In this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove the existence of a pullback attractor for a non-autonomous 2D Navier-Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work since the non-autonomous term is quite general

Caraballo Garrido, Tomás

Lukaszewicz, Grzegorz

Real Anguas, José

idUS. Depósito de Investigación Universidad de Sevilla

Pullback attractors for non-autonomous 2D-Navier-Stokesequations in some unbounded domainsT. Caraballo a,∗, G. ÃLukaszewicz b,∗∗, and J. Real a,∗aDpto. de Ecuaciones Diferenciales y Ana´lisis Nume´rico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla,SpainbUniversity of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, PolandAbstractIn this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuringthe existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions ofpullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove theexistence of a pullback attractor for a non-autonomous 2D Navier-Stokes model in an unbounded domain, a casein which the theory of uniform attractors does not work since the non-autonomous term is quite general.Re´sume´Dans cette Note, on pre´sente d’abord la notion de compacite´ asymptotique pullback. On e´tablit ensuite unre´sultat d’existence d’un attracteur pullback pour un syste`me dynamique non autonome, sous les hypothe`ses decompacite´ asymptotique pullback et d’existence d’une famille d’ensembles absorbants au sens pullback. On prouvefinalement l’existence d’un attracteur pullback pour un syste`me de Navier-Stokes bidimensionel non autonomedans un domaine non borne´, une situation dans laquelle, e´tant donne´e la ge´ne´ralite´ du terme non autonome, lathe´orie des attracteurs uniformes ne peut pas eˆtre applique´e.Version franc¸aise abre´ge´eLa nume´rotation, les notations et la terminologie que nous utilisons se re´fe`rent directement a` lanume´rotation, aux notations et a` la terminologie de la version anglaise.On appelle processus sur un espace me´trique X toute famille {U(t, τ); −∞ < τ ≤ t < +∞} d’appli-cations continues U(t, τ) : X → X, telle que U(τ, τ)x = x, et (3) est satisfaite. On suppose donne´e unefamille non vide D d’ensembles D̂ = {D(t); t ∈ R} ⊂ P(X), ou` on de´note par P(X) la classe de tous les∗ Supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (European Community) grant BFM2002-03068∗∗Supported by Polish Government grant KBN 2 P303A 030 22Email addresses: caraball@us.es (T. Caraballo), glukasz@mimuw.edu.pl (G. ÃLukaszewicz), jreal@us.es (J. Real).Preprint submitted to Comptes Rendus Mathematique 21 novembre 2005sous-ensembles non vides de X. On introduit les notions de processus D-asymptotiquement compact ausens pullback (Definition 1), de famille D-absorbante pullback (Definition 2) et de D-attracteur pullback(Definition 3). Notre re´sultat principal est le suivant :The´ore`me 0.1. Supposons que le processus U(·, ·) est D-asymptotiquement compact au sens pullback, etque B̂ ∈ D est une famille D-absorbante pullback. Alors, la famille Â = {A(t); t ∈ R} ⊂ P(X) definiepar A(t) = Λ(B̂, t), t ∈ R, ou` pour chaque D̂ ∈ DΛ(D̂, t) =⋂s≤t⋃τ≤sU(t, τ)D(τ) ,est un D-attracteur pullback pour U(·, ·), et satisfait que A(t) = ⋃D̂∈D Λ(D̂, t), pour tout t ∈ R. De plus,Â est minimal au sens que si Ĉ = {C(t); t ∈ R} ⊂ P(X) est une famille d’ensembles ferme´s telle quelimτ→−∞ dist(U(t, τ)B(τ), C(t)) = 