Existence,Regularity And Stability Of Exponential Attractors For 2D Quasi-geostrophic Equations And Delay Lattice Systems

The fractionally dissipative 2D quasi-geostrophic equation is an important model in geophysical fluid dynamics.In recent years,there are few literature on the long-time dynamic behavior of solutions of 2D quasi-geostrophic equations with fractional dissipation.In this paper,we investigate the existence of global and random attractors in Banach space the existence and the regularity of global and exponential attractors in Hilbert space to fractionally dissipative 2D quasi-geostrophic equations.Moreover,the existence and robustness of exponential attractors for infinite dimen-sional autonomous and non-autonomous dynamical systems with delays are presented in this paper.The existence of pullback exponential attractors for an infinite lattice model of non-autonomous recurrent neural networks with discrete and distributed time varying delays is established.The existence and the robustness of exponential attrac-tors for two-dimensional nonlocal diffusion lattice systems with delays and small delays are proved,respectively.The thesis consists of eight chapters.Chapter one summarizes the research background and significances of fractionally dissipative 2D quasi-geostrophic equations,analyzes the current research situation of the fractionally dissipative 2D quasi-geostrophic equation,introduces the recent devel-opment of exponential attractors for dynamical systems with delays,and illustrates the main content,research methods and innovations of this paper.The second chapter reviews some definitions and famous theoretical results.In the third chapter,the asymptotic behavior of stochastic modified quasi-geostrophic equations with damping driven by colored noise is analyzed.In fact,the existence of random attractors in W2?,p(R2)is established,where ??(1/2 1],?-is the number strictly less than ? but close to it,and p satisfies 2?-2/p>1.We prove also the ex-istence of a global compact attractor in W2?-,P(R2)for autonomous quasi-geostrophic equations with damping.Here we do not add any modifying factor on the nonlinear term.In the fourth chapter,we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in H2?+s(T2)with ?>1/2 and s>1.We prove the existence of(H2?+s(T2),H2?+s(T2))-global attractor A,that is,A is compact in H2?+s(T2)and attracts all bounded subsets of H2?+s(T2)with respect to the norm of H2?+s(T2).The asymptot-ic compactness of solutions in H2?+s(T2)is established by using commutator estimates for nonlinear terms,the spectral decomposition of solutions and new estimates of high-er order derivatives.Furthermore,we show the existence of the exponential attractor in H2?+s(T2),whose compactness,boundedness of the fractional dimension and expo-nential attractiveness for the bounded subset of H2?-+s(T2)are all in the topology of H2?+s(T2).In the fifth chapter,first some sufficient conditions are presented for the exis-tence and the construction of pullback exponential attractors for infinite dimensional non-autonomous dynamical systems with delays.This abstract result is then used to establish the existence of pullback exponential attractors for an infinite lattice model of non-autonomous recurrent neural networks with discrete and distributed time-varying delays.In the sixth chapter,we study the long term dynamical behavior of a two-dimensional nonlocal diffusion lattice system with delay.First some sufficient conditions for the con-struction of an exponential attractor are presented for infinite dimensional autonomous dynamical systems with delay.Then,the existence of exponential attractors for the two-dimensional nonlocal diffusion delay lattice system is established by using the new method of tail-estimates of solutions and overcoming the difficulties caused by the nonlocal diffusion operator and the multi-dimensionality.In the seventh chapter,first some sufficient conditions are given for constructing a robust family of exponential attractors for infinite dimensional autonomous dynamical systems with small delays.As an application of our abstract results,we then show the existence of a family of robust exponential attractors for two-dimensional nonlocal diffusion lattice systems with small delaysChapter eight covers conclusion and prospect for future study.