José Real

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.

Asymptotic behaviour of two–dimensional 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 on a Razumikhin–type method.The Navier-Stokes equations govern the motion of usual fluids like water, air, oil, etc. These equations have been the subject of numerous works since the first paper of Leray was published in 1933 (see Constantin & Foias 1988; Lions 1969; Temam 1979, and the references therein). In our recent work Caraballo & Real (2001) we consider a Navier-Stokes model in which the external force contains some hereditary features and prove the existence of weak solutions. These situations containing delays may appear when we want to control the system (in certain sense) by applying a force which takes into account not only the present state of the system but the history of the solution. Another interesting problem concerns the asymptotic behaviour of the systems, since this analysis can provide useful information on the future evolution of the system. This will be the main aim of this paper. To this end, let Ω ⊂ R be an open bounded set with regular boundary Γ, and consider the following functional 2D−Navier-Stokes problem (for further details and notations see Lions 1969 and Temam 1979):    ∂u

Pullback and Forward Attractors for a Damped Wave Equation with Delays

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of sine–Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

Stochastic stabilization of differential systems with general decay rate

Some sufficient conditions concerning stability of solutions of stochastic differential evolution equations with general decay rate are first proved. Then, these results are interpreted as suitable stabilization ones for deterministic and stochastic systems. Also, they permit us to construct appropriate linear stabilizers in some particular situations.

On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier-Stokes equations

We prove the existence and uniqueness of solutions for a stochastic version of the three-dimensional Lagrangian averaged Navier–Stokes equation in a bounded domain. To this end, we previously prove some existence and uniqueness results for an abstract stochastic equation and justify that our model falls within this framework.

Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations

Abstract We prove the existence and uniqueness of strong solutions of a three dimensional system of globally modified Navier-Stokes equations. The flattening property is used to establish the existence of global V -attractors and a limiting argument is then used to obtain the existence of bounded entire weak solutions of the three dimensional Navier-Stokes equations with time independent forcing.

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

On the pathwise exponential stability of non–linear stochastic partial differential equations

Sufficient conditions to get exponential stability for the sample paths (with probability one) of a non–linear monotone stochastic Partial Differential Equation are proved. In fact, we improve a stability criterion established in Chow [3] since, under the same hypotheses, we get pathwise exponential stability instead of stability of sample paths

Existence and uniqueness of solutions for delay stochastic evolution equations

Some results on the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved. In fact, our theory is developed from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation.

On the approximate controllability of a stochastic parabolic equation with a multiplicative noise

Abstract In this Note, we present some results concerning the approximate controllability for a stochastic parabolic equation with a multiplicative noise. For simplicity, we only consider the distributed control case.