0, alors A(t) ⊂ C(t).On applique le The´ore`me 0.1 au syste`me de Navier-Stokes bidimensionel non autonome (2) dans undomaine non borne´ satisfaisant (1). On denote σ = νλ1, et Rσ l’ensemble de toutes les fonctions r : R→(0,+∞) qui satisfont (6). On definit Dσ comme e´tant la classe de toutes les familles D̂ = {D(t); t ∈R} ⊂ P(H) pour lesquelles il y a une fonction rD̂∈ Rσ telle que D(t) ⊂ B(0, rD̂(t)), et on de´montre lere´sultat suivant :The´ore`me 0.2. Supposons que f ∈ L2loc(R;V ′) et satisfait∫ t−∞ eσξ‖f(ξ)‖2 dξ < +∞ pour tout t ∈ R.Alors, il existe un unique Dσ-attracteur pullback, appartenant a` Dσ, pour le processus U(·, ·) defini par(4).1. IntroductionLet O ⊂ R2 be an open set, not necessarily bounded, with boundary ∂O, and suppose that O satisfiesthe Poincare´ inequality, i.e., there exists a constant λ1 > 0 such thatλ1∫Oφ2 dx ≤∫O|∇φ|2 dx for all φ ∈ H10 (O). (1)Consider the following 2D−Navier-Stokes problem (for further details see Lions [8] and Temam [14]):∂u∂t− ν∆u+2∑i=1ui∂u∂xi= f(t)−∇p in (τ,+∞)×O,div u = 0 in (τ,+∞)×O,u = 0 on (τ,+∞)× ∂O,u(τ, x) = u0(x), x ∈ O.This problem is set in a suitable abstract framework by considering the usual abstract spaces:V ={u ∈ (C∞0 (O))2 ; div u = 0},2H = the closure of V in (L2(O))2 with norm |·| , and inner product (·, ·) where(u, v) =2∑j=1∫Ouj(x)vj(x) dx, for u, v ∈ (L2(O))2.V = the closure of V in (H10 (O))2 with norm ‖·‖ , and associated scalar product ((·, ·)), where((u, v)) =2∑i,j=1∫O∂uj∂xi∂vj∂xidx, for u, v ∈ (H10 (O))2.It follows that V ⊂ H ≡ H ′ ⊂ V ′, where the injections are continuous and dense.Finally, we will use ‖·‖∗ for the norm in V ′ and 〈·, ·〉 for the duality pairing between V and V ′.Consider the trilinear form b on V × V × V given byb(u, v, w) =2∑i,j=1∫Oui∂vj∂xiwj dx, u, v, w ∈ V,define B : V × V → V ′ by 〈B(u, v), w〉 = b(u, v, w), for u, v, w ∈ V, and denote B(u) = B(u, u). Assumenow that u0 ∈ H, f ∈ L2loc(R;V ′). For each τ ∈ R we consider the problem:u ∈ L2(τ, T ;V ) ∩ L∞(τ, T ;H) for all T > τ,ddt(u(t), v) + ν((u(t), v)) + 〈B(u(t)), v〉 = 〈f(t), v〉, for all v ∈ V,in the sense of scalar distributions on (τ,+∞),u(τ) = u0.(2)It follows [14] that problem (2) has a unique solution, u(·; τ, u0), that moreover belongs to C0([τ,+∞);H).It is known that the theory of global attractors previously developed for autonomous systems is nolonger valid to analyse the long-term behaviour of this kind of problems, and it is necessary to use sometechniques for non-autonomous systems. Having several possible options (e.g. kernel sections [4], skew-product semiflows [13], etc..), we will use the already well established theory of pullback attractors (Crauelet al. [5], Langa and Schmalfuss [7], Kloeden and Schmalfuss [6], Schmalfuss [12]) since it allows to handlemore general non-autonomous terms, and it works under the presence of random environments as well.In order to prove the existence of the attractor we will exploit an approach which is based on the useof the energy equations which are in direct connection with the concept of asymptotic compactness, andwhich permits us to obtain informations on the asymptotic behaviour of our systems when the injectionV ↪→ H is not compact (as in our case of unbounded domain). This method was first used by Ball (seee.g. [1]) in the context of autonomous dynamical systems, and by ÃLukaszewicz and Sadowski