Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

This paper proves the existence of the pullback exponential attractor for the process associated to the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations

In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditions for concrete stochastic partial differential equations. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theore

Análisis de sistemas dinámicos infinito-dimensionales asociados a ecuaciones en derivadas parciales funcionales.

Based on the theory of functional diferential equations, theory of semigroup, theory of random dynamical systems and theory of in nite dimensional dynamical systems, this thesis studies the long time behavior of several kinds of in nite dimensional dynamical systems associated to partial diferential equations containing some kinds of hereditary characteristics (such as variable delay, distributed delay or memory, etc), including existence and upper semicontinuity of pullback/random attractors and the stability analysis of stationary (steady-state) solutions. Three important mathematical-phyiscal models are considered, namely, reaction-di usion equation, 2D-Navier-Stokes equation as well as in-compressible non-Newtonian uids. Chapter 1 is devoted to the dynamics of an integer order stochastic reaction-difusion equation with thermal memory when the nonlinear term is subcritical or critical. Notice that our model contains not only memory but also white noise, which means it is not easy to prove the existence and uniqueness of solutions directly. In order to deal with this problem, we need introduce a new variable to transform our model into a system with two equations, and we use the Ornstein-Uhlenbeck to transfer this system into a deterministic ones only with random parameter. Then a semigroup method together with the Lax-Milgram theorem is applied to prove the existence, uniqueness and continuity of mild solutions. Next, the dynamics of solutions is analyzed by a priori estimates, and the existence of pullback random attractors is established. Besides, we prove that this pullback random attractors cannot explode, a property known as upper semicontinuity. But the dimension of the random attractor is still unknown. On the other hand, it has been proved that sometimes, especially when self-orgnization phenomena, anisotropic di usion, anomalous difusion occurs, a fractional order diferential equation can model this phenomena more precisely than a integer one. Hence, in Chapter 2, we focus on the asymptotical behavior of a fractional stochastic reaction-difusion equation with memory, which is also called fractional integro-diferential equation. First of all, the Ornstein-Uhlenbeck is applied to change the stochastic reaction-difusion equation into a deterministic ones, which makes it more convenient to solve. Then existence and uniqueness of mild solutions is proved by using the Lumer-Phillips theorem. Next, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining some uniform estimates and solutions decomposition. Moreover, the random attractor is shown to have nite Hausdorf dimension, which means the asymptotic behavior of the system is determined by only a nite number of degrees of freedom, though the random attractor is a subset of an in nite-dimensional phase space. But we still wonder whether this random attractor has inertial manifolds, which means this random attractor needs to be exponentially attracting. Besides, the long time behavior of time-fractional reaction-difusion equation and fractional Brownian motion are still unknown. The rst two chapters consider an important partial function diferential equations with in nite distributed delay. However, partial functional diferential equations include more than only distributed delays; for instance, also variable delay terms can be considered. Therefore, in the next chapter, we consider another signi cant partial functional diferential equation but with variable delay. In Chapter 3, we discuss the stability of stationary solutions to 2D Navier-Stokes equations when the external force contains unbounded variable delay. Notice that the classic phase space C which is used to deal with diferential equations with in nite delay does not work well for our unbounded variable delay case. Instead, we choose the phase space of continuous bounded functions with limits at1. Then the existence and uni- queness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solutions is established by means of the Lax-Milgram theorem and the Schauder xed point theorem. Afterward, the local stability analysis of stationary solutions is carried out by three diferent approaches: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. It worths mentioning that the classical Lyapunov function method requires diferentiability of delay term, which in some extent is restrictive. Fortunately, we could utilize Razumikhin-Lyapunov argument to weak this condition, and only requires continuity of every operators of this equation but allows more general delay. Neverheless, by these methods, the best result we can obtain is the asymptotical stability