Existence And Decay Estimates Of The Global Solution For The Landau-Lifshitz-Gilbert Equation

Landau-Lifshitz equation is the equation describing the phenomenon of dynamic magnetization about magnetic material. Just as the Navier-Stokes equations does in that of fluid mechanics, Landau-Lifshitz equation plays a very important role in the nonequilibrium magnetic studies. Moreover, Landau-Lifshitz equation has an extremely important applications in materials science, ferromagnetic and electromagnetic medium. In recent yeas, mathematicians made a lot of achievements in the theory of dynamical systems, soliton solutions and numerical computing for this equation.In this paper, we mainly consider the Cauchy problem for the Landau-Lifshitz-Gilbert equation in R3:First of all, by energy estimate and standard continuity argument, we obtain the global existence and uniqueness of smooth solution to the 3D Landau-Lifshitz-Gilbert equation under the suitable small initial data. Secondly, we built the time decay rates for the solution.In this paper, we let λ1=λ2=1. Then we consider the above problem about the global existence of smooth solution in Hm(R3) and decay estimates.This article is divided into four chapters.The first chapter is the introduction. This chapter introduces the significance, back-ground and the research results of Landau-Lifshitz equation.The second chapter is the basic knowledge.The third chapter is the energy estimate. We study the global existence of smooth solutions to 3D Landau-Lifshitz-Gilbert equation in Sobolev space Hm(R3).The fourth chapter is the decay estimates. We study the decay of solutions to 3D Landau-Lifshitz-Gilbert equation.