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.

Contributor: Boling GUO

Boling GUO is a Member of the Chinese Academy of Sciences (CAS) and researcher at the Beijing Institute of Applied Physics and Computational Mathematics. His research interests include nonlinear evolution equations, soliton theory, infinite-dimensional dynamical systems, stochastic infinite-dimensional dynamical systems, and fractional partial differential equations. He has authored more than 800 journal papers and over 10 books.