of stationary solutions by constructing a suitable Lyapunov functionals. Fortunately, we could obtain polynomial stability of the steady-state in a particular case of unbounded variable delay, namely, the proportional delay. However, the exponential stability of stationary solutions to Navier-Stokes equation with unbounded variable delay still seems an open problem. We can also wonder about the stability of stationary solutions to 2D Navier-Stokes equations with unbounded delay when it is perturbed by random noise. Therefore, in Chapter 4, a stochastic 2D Navier-Stokes equation with unbounded delay is analyzed in the phase space of continuous bounded functions with limits at1. Because of the perturbation of random noise, the classical Galerkin approximations alone is not enough to prove the existence and uniqueness of weak solutions. By combing a technical lemma and Faedo-Galerkin approach, the existence and uniqueness of weak solutions is obtained. Next, the local stability analysis of constant solutions (equilibria) is carried out by exploiting two methods. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. Although it is not possible, in general, to establish the exponential convergence of the stationary solutions, the polynomial convergence towards the stationary solutions, in a particular case of unbounded variable delay can be proved. We would like to point out that the Razumikhin argument cannot be applied to analyze directly the stability of stationary solutions to stochastic equations as we did to deterministic equations. Actually, we need more technical, and this will be our forthcoming paper. We also would like to mention that exponential stability of other special cases of in nite delay remains as an open problem for both the deterministic and stochastic cases. Especially, we are interested in the pantograph equation, which is a typical but simple unbounded variable delayed diferential equation.We believe that the study of pantograph equation can help us to improve our knowledge about 2D{Navier-Stokes equations with unbounded delay. Notice that Chapter 3 and Chapter 4 are both concerned with delayed Navier-Stokes equations, which is a very important Newtonian uids, and it is extensively applied in physics, chemistry, medicine, etc. However, there are also many important uids, such as blood, polymer solutions, and biological uids, etc, whose motion cannot be modeled pre- cisely by Newtonian uids but by non-Newtonian uids. Hence, in the next two chapters, we are interested in the long time behavior of an incompressible non-Newtonian uids ith delay. In Chapter 5, we study the dynamics of non-autonomous incompressible non-Newtonian uids with nite delay. The existence of global solution is showed by classical Galerkin approximations and the energy method. Actually, we also prove the uniqueness of solutions as well as the continuous dependence of solutions on the initial value. Then, the existence of pullback attractors for the non-autonomous dynamical system associated to this problem is established under a weaker condition in space C([h; 0];H2) rather than space C([h; 0];L2), and this improves the available results that worked on non-Newtonian uids. However, we still would like to analyze the Hausdor dimension or fractal dimension of the pullback attractor, as well as the existence of inertial manifolds and morsedecomposition. Finally, in Chapter 6, we consider the exponential stability of an incompressible non Newtonian uids with nite delay. The existence and uniqueness of stationary solutions are rst established, and this is not an obvious and straightforward work because of the nonlinearity and the complexity of the term N(u). The exponential stability of steady state solutions is then analyzed by means of four diferent approaches. The rst one is the classical Lyapunov function method, which requires the diferentiability of the delay term. But this may seem a very restrictive condition. Luckily, we could use a Razumikhin type argument to weaken this condition, but allow for more general types of delay. In fact, we could obtain a better stability result by this technique. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. We would like to emphasize that by using a Gronwall-like lemma, only the measurability of delay term is demanded, but still ensure the exponential stability. Furthermore, we also would like to discuss the dynamics of stochastic non-Newtonian uids with both nite delay and in nite delay. All the problems deserve our attraction, and actually, these are our forthcoming work

Entropic Fluctuations in Statistical Mechanics I. Classical Dynamical Systems

Within the abstract framework of dynamical system theory we describe a general approach to the Transient (or Evans-Searles) and Steady State (or Gallavotti-Cohen) Fluctuation Theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. Besides its conceptual simplicity, another advantage of our approach is its natural extension to quantum statistical mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.Comment: 72 pages, revised version 12/10/2010, to be published in Nonlinearit