Stability And Numerical Approximation Of Attractors For Several Types Of Stochastic Equation

Random dynamical system is the generalization of dynamical system in random cases.They have wide applications in physics,mechanics,biology,oceanography,engineering and other scientific fields.Random attractor is one of the important tools to depict the asymptotic behavior of random dynamic systems.In this thesis,we mainly study the dynamic behavior for several classes of stochastic equations by random attractors,weak pullback mean random attractors,numerical attractors and stationary solutions.We first establish the theoretical criteria for asymptotically autonomous(or forward semi-convergent)of a pullback random attractor of random lattice systems.As an application,a non-autonomous stochastic lattice equation perturbed by random viscosity is considered.It is worth pointing out that the main difficulty of considering asymptotic autonomy of pullback random attractors arises from the measurability of the attractor.Since the absorbing set is an uncountable union of random sets,its measurability is unknown.In this case,we introduce two different universes,one is the usual universe of all tempered sets,the other is the universe of all forward tempered sets.We then prove that the existence and equality of a pullback random attractor and a forward pullback attractor in the two universes,respectively.The measurability of the forward pullback attractor fulfills due to the measurability of the pullback random attractor.Next,we investigate the mean dynamics and invariant measures of multi-stochastic sine-Gordon lattice equations with stochastic viscosity and nonlinear noise.The main difficulty in proving the well-posedness of the system is how to deal with the locally Lipschitz continuous diffusion term of noise.To solve this problem,we use a globally Lipschitz continuous cut-off function to approximate the diffusion term of noise.Thus,the well-posedness of the system is obtained.We then define a mean random dynamical system via the solution operators.Moreover,we prove that such a mean random dynamical system possesses a unique weak pullback mean random attractor in the Bochner space.Furthermore,we also discuss the existence of invariant measures,whose main task is to prove the tightness of distribution laws of solutions.Due to the non-compactness of Sobolev embeddings on unbounded domains,we use the idea of uniform tail-estimates to prove that the tails of solutions are uniformly small.Combining with the absorption estimates,we show the existence of invariant measures for the multi-stochastic lattice system.We then consider numerical attractors and approximations for stochastic or deterministic sine-Gordon lattice equations.We use the implicit Euler scheme for the time variable to discretize the equation,and prove the existence,uniqueness and upper semi-continuity of numerical attractors for the time-discrete sine-Gordon lattice equation with small step sizes.Besides,we prove the upper semi-convergence from the random attractor of the stochastic sine-Gordon lattice equation to the global attractor as the intensity of noise approaches zero.Moreover,we establish the finitely dimensional approximations of the three(numerical,random and global)attractors as the dimension of the state space tends to infinity.However,for the numerical attractors of non-autonomous(or stochastic)lattice systems,it is still an open problem.Eventually,we discuss the stochastic dynamics of non-autonomous stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay and nonlinear hereditary noise.Using Galerkin’s approximations method,we prove the well-posedness of systems.We show the existence of a unique stationary solution to the corresponding deterministic equation via the Lax-Milgram and the Schauder theorems.We study the local stability of stationary solutions for systems with general delay terms by using a direct method and then apply the abstract results to systems with two kinds of infinite delays.The exponential stability of stationary solutions is established in the case of unbounded distributed delay.We also investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals.We further prove the polynomial asymptotic stability of stationary solutions for the special case of proportional delay.