Resumen de Dynamics of stochastic systems with delay and applications to real models
Shuang Yang
In this thesis we investigate the long time behavior of random dynamical systems associated to several kinds of stochastic equations with delays in terms of stability for stationary solutions, weak pullback mean random attractors, random attractors and numerical attractors. The thesis consists of three parts, where the first part covers Chapters 1-3, the last two cover Chapters 4 and 5, respectively.
Chapters 1-3 are devoted to the random dynamics of 3D Lagrangian-averaged NavierStokes equations with infinite delay in three cases.
In Chapter 1 we consider the stability analysis of such systems in the case of bounded domains. We first use Galerkin’s approximations to prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. We then prove the existence of a unique stationary solution to the corresponding deterministic equation via the LaxMilgram and the Schauder theorems. The stability and asymptotic stability of stationary solutions (equilibrium solutions) are also established. The local stability of stationary solutions for general delay terms is carried out by using a direct method and then apply the abstract results to two kinds of infinite delays. It is worth mentioning that all conditions are general enough to include several kinds of delays, where we mainly consider unbounded variable delays and infinite distributed delays. As we know, it is still an open and challenging problem to obtain sufficient conditions ensuring the exponential stability of solutions in case of unbounded variable delay. Fortunately, we obtained the exponential stability of stationary solutions in the case of infinite distributed delay. However, we are able to further investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing suitable Lyapunov functionals. Besides, we proved the polynomial asymptotic stability of stationary solutions for the particular case of proportional delay.
In Chapter 2, we further discuss mean dynamics and stability analysis of stochastic systems in the case of unbounded domains. We first prove the well-posedness of systems with infinite delay when the non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and the nonlinear diffusion term is locally Lipschitz continuous, which leads to the existence of a mean random dynamical system. We then obtain that such a dynamical system possesses a unique weak pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting set. Moreover, we prove the existence and uniqueness of stationary solutions to the corresponding deterministic equation via the classical Galerkin method, the Lax-Milgram and the Brouwer fixed theorems.
We discuss in the last part of Chapter 2 with those stability results concerning stationary solutions discussed in Chapter 1.
The last case is concerned with the invariant measures for the autonomous version of stochastic equations in Chapter 3 by using the method of generalized Banach limit. We first use Galerkin approximations, a priori estimates and the standard Gronwall lemma to show the well-posedness for the corresponding random equation, whose solution operators generate a random dynamical system. Next, the asymptotic compactness for the random dynamical system is established via the Ascoli-Arzel`a theorem. Besides, we derive the existence of a global random attractor for the random dynamical system. Moreover, we prove that the random dynamical system is bounded and continuous with respect to the initial values.
Eventually, we construct a family of invariant Borel probability measures, which is supported by the global random attractor.
It is well-known that lattice dynamical systems have wide applications in physics, chemistry, biology and engineering such as pattern formation, image processing, propagation of nerve pulses, electric circuits and so on. The theory of attractors for deterministic or stochastic lattice systems has been widely developed. Therefore, we focus on the asymptotical behavior of attractors for lattice dynamical systems in the last two chapters.
Two problems related to FitzHugh-Nagumo lattice systems are analyzed in Chapter 4. The first one is concerned with the asymptotic behavior of random delay FitzHughNagumo lattice systems driven by nonlinear Wong-Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation parameter of Wong-Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delay lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli-Arzel`a theorem and uniform tailestimates. We then show the upper semi-continuity of attractors as the correlation parameter tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model, and study the convergence of attractors when the delay approaches zero. Namely, the upper semicontinuity of attractors for the delay system to the nondelay one is proved.
Eventually, existence and connection of numerical attractors for discrete-time p-Laplace lattice systems via the implicit Euler scheme are proved in Chapter 5. So far, it remains open to obtain a numerical attractor for a non-autonomous (or stochastic) lattice system, and thus we can at least investigate numerical attractors for the deterministic and nondelayed version of p-Laplace lattice equations. The numerical attractors are shown to have an optimized bound, which leads to the continuous convergence of the numerical attractors when the graph of the nonlinearity closes to the vertical axis or when the external force vanishes. A new type of Taylor expansions without Fr´echet derivatives is established and applied to show the discretization error of order two, which is crucial to prove that the numerical attractors converge upper semi-continuously to the global attractor of the original continuous-time system as the step size of the time goes to zero. It is also proved that the truncated numerical attractors for finitely dimensional systems converge upper semicontinuously to the numerical attractor and the lower semi-continuity holds in special cases.
