Tesis Doctoral
Dynamics of stochastic systems with delay and applications to real models
Autor/es | Yang, Shuang |
Director | Caraballo Garrido, Tomás |
Departamento | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Fecha de publicación | 2023-06-22 |
Fecha de depósito | 2023-07-12 |
Resumen | In this thesis we investigate the long time behavior of random dynamical systems associ-
ated to several kinds of stochastic equations with delays in terms of stability for stationary
solutions, weak pullback mean random ... In this thesis we investigate the long time behavior of random dynamical systems associ- ated to several kinds of stochastic equations with delays in terms of stability for stationary solutions, weak pullback mean random attractors, random attractors and numerical attrac- tors. The thesis consists of three parts, where the rst 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 Navier- Stokes equations with in nite delay in three cases. In Chapter 1 we consider the stability analysis of such systems in the case of bounded domains. We rst 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 Lax- Milgram and the Schauder theorems. The stability and asymptotic stability of stationary solutions (equilibrium solutions) are also established. The local stability of stationary so- lutions for general delay terms is carried out by using a direct method and then apply the abstract results to two kinds of in nite 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 in nite distributed delays. As we know, it is still an open and challenging problem to obtain su cient 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 in nite 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 sys- tems in the case of unbounded domains. We rst prove the well-posedness of systems with in nite delay when the non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and the nonlinear di usion term is locally Lipschitz continu- ous, 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 equa- tion via the classical Galerkin method, the Lax-Milgram and the Brouwer xed theorems. 1 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 rst 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, chem- istry, 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 s- tochastic 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 rst one is concerned with the asymptotic behavior of random delay FitzHugh- Nagumo 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 in nity rather than to zero. We rst prove the existence of tempered random attractors for the random delay lattice systems with a nonlinear drift function and a nonlinear di usion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli-Arzel a theorem and uniform tail- estimates. We then show the upper semi-continuity of attractors as the correlation parameter tends to in nity. 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 non- delayed 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 2 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 nitely dimensional systems converge upper semi- continuously to the numerical attractor and the lower semi-continuity holds in special cases. The results of our investigation in this thesis are included in the following papers: S. Yang, Y. Li, Q. Zhang and T. Caraballo, Stability analysis of stochastic 3D Lagrangian- averaged Navier-Stokes equations with in nite delay, J. Dynam. Di erential Equations, Published Online, (2023), Doi: 10.1007/s10884-022-10244-0. S. Yang, T. Caraballo and Y. Li, Dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier-Stokes equations with in nite delay on unbounded do- mains, Appl. Math. Optim., (submitted). S. Yang, T. Caraballo and Y. Li, Invariant measures for stochastic 3D Lagrangian- averaged Navier-Stokes equations with in nite delay, Commun. Nonlinear Sci. Nu- mer. Simul., 118 (2023), pp. 107004, 21. S. Yang, Y. Li and T. Caraballo, Dynamical stability of random delayed FitzHugh- Nagumo lattice systems driven by nonlinear Wong-Zakai noise, J. Math. Phys., 63 (2022), pp. 111512, 32. Y. Li, S. Yang and Tom as Caraballo, Optimization and convergence of numerical at- tractors for discrete-time quasi-linear lattice system, SIAM J. Numer. Anal., (to ap- pear), 2023. 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 ... 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 Navier-Stokes 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 Lax-Milgram 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 su cient 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 di usion 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- Arzelà 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 FitzHugh-Nagumo 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 di usion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli-Arzelà theorem and uniform tail-estimates. 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 non-delayed 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 Fréchet 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 semi-continuously to the numerical attractor and the lower semi-continuity holds in special cases. The results of our investigation in this thesis are included in the following papers: S. Yang, Y. Li, Q. Zhang and T. Caraballo, Stability analysis of stochastic 3D Lagrangianaveraged Navier-Stokes equations with infinite delay, J. Dynam. Di erential Equations, Published Online, (2023), Doi: 10.1007/s10884-022-10244-0. S. Yang, T. Caraballo and Y. Li, Dynamics and stability analysis for stochastic 3D Lagrangianix averaged Navier-Stokes equations with infinite delay on unbounded domains, Appl. Math. Optim., (submitted). S. Yang, T. Caraballo and Y. Li, Invariant measures for stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay, Commun. Nonlinear Sci. Numer. Simul., 118 (2023), pp. 107004, 21. S. Yang, Y. Li and T. Caraballo, Dynamical stability of random delayed FitzHugh-Nagumo lattice systems driven by nonlinear Wong-Zakai noise, J. Math. Phys., 63 (2022), pp. 111512, 32. Y. Li, S. Yang and Tomás Caraballo, Optimization and convergence of numerical attractors for discrete-time quasi-linear lattice system, SIAM J. Numer. Anal., (to appear), 2023. |
Cita | Yang, S. (2023). Dynamics of stochastic systems with delay and applications to real models. (Tesis Doctoral Inédita). Universidad de Sevilla, Sevilla. |
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