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《The Ferromagnetic Chain Equations at High Temperatures Analysis and Applications of the Landau-Lifshitz-Bloch Equations（高温下的铁磁链方程：Landau-Lifshitz-Bloch方程的分析与应用）》收集了与LLB模型有关的物理背景和研究成果，对LLB方程的弱解和光滑解，Maxwell-LLB方程的光滑解及其整体吸引子，具温度效应的LLB方程和非线性电子极化LLB方程方程的光滑解，随机和分数阶LLB的光滑解，多种广义LLB方程和LLB方程组的整体解，Maxwell-LLB方程的周期解等方面进行了深入、系统的研究，取得了一系列具有创新性的丰富的成果，并对各种最新研究成果给予了深入浅出的证明。
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ContentsPrefaceCHAPTER 1 The 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 2 Smooth 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 3 The 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 4 Landau–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 5 The 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–Bloch 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 6 Weak 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 Priori Estimates Uniform in ε and Existence of Global Weak Solutions 1026.4 Global Smooth Solution for Problem (6.1.1)–(6.1.4) 106CHAPTER 7 Smooth 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 8 Well-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 9 The 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
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Chapter 1 The Physics Background of Landau-Lifshitz-Bloch Equation 1.1 Landau-Lifshitz Equation In 1935， Landau and Lifshitz proposed the famous equation of motion in ferromagnetic media， also known as the Landau-Lifshitz equation (1.1.1) where denotes the magnetization vector， 2 are constants， Heff is an effective magnetic field strength and (1.1.2) where emag(s) denotes the overall magnetization energy density and (1.1.3) san(s) represents energy in different directions and can be written as (1.1.4) is a convex function and depends on the crystal structure of the material; we take the first-order approximation of: (1.1.5) where is a symmetric positive definite tensor， is an energy. The exchange energy is For steady-state cases， Maxwell’s equations can be written as (1.1.7) (1.1.8) For unsteady cases， Maxwell’s equations can be written as (1.1.9) (1.1.10) Landau-Lifshitz equations have been widely used in physics and mathematics， and have achieved a series of important results. It is objectively related to the heat flow of harmonic mappings and is an important basis and application of harmonic mappings in physics. The Landau-Lifshitz equation describes situations at low temperatures (below the Curie temperature). For high temperatures， it does not conform to physical laws， and in such cases， it is replaced by the Landau-Lifshitz-Bloch equation (hereinafter referred to as the “LLB equation”for high temperatures. 1.2 Landau-Lifshitz-Bloch Equation Consider the following stochastic Landau-Lifshitz equation: (1.2.1) The Langevin field is (1.2.2) where cdp denotes the ， or z components of 8ap. The Fokker-Planck equation corresponding to (1.2，1) is formulated for the distribution function on the sphere，differentiating over t with the use of (1.2.1) and calculating the right part with the methods of stochastic theory， one comes to the Fokker-Planck equation (1.2.3) It is easy to see the distribution function (1.2.4) satisfies the Fokker-Planck equation at an equilibrium. From equation (1.2.3) and (1.2.5) ones can derive the equation of motion for the spin， polarization of the distribution function which has the form (1.2.6) where y is the characteristic diffusion relaxation rate (1.2.7) Neglecting some features of the equation (1.2.6)， choosing the following distribution function (1-2.8) where ^(t) is chosen so that the equation (1.2.6) is satisfied. Then by (1.2.6) and (1.2.8)， one arrives where the longitudinal and transverse relaxation rates are The asymptotic forms of Fi and r2 are given by (1.2.9) (1.2.10) (1.2.12) When，the equation (1.2.9) has the equilibrium solution. Using m/ms instead of one can derive the following LLB equation (1.2.13) where， and L2 are respectively the longitudinal and transverse damping parameters. Here， (1.2.15) where 6C is the Curie temperature. Hm， Han correspond to smag，sex， in (1.1.3) respectively， A， B， ly fi0 are constants， and Heff can be represented as where Xu is a constant. Using the vector operation rule， we get (1.2.13) (1.2.16) (1.2.17) 1.3 Landau-Lifshitz-Bloch Equation with Temperature Effect From the previous sections，there exists a temperature phase transition process from the Landau-Lifshitz equation (low temperature) to the Landau-Lifshitz-Bloch equation (High temperature)， from which the Landau-Lifshitz-Bloch equation with temperature effects can be derived. Consider a rigid ferromagnetic conductor occupying a domain M3 with boundary， and denoting by E，H，D，B the electric field， the magnetic field， the electric displacement， and the magnetic induction， the behavior of the material is ruled by Maxwell’s equations (1.3.2) where J is the current density and pe is the free charge density. Assume the electromagnetic isotropy of the material and the following constitutive equations (1.3.3) where s，p，cr are respectively the dielectric constant， the magnetic permeability， and the conductivity， and M is the magnetization vector. From the phase transition theory and the first law of thermodynamics， the internal energy can be chosen in the following form (1.3.4) Hence the heat conduction equation for the temperature is (1.3.5) (1.3.6) where. The fields ， are small enough so that the quadratic terms (1.3.7) are negligible if compared to other contributions in (1，3-5)， Within this approximation scheme， the energy balance reduces to (1.3.8) Combining equations (1.3.1) and (1.3.2)，we obtain (1.3.9) (1.3.10) so the Landau-Lifshitz-Bloch equation with temperature effects are (1.3.11) (1.3.12) (1.3.13) (1.3.14) where we assume here that f is a known function of. The bound
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