximate Solution to the Periodic Initial Value Problem . . . . 685.3 The Estimation of the Approximate Solution . . . . . . . . . . . . . . . . . . . 695.4 Existence of Global Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 735.5 The Solution to the Initial Value Problem of the High-Dimensional Generalized Landau–Lifshitz–Bolch Equation . . . . . . . . . . . . . . . . . . . 745.5.1 The Approximate Solutions are Uniformly Bounded and Convergent . . .. . . 755.5.2 The Global Weak Solution for an Infinitely Long Cylinder . . . . 775.5.3 Uniqueness of Smooth Solutions . . . . . . . . . . . . . . . . . . . . . . . . 78CHAPTER 6Weak and Strong Solutions to Landau–Lifshitz–Bloch–Maxwell Equations with Polarization .. 856.1 Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Solutions to the Viscosity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.1 Global Solutions to the ODE (6.2.24)–(6.2.32) . . . . . . . . . . . . . 916.2.2 Existence of Weak Solution for the Viscosity Problem . . . . . . . 996.3 A Prior Estimates Uniform in   and Existence of Global Weak Solutions .. . 1026.4 Global Smooth Solution for Problem (6.1.1)–(6.1.4) . . . . . . . . . . . . . . . 106CHAPTER 7Smooth Solutions of the Fractional Order Landau–Lifshitz–Bloch Equation . 1157.1 A Priori Estimates for Local Smooth Solutions . . . . . . . . . . . . . . . . . . 1167.2 Proof of Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121CHAPTER 8Well-Posedness and Ergodicity of Solutions for Stochastic Landau–Lifshitz–Bloch Equations .. . . . 1238.1 Smooth Solutions of Stochastic Landau–Lifshitz–Bloch Equation . . . . 1238.1.1 A Priori Estimates of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 1268.1.2 The Uniqueness of the Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.2 Ergodicity of Stochastic Landau–Lifshitz–Bloch Equation . . . . . . . . . . 1318.2.1 Relevant Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.2.2 The Main Results of This Section. . . . . . . . . . . . . . . . . . . . . . . 1358.3 The Existence of an Invariant Measurable Set . . . . . . . . . . . . . . . . . . . 1388.3.1 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388.3.2 The Pathwise Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.3.3 Higher Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.3.4 Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.4 Ergodicity: The Uniqueness of the Invariant Measure Set . . . . . . . . . . 1538.4.1 Asymptotic Strong Feller Property . . . . . . . . . . . . . . . . . . . . . . 1538.4.2 The Compact Property of Invariant Measures . . . . . . . . . . . . . 1608.4.3 Proof of the Gradient Flow Equation . . . . . . . . . . . . . . . . . . . . 163CHAPTER 9The Initial Value Problem of the Landau–Lifshitz–Bloch Equation Coupled with Spin Polarization Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.1 Landau–Lifshitz–Bloch Equation Coupled with Spin Polarization Transport Equation .. . . . 1679.2 Existence of Global Smooth Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 1699.3 Uniqueness for the Global Smooth Solution. . . . . . . . . . . . . . . . . . . . . 180Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183<div> </div> 21 00 06 01 https://laboutique.edpsciences.fr/produit/1512/9782759838486/the-ferromagnetic-chain-equations-at-high-temperatures 01 00 03 02 https://laboutique.edpsciences.fr/system/product_pictures/data/009/983/433/original/9782759838479-The_Ferromagnetic_Chain_Equations_at_High_Temperatures_couv-sofedis.jpg 17 14 20250901T150442+0200 01 P15 EDP 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