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### Open Access Journals

DCDS

Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing

This paper treats the existence of pullback attractors for the non-autonomous 2D Navier--Stokes equations in two different spaces, namely $L^2$ and $H^1$. The non-autonomous forcing term is taken in $L^2_{\rm loc}(\mathbb R;H^{-1})$ and $L^2_{\rm loc}(\mathbb R;L^2)$ respectively for these two results: even in the autonomous case it is not straightforward to show the required asymptotic compactness of the flow with this regularity of the forcing term. Here we prove the asymptotic compactness of the corresponding processes by verifying the flattening property -- also known as ``Condition (C)". We also show, using the semigroup method, that a little additional regularity -- $f\in L^p_{\rm loc}(\mathbb R;H^{-1})$ or $f\in L^p_{\rm loc}(\mathbb R;L^2)$ for some $p>2$ -- is enough to ensure the existence of a compact pullback absorbing family (not only asymptotic compactness). Even in the autonomous case the existence of a compact absorbing set for this model is new when $f$ has such limited regularity.

DCDS

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.

DCDS-B

The existence of a pullback attractor for a reaction-diffusion
equations in an unbounded domain containing a non-autonomous
forcing term taking values in the space $H^{-1}$, and with a
continuous nonlinearity which does not ensure uniqueness of
solutions, is proved in this paper. The theory of set-valued
non-autonomous dynamical systems is applied to the problem.

DCDS

The existence and uniqueness of solutions for a stochastic
reaction-diffusion equation with infinite delay is proved.
Sufficient conditions ensuring stability of the zero solution are
provided and a possibility of stabilization by noise of the
deterministic counterpart of the model is studied.

DCDS

We analyze the asymptotic behaviour of a 3D Lagrangian averaged
Navier-Stokes $\alpha$-model (3D LANS$-\alpha$) with delays. In
fact, we apply the theory of pullback attractors to ensure the
existence of a pullback attractor, and at the same time, we also
prove the existence of a uniform (forward) attractor in the sense
of Chepyzhov and Vishik. Instead of working directly with the 3D
LANS$-\alpha$ model, we establish a general theory for an abstract
delay model and then we apply the general results to our
particular situation.

DCDS

In this paper we analyze the almost sure exponential stability and
ultimate boundedness of the solutions to a class of neutral
stochastic semilinear partial delay differential equations. These
kind of equations arise in problems related to coupled oscillators
in a noisy environment, or in viscoeslastic materials under random
or stochastic influences.

CPAA

The aim of this paper is to consider the robustness of exponential
attractors for non-autonomous dynamical systems with line memory
which is expressed through convolution integrals. Some properties
useful for dealing with the memory term for non-autonomous case
are presented. Then, we illustrate the abstract results by
applying them to the non-autonomous strongly damped wave equations
with linear memory and critical nonlinearity.

CPAA

A new proof of existence of solutions for the three dimensional
system of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This is
then used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcing
term. Indeed, we are able to prove that a stationary statistical solution is also
an invariant probability measure under suitable assumptions.

CPAA

The existence and finite fractal dimension of a pullback attractor
in the space $V$ for a three dimensional system of the
nonautonomous Globally Modified Navier-Stokes Equations on a
bounded domain is established under appropriate properties on the
time dependent forcing term. These equations were proposed
recently by Caraballo

*et al*and are obtained from the Navier- Stokes Equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.
DCDS-B

Existence and uniqueness of solution for a globally modified
version of Navier-Stokes equations containing infinite delay terms
are established. Moreover, we also analyze the stationary problem
and, under suitable additional conditions, we obtain global
exponential decay of the solutions of the evolutionary problem to
the stationary solution.

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