Some Results on Stochastic Differential Equations with Reflecting Boundary Conditions

Some results related to stochastic differential equations with reflecting boundary conditions (SDER) are obtained. Existence and uniqueness of strong solution is ensured under the relaxation on the drift coefficient (instead of the Lipschitz character, a monotonicity condition is supposed)

Pullback Attractors for 2D-Navier-Stokes Equations with Delays In Continuous and Sub-Linear Operators

We obtain a result of existence of solutions to the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sub-linear and only continuous. As a consequence of the continuity assumption the uniqueness of solutions does not hold in general. We use then the theory of multi-valued dynamical system to establish the existence of attractors for our problem in several senses and establish relations among them

Attractors for 2D-Navier-Stokes models with delays

The existence of an attractor for a 2D-Navier-Stokes system with delay is proved. The theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous. However, on some occasions, the attractors may attract not only in the pullback sense but in the forward one as well. Also, this formulation allows to treat, in a unified way, terms containing various classes of delay features (constant, variable, distributed delays, etc.). As a consequence, some results for the autonomous model are deduced as particular cases of our general formulation

Navier-Stokes Equations with Delays

Some results on the existence and uniqueness of solutions to Navier-Stokes equations when the external force contains some hereditary characteristics are proved

Probabilistic Representation of Solutions for Quasi-Linear Parabolic Pde Via Fbsde with Reflecting Boundary Conditions

A probabilistic representation of the solution (in the viscosity sense) of a quasi-linear parabolic PDE system with non-lipschitz terms and a Neumann boundary condition is given via a fully coupled forward-backward stochastic differential equation with a reflecting term in the forward equation. The extension of previous results consists on the relaxation on the Lipschitz assumption on the drift coefficient of the forward equation, using a previous result of the authors

Partial Differential Equations with Delayed Random Perturbations: Existence, Uniqueness and Stability of Solutions

We consider a stochastic non–linear Partial Differential Equation with delay which may be regarded as a perturbed equation. First, we prove the existence and the uniqueness of solutions. Next, we obtain some stability results in order to prove the following: if the unperturbed equation is exponentially stable and the stochastic perturbation is small enough then, the perturbed equations remains exponentially stable. We impose standard assumptions on the differential operators and we use strong and mild solutions

Asymptotic Behaviour of 2D-Navier-Stokes Equations with Delays

Some results on the asymptotic behaviour of solutions to Navier-Stokes equations when the external force contains some hereditary characteristics are proved. We show two different approaches to prove the convergence of solutions to the stationary one, when this is unique. The first is a direct method while the second is based in the Razumikhin type one