Global Smooth Solutions Of The Modified Landau-Lifshitz-Bloch Equation

Magnetic theory is widely used in physics,chemistry,medicine and interdisciplinary fields.The existence,regularity and singularity of the solutions of the Landau-Lifshitz equation enrich the theory of magnetism.The Landau-LifshitzBloch equation is considered for the case below Curie temperature;however,many physical phenomena occur in the environment above Curie temperature,so the Landau-Lifshitz-Bloch equation is needed to deal with them.The LandauLifshitz-Bloch equation describes the magnetization dynamics and is commonly used to model micromagnetic phenomena at high temperatures.The analysis of the Landau-Lifshitz-Bloch equation needs to be further explored as it remains to be developed.In this paper,we focus on the following modified Landau-LifshitzBloch equation:This equation has a global smooth solution,and this solution is unique.This paper analyzes and proves this feature of the equation.The arrangements of the paper are as follow:The first chapter first briefly describes the background of the development of partial differential equations and their importance;secondly,it introduces the current status of the Landau-Lifshitz equation and the Landau-Lifshitz-Bloch equation is introduced.Besides,the main research contents and chapters of this paper are introduced.Chapter 2 gives the basic preparatory knowledge needed in the course of this paper,such as the important Gagliardo-Nirenberg inequality,embedding theorem of Sobolev space,differential and integral Gronwall inequality,H¨older inequality,etc.Chapter 3 is the main part of this paper.The main results of the modified Landau-Lifshitz-Bloch equation are given,and a-priori estimates are obtained us-ing the energy method,showing the existence of a global smooth solution of the modified Landau-Lifshitz-Bloch equation.Besides,the uniqueness of the global smooth solution of this equation is proved.