The Study On Dissipation And Dispersion Mechanisms Of Some Nonlinear Evolution Equations

Dissipation and dispersion are two completely different mechanisms that ap-pear in nonlinear evolution equations.These two mechanisms play an important role in the study of well-posedness and behavior of solutions for nonlinear evolu-tion equations.In this dissertation,the well-posedness and long time behavior of solutions of some nonlinear evolution equations are established by studying the dissipation or dispersion mechanism.In particular,first,the dissipation mecha-nisms of two kinds of fluid equations are analyzed.With the dissipations result-ing in loss of energy in L2 framework,we apply the energy method to obtain the well-posedness and relaxation limit of solutions for the three dimensional Cauchy problem of magnetic fluid equation with Cattaneo law,which reveal the relation-ship between the magnetic fluid equation with Cattaneo law and the magnetic fluid equation with Fourier law,and obtain the pullback dynamic behavior-the existence of pullback exponential attractor of solutions for the two dimensional initial boundary value problem of a non-Newtonian fluid equation,which reveals the existence of pullback attractor with exponential attraction and finite fractal dimensions.Then the dispersion mechanism of a double dispersion equation is analyzed.With the dispersion resulting in time decay in L? framework,we ap-ply the Green function method to obtain the global well-posedness and long time behavior of solutions for the Cauchy problem of a double dispersion equation,which shows the time decay rates of solutions for the double dispersion equation are slower than that for the single dispersion equation.Finally,the coupling mechanism of dissipation and dispersion of the damped Boussinesq equation is analyzed.With the dissipation and dispersion resulting in time decay in L?framework simultaneously,we apply the Green function method to obtain the global well-posedness,long time behavior and the viscosity limit of solutions for the Cauchy problem of the damped Boussinesq equation,which shows the time decay rates of solutions when we consider the dissipation and dispersion simultaneously are faster than that when we consider only the dissipation or the dispersion and the solutions of damped Boussinesq equation will approach the solution of Boussinesq equation when the dissipation asymptotically vanishes.The specific content is summarized as follows:In chapter 1,we will introduce the relevant background,current research status,problems to be considered of some nonlinear evolution equations and some preliminaries.In chapter 2,we will study the dissipation mechanisms of two kinds of fluid equations.This chapter is divided into two parts.In the first part,the three dimensional Cauchy problem of the magnetic fluid equation with Cattaneo law will be considered.The local well-posedness with large initial data,global well-posedness with small initial data and the relaxation limit will be obtained.In the second part,the two dimensional initial boundary value problem of a non-Newtonian fluid equation will be considered.The existence of pullback exponen-tial attractor with large initial data will be obtained.In chapter 3,we will study the dispersion mechanism of a double dispersion equation.The Cauchy problem of a double dispersion equation in any finite dimensions will be considered.The global well-posedness,long time behavior and scattering with small initial data will be obtained.In Chapter 4,we will study the coupling mechanism of dissipation and dis-persion of the damped Boussinesq equation.The Cauchy problem of the damped Boussinesq equation in high dimensions will be considered.The global well-posedness,long time behavior and the viscosity limit with small initial data will be obtained.