Researches On The Global Well-posedness And Attractors For Some Classes Of Nonlinear Evolutionary Equations

In the field of the research and application of modern science and technology, especially in applied mathematics, physics, control engineering, biology and other re-lated subject, the basic mathematical models are partial differential equations. Nonlin-ear evolution equations usually means a partial differential equations with one of the independent variables being time t. We mainly investigate the global well-posedness theory of some classes of nonlinear evolution equations, including the global well-posedness and existence of attractors of solutions to radiative fluids equations, liquid crystal equations and viscoelastic equation in this dissertation. We have obtained some meaningful results.This dissertation is divided into the following six chapters.Chapter1is the preface and preliminary knowledge, in which we mainly intro-duce the relative research background, present research situation and introduce what we shall do in this dissertation. At last we give some needed definitions and lemmas.Chapter2studied the compressible infrarelativistic model and proves the global existence and large-time behavior of solutions to this model. Novelties of this chapter are:(1) Using a suitable expression of specific volume and the delicate prior estimates, we establish the positively lower bound and upper bound of the specific volume.(2) Using the embedding theorems and the delicate interpolation inequalities, we have overcome the mathematical difficulties caused by the higher order of partial deriva-tives and proved the global existence and large-time behavior of solutions in higher regular spaces.In Chapter3, we obtain the the large-time behavior of solutions to the compress-ible radiative fluid equations:the pure scattering case, and correct some defects in [27]. Furthermore, we prove the global existence and large-time behavior of the solutions to the system in more regular spaces under new assumptions. Novelties of this chapter are:(1) We correct some defects about the global existence and asymptotic behavior of solutions to the system in [27].(2) By the new assumption0<σs<C|θ-θ|αk(v) and within the suitable range of the constant a, we prove the global existence and large-time behavior of solutions in higher regular spaces.(3) We prove the global existence and large-time behavior of classical solutions.In Chapter4, we first establish the large-time behavior of solutions to a one-dimensional compressible liquid crystal fluid equations. The novelty in this chapter is that using a suitable expression of the specific volume, we establish uniform bound of the specific volume by the embedding theorems and a sequence of delicate interpo-lation techniques and then prove the large-time behavior of solutions to the system by using Shen-Zheng inequality.Chapter5investigates a non-autonomous viscoelastic equation with a past history. In this chapter, under the necessary assumption on history kernel, we establish the suitable energy functional and Lyapunov functional and first prove the existence of the uniform attractors by the energy perturbation and multiplier techniques method. Our results improve those results by R. O. Araujo et al.[4].At last, we summarize our work and prospect the future research in this disserta-tion in Chapter6.