Researches On The Global Well-posedness For Some Classes Of Dynamical Systems

With the development of science, there are more and more mathematical mod-els, which come from different disciplines and represent different application back-grounds. Meanwhile, these mathematical models also inspire researchers engaging in mathematics. In practical applications, there are many models including the time variable t. This kind of models is called evolutionary models. Evolutionary models together with their boundary conditions are called dynamical system. This disserta-tion studies the well-posedness of some classes of dynamical systems including three-dimensional Navier-Stokes-Vioght equations, incompressible non-Newtonian fluid and thermo-diffusion equations with second sound. Some meaningful results are also ob-tained.The dissertation is divided into the following five chapters.Chapter1mainly introduces the research background and presents research situa-tion, then describes what we shall do in this dissertation and at last gives some needed basic theories and inequalities in this dissertation.Chapter2studies the global well-posedness for three-dimensional incompressible Navier-Stokes-Voight equations with delays. When the external force term g(t,u(t-p(t))) is continuous and sub-linear with respect to the velocity u. The solution for this model is not unique caused by the continuity assumption. We apply the theory of multi-valued dynamical system to establish the existence of pullback attractors in two phase spaces and obtain relations among them.In Chapter3, we discuss the relationship between the non-autonomous systems and autonomous systems by analyzing the infinite dimensional dynamical systems of three-dimensional Navier-Stokes-Voight equations and the two-dimensional non-Newtonian fluid equations respectively. By theoretical analysis and calculation, we obtain the existence of pullback attractors after non-autonomous perturbation and the upper semi-continuity of pullback attractors for the dynamical system. The in-novations in this chapter include:(1) This chapter obtains the existence of pullback attractors Aε={-Aε(t)}t∈R for Navier-Stokes-Voight equations by using an idea of de-composition;(2) This chapter verifies that the pullback attractors {Aε(t)}t∈R after non-autonomous perturbation and the global attractor A satisfy limεâ†'0+distx(Aε(t),A)=0;(3) For the two-dimensional non-Newtonian fluid model, the same results also hold.Chapter4investigates the quasi-linear thermo-diffusion equations with second sound. In the process of the proof, the properties of the strong positive definite kernel are fully used. Concretely, the quasi-linear thermol-diffusion equations with second sound are transformed into the linear thermol-diffusion equations with second sound, then combining with properties of quasi-linear functions and the relationship of coef-ficients, some better results are obtained. The innovations in this chapter include:(1) This chapter explores a method for determining strong positive definite kernel;(2) This chapter obtains the global existence and exponential decay of solutions to the system by constructing energy functional and using multiplier techniques.Chapter5finally summaries the work done in this dissertation, and raises the prospect for our future research.