Global Classical Solutions To An Evolutionary Model For Magnetoelasticity

In this paper,we first prove the local-in-time existence of the evolutionary model for magnetoelasticity with finite initial energy by employing the nonlinear iterative approach given in[27]to deal with the geometric constraint M?S^d-1in the Landau-Lifshitz-Gilbert(LLG)equation.Inspired by[14,15],we reformulate the evolutionary model for magnetoelasticity with vanishing external magnetic field H_ext,so that a further dissipative term will be sought from the elastic stress.We thereby justify the global well-posedness to the evolutionary model for magnetoelasticity with zero external magnetic field under small size of initial data.This paper is divided into the following five chapters:The first chapter is divided into three sub-sections.The first section mainly introduces the background of the evolutionary model for magnetoelasticity,the development history and its research progress.Besides,we also introduce what we want to study.The second subsection mainly introduces difficulties and its solutions that we have in researching this article.In the third subsection,we mainly explain the results that we obtained in this paper(the main theorem).In Chapter 2,we will use the energy method to derive the a priori estimate of the system(1.1.2).The third chapter is divided into two subsections.In the first subsection we prove the lemma on the relation between the Lagrangian multiplier?(M)and the geometric constraint|M|=1.In the second subsection we will prove the existence of the local-in-time classical solutions to the LLG equation(3.0.1)with a given bulk velocity v by adopting the mollifier approximate method.The fourth chapter is divided into three subsections.In the first subsection,we construct the approximate system by iteration.The second subsection mainly proves the uniform energy bound of the iterative approximation system.And the third subsection gives the proof of the theorem 1.3.1.In the last chapter,we will construct a global classical solution to the magnetoelasticity model(1.1.2)under the external magnetic field H_ext=0 with small initial data.