Analysis of a stochastic 2D–Navier-Stokes model with infinite delay

Some results concerning a stochastic 2D Navier-Stokes system when the external forces contain hereditary characteristics are established. The existence and uniqueness of solutions in the case of unbounded (infinite) delay are first proved by using the classical technique of Galerkin approximations. The local stability analysis of constant solutions (equilibria) is also carried out by exploiting two approaches. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. The asymptotic stability and hence, the uniqueness of equilibrium solution are obtained by constructing Lyapunov functionals. Moreover, some sufficient conditions ensuring the polynomial stability of the equilibrium solution in a particular case of unbounded variable delay will be provided. Exponential stability for other special cases of infinite delay remains as an open problem.Ministerio de Economía y Competitividad (MINECO). EspañaJunta de AndalucíaNational Natural Science Foundation of ChinaScience and Technology Commission of Shanghai Municipalit

Analysis of a stochastic SIR model with fractional Brownian motion

In this article, a stochastic version of a SIR nonautonomous model previously introduced in Kloeden and Kozyakin (2011) is considered. The noise considered is a fractional Brownian motion which satisfies the property of long range memory, which roughly implies that the decay of stochastic dependence with respect to the past is only subexponentially slow, what makes this kind of noise a realistic choice for problems with long memory in the applied sciences. The stochastic model containing a standard Brownian motion has been studied in Caraballo and Colucci (2016). In this paper, we analyze the existence and uniqueness of solutions to our stochastic model as well as their positiveness.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí

Recent results on stabilization of PDEs by noise

This paper is intended to be a brief review on some recent results on the stabilization effect produced by noise in phenomena modelled by partial differential equations. We emphasyse the different effects that distinct interpretations of the noise may cause on the same system, and we focus on two classical and canonical interpretations (Itò versus Stratonovich). Finally, we comment on some open problems.Ministerio de Ciencia y Tecnologí

Existence and Uniqueness of Solutions for Non-Linear Stochastic Partial Differential Equations

We state some results on existence and uniqueness for the solution of non linear stochastic PDEs with deviating arguments. In fact, we consider the equation dx(t) + (A(t; x(t)) + B(t; x(¿ (t))) + f(t)) dt = (C(t; x(½(t))) + g(t)) dwt ; where A(t; :) ; B(t; :) and C(t; :) are suitable families of non linear operators in Hilbert spaces, wt is a Hilbert valued Wiener process, and ¿ ; ½ are functions of delay. If A satisfies a coercivity condition and a monotonicity hypothesis, and if B ; C are Lipschitz continuous, we prove that there exists a unique solution of an initial value problem for the precedent equation. Some examples of interest for the applications are given to illustrate the results.Solutions, Non–Linear Stochastic Partial Differential Equation

Non-Linear Stochastic Partial Differential Equations with Delays: Existence and Uniqueness of Solutions

The main aim of this paper is to study stochastic PDE's with delay terms. In fact, we prove existence and uniqueness of solutions (in Itô's sense) for a rather general type of stochastic PDE's with non-linear monotone operators and with delays

Condiciones suficientes de estabilidad para Ecuaciones en Derivadas Parciales estocásticas con retardos

Sufficient conditions for pathwise exponential stability of the zero solution of stochastic PDE with deviating argument dxt = Axt dt + Bx (t) dwt are given. The assumptions on the operators A and B are the same that in the case without delay, but the proof is different. In fact, our method shows an alternative proof for the results in the particular case (t) = t : First, we obtain su cient conditions for the second moment of xt to decay exponentially. Next, asymptotic exponential stability of paths (with probability one) is deduced. Finally,an example is given in order to illustrate our theory

Nonlinear Partial Functional Differential Equations: Existence and Stability

Existence and uniqueness of solutions for a class of nonlinear functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of exponential stability from partial differential equations to partial functional differential equations are studied. The stability results derived are also applied to ordinary differential equations with hereditary characteristics

Pullback attractor for a dynamic boundary non-autonomous problem with Infinite delay

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned. Finally, we also prove the existence of a more general attractor for the problem known as D-pullback attractor.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí