Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing
Julia García-Luengo Pedro Marín-Rubio José Real James C. Robinson
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 203-227 doi: 10.3934/dcds.2014.34.203
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.
keywords: pullback attractors pullback flattening property 2D Navier--Stokes equations compact absorbing set.
Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators
Pedro Marín-Rubio José Real
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 989-1006 doi: 10.3934/dcds.2010.26.989
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.
keywords: delay terms Navier-Stokes equations pullback attractors tempered attractors.
Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions
María Anguiano Tomás Caraballo José Real José Valero
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 307-326 doi: 10.3934/dcdsb.2010.14.307
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.
keywords: asymptotic compactness multivalued evolution process Pullback attractor non-autonomous reaction-diffusion equation.
Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory
Tomás Caraballo I. D. Chueshov Pedro Marín-Rubio José Real
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 253-270 doi: 10.3934/dcds.2007.18.253
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.
keywords: materials with memory stabilization. mean square exponential stability Stochastic heat equation
Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays
Julia García-Luengo Pedro Marín-Rubio José Real
Communications on Pure & Applied Analysis 2015, 14(5): 1603-1621 doi: 10.3934/cpaa.2015.14.1603
In this paper we strengthen some results on the existence and properties of pullback attractors for a 2D Navier-Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271--297]. Actually, we prove that under suitable assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Moreover, thanks to regularity results, the attraction from different phase spaces also happens in $C([-h,0];V)$. Finally, from comparison results of attractors, and under an additional hypothesis, we establish that all these families of attractors are in fact the same object.
keywords: 2D Navier-Stokes equations pullback attractors. delay terms
Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay
Tomás Caraballo Antonio M. Márquez-Durán José Real
Discrete & Continuous Dynamical Systems - A 2006, 15(2): 559-578 doi: 10.3934/dcds.2006.15.559
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.
keywords: 3D Lagrangian averaged Navier-Stokes equations distributed delay. pullback attractor forward attractor variable delay
The exponential stability of neutral stochastic delay partial differential equations
Tomás Caraballo José Real T. Taniguchi
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 295-313 doi: 10.3934/dcds.2007.18.295
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.
keywords: existence of solutions exponential stability. ultimate boundedness Neutral stochastic partial differential equations
Robust exponential attractors for non-autonomous equations with memory
Peter E. Kloeden José Real Chunyou Sun
Communications on Pure & Applied Analysis 2011, 10(3): 885-915 doi: 10.3934/cpaa.2011.10.885
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.
keywords: wave equations. robust exponential attractor Non-autonomous systems with memory
Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations
P.E. Kloeden Pedro Marín-Rubio José Real
Communications on Pure & Applied Analysis 2009, 8(3): 785-802 doi: 10.3934/cpaa.2009.8.785
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.
keywords: Modified Navier-Stokes model statistical solutions invariant measures.
Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations
P.E. Kloeden José A. Langa José Real
Communications on Pure & Applied Analysis 2007, 6(4): 937-955 doi: 10.3934/cpaa.2007.6.937
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.
keywords: weak solutions flattening property existence and uniqueness of strong solutions 3-dimensional Navier-Stokes equations nonautonomous and pullback attractors.

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