Asymptotic Behavior For Non-Newtonian Fluid And Lattice Systems

This dissertation is divided into two parts.The first one(from Chapter 1 to Chapter 6) is related to the asymptotie behavior of solutions for non-Newtonian fluid.The second one(from Chapter 7 to Chapter 11) is concerned with the asymptotic behavior of solutions for lattice systems.Fluid dynamics occurs in physics,biology,atmosphere and ocean.ete. Non-Newtonian fluid is an important branch of modern fluid dynamics.The first part of this dissertation studies the existence and regularity of trajectory attractor,uniform attractor and pullback attractor for the Ladyzhenskaya model on non-Newtonian fluid. Also we prove the stability of the uniform attractor for the non-Newtonian fluid with rapidly oscillating external forces,and establish the existence and regularity of pullback attractor for the non-Newtonian fluid with delays.Discretization and continuity are the two forms of motion for the objective substance.Lattice dynamical systems(LDSs) are the spatiotemporal systems with discretization in some variables including coupled ODEs and coupled map lattices and cellular automata[26,36].In some cases,LDSs occur as spatial discretizations of partial differential equations(PDEs).In the second part of the dissertation,we first establish the sufficient and necessary conditions for the existence of global attractor,kernel sections and uniform attractor for the lattice systems with delays. Then we apply the results to retarded lattice reaction-diffusion equations and verily the existence,upper semicontinuity and limiting behavior of the global attractor.Thirdly,we prove the existence,upper semicontinuity and boundedness of Kohnogorovε-entropy of compact kernel sections for the process associated to Klein-Gordon-Schrodinger equations and long-wave-short-wave resonance equations on infinite lattices.At the same time,we consider the limiting behavior of global attractors for the nonclassical parabolic equation and complex Ginzburg-Landau equation on infinite lattices.We also prove a criteria of finite fractal dimensionality for compact set in Hilbert space and apply this criteria to concrete lattice systems.Finally,we establish a sufficient condition for the existence of stochastic global attractor for stochastic lattice dynamical system and apply this result to the stochastic lattice sine-Gordon equation.The dissertation is arranged as follows.In Chapter 1,we first summarize the background of infinite dimensional dynamical system,as well as the concept and classical result related the infinite dimensional dynamical system.Then we introduce the physical significance of the non-Newtonian fluid and summarize the related researching surveys.Also,we summarize the main result on the non-Newtonian fluid within this dissertation.Lastly,we introduce the origin of the infinite lattice systems and the researching surveys,and also we summarize the main result on infinite lattice systems within this dissertation.In Chapter 2,we consider the existence of trajectory attractor for the non-Newtonian fluid.When the uniqueness of solutions is unknown,we consider the natural translation semigroup acting on the trajectory space and prove the existence of compact absorbing set.Then we establish the existence of trajectory attractor and general global attractor.In Chapter 3,we consider the existence of uniform attractor for the non-Newtonian fluid.Firstly,we prove the existence of the uniform attractor in space H,via some delicate a priori estimations.Then we use the analysis of spectrum to elliptic operator and establish the existence of the uniform attractor in space V.Lastly,we use the uniform Gronwall inequality and the character of the equations to verify that the obtained two uniform attractors coincide with each other,i.e.,we prove the regularity of the uniform attractor, which implies the asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.In Chapter 4,we consider the existence of pullback attractor for the non-Newtonian fluid.We first prove the existence of the uniform attractor in space H,via some delicate a priori estimations.Then we use the analysis of spectrum to elliptic operator and establish the existence of the pullback attractor in space V.Finally,we use the uniform Gronwall inequality to verify that the obtained two pullback attractors coincide with each other, which reveals the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.In Chapter 5,we consider the non-Newtonian fluid driven by external forces that are rapidly oscillating in time but have a smooth average.We mainly prove that the uniform attractor of the oscillating equations could be approximated by the one of the averaged equations.Chapter 6 discusses the non-Newtonian fluid with delays.We first prove the existence of pullback attractor for the process defined on different state space.Then we verify the relation between these pullback attractors via energy method.Finally,we prove the regularity of the pullback attractor by using the obtained relation.In Chapter 7.we first give a sufficient and necessary condition for the existence or a global attractor associated to retarded lattice systems.Then we apply this result to lattice reaction-diffusion equations and obtain the existence of global attractors,and then we consider the singular limiting behavior of the global attractor as the length of the delayed interval tend to zero.Lastly,we remark that some similar results holds for the compact kernel sections and uniform attractors.In Chapter 8,we prove the existence,upper semi-continuity and an upper bound of Kolmogorov-εentropy of compact kernel sections for the process associated to Klein-Gordon-Schrodinger equations and long-wave-short-wave resonance equations on infinite lattice.In Chapter 9.we consider the singular limiting hehavior of the global attractor for the nonclassical parabolic equation and complex Ginzburg-Landau equation on infinite lattice.In Chapter 10,we prove a criteria of finite fractal dimensionality for compact set in Hilbert space and apply this criteria to first-order lattice systems.In Chapter 11,we establish a sufficient condition for the existence of stochastic global attractor for stochastic lattice dynamical system and apply this result to the stochastic lattice sine-Gordon equation.