Researches On The Global Well-posedness And Infinite-Dimensional Dynamical Systems For Some Classes Of Nonlinear Evolutionary Equations

In this doctoral dissertation, we mainly consider the global well-posedness and infinite-dimensional dynamical systems of global solutions for some classes of nonlin-ear evolutionary equations, we derive some significant conclusions for some types of concrete models. This dissertation is divided into the following eight chapters.Chapter 1 is preface, in which we mainly introduce the fundamental theories for the global well-posedness and the infinite-dimensional dynamical systems for some classes of evolutionary equations.In Chapter 2, we consider the blowup criteria for the solutions of 3D Boussinesq equations, which is deduced exactly in Lorentz space and Besov space.In Chapter 3, some types of nonhomogeneous nonlinear or semilinear Breese ther-moelastic systems are considered. By semigroup method, we verify the dissipation of the system and further obtain the global existence of solutions. Such results include those results due to Zhuangyi Liu and Bopeng Rao [63].In Chapter 4, the existence of the uniform attractor for 3D non-autonomous Navier-Stokes-Voight equations is established by the weak convergence methods and delicate choice of parameters. When the external force term is independent of time, the uni-form attractor we derive is the global attractor for autonomous system, i.e., our work is an extension of Titi’s result in [53]. Novelties in this chapter are:(1) how to get the estimates of convectional terms ((u·(?))u, u),((u·(?))u. (?)u) in the spaces H1 and H2 respectively; (2) how to establish the weak continuity.In Chapter 5, we first establish the existence of pullback attractors for 3D non-autonomous Navier-Stokes-Voight equations by proving the pullback asymptotical compactness of the process after choosing right parameters. The main difficulties and novelties include:(1) how to obtain the existence of uniform absorbing ball by choosing appropriate parameters; (2) how to estimate of nonlinear convectional terms ((u·(?))u, u), ((u·(?))u,(?)u) with parameters.In Chapter 6, we investigate the 3D non-autonomous Navier-Stokes-Voight equa-tions, obtain that the uniform bounded (independent ofε) of corresponding uniform attractors Aεwhen the singularly oscillating forces fε(0<ε<1) tends to fâ– asεâ†'0, and further prove the convergence of corresponding uniform attractors which is a dif-ficulty and novelty in this chapter.In Chapter 7, using some delicate estimates and appropriately selecting suitable parameter, we deduce the existence of pullback attractor in H2 for the 3D non-autonomous Benjamin-Bona-Mahony equations, which has improved the results of Park [80].In Chapter 8, we summarize main results and look into the future research in this dissertation.