in [9] (andlater also in [10]) to extend to the non-autonomous situation the corresponding one in the autonomousframework (see Rosa [11]), but related to uniform asymptotic compactness. Now we are interested in thecase without uniformity properties.In Section 2 we introduce the concept of asymptotically compact non-autonomous dynamical system,and establish a general result ensuring the existence of a minimal pullback attractor under assumptionsof asymptotic compactness and existence of a family of absorbing sets. Then, we prove the existenceof a pullback attractor for our 2D Navier-Stokes model in an unbounded domain in which the externalforce needs not be bounded, neither almost periodic nor translation compact. It is enough that this termsatisfies an appropriate integrability condition.32. Pullback attractors for asymptotically compact non-autonomous dynamical systemsAlthough it is possible to state all our analysis using the cocycle formalism (see [3]), which has provenalso very useful, in particular, in the case of random dynamical systems, we prefer to use the language ofevolutionary processes since it seems to be more appropriate for our situation.Let us consider a process (also called a two-parameter semigroup) U on a metric space X, i.e., a family{U(t, τ); −∞ < τ ≤ t < +∞} of continuous mappings U(t, τ) : X → X, such that U(τ, τ)x = x, andU(t, τ) = U(t, r)U(r, τ) for all τ ≤ r ≤ t. (3)Suppose D is a nonempty class of parameterized sets D̂ = {D(t); t ∈ R} ⊂ P(X), where P(X) denotesthe family of all nonempty subsets of X.Definition 1 The process U(·, ·) is said to be pullback D-asymptotically compact if for any t ∈ R, anyD̂ ∈ D, any sequence τn → −∞, and any sequence xn ∈ D(τn), the sequence {U(t, τn)xn} is relativelycompact in X.Definition 2 It is said that B̂ ∈ D is pullback D-absorbing for the process U(·, ·) if for any t ∈ R andany D̂ ∈ D, there exists a τ0(t, D̂) ≤ t such thatU(t, τ)D(τ) ⊂ B(t) for all τ ≤ τ0(t, D̂).Definition 3 The family Â = {A(t); t ∈ R} ⊂ P(X) is said to be a pullback D-attractor for U(·, ·) if(i) A(t) is compact for all t ∈ R,(ii) Â is pullback D-attracting, i.e.,limτ→−∞ dist(U(t, τ)D(τ), A(t)) = 0 for all D̂ ∈ D, and all t ∈ R,(iii) Â is invariant, i.e.,U(t, τ)A(τ) = A(t), for −∞ < τ ≤ t < +∞.Remark 4 Observe that Definition 3 does not guarantee the uniqueness of pullback D-attractors (seeCaraballo and Langa [2] for a discussion on this point). In order to ensure uniqueness one needs toimpose additional conditions as, for instance, the condition that the attractor belongs to the same familyD, or some kind of minimality. However, we do not include this stronger assumptions in the definitionsince, as we will show in Theorem 5, under very general hypotheses, it is possible to ensure the existenceof a global pullback D-attractor which is minimal in an appropriate sense.We have the following resultTheorem 5 Suppose that the process U(·, ·) is pullback D-asymptotically compact and that B̂ ∈ D is afamily of pullback D-absorbing sets for U(·, ·).Then, the family Â = {A(t); t ∈ R} ⊂ P(X) defined by A(t) = Λ(B̂, t), t ∈ R, where for each D̂ ∈ DΛ(D̂, t) =⋂s≤t⋃τ≤sU(t, τ)D(τ) ,is a pullback D-attractor for U(·, ·) which satisfies in addition that A(t) = ⋃D̂∈D Λ(D̂, t), for t ∈ R.Furthemore, Â is minimal in the sense that if Ĉ = {C(t); t ∈ R} ⊂ P(X) is a family of closed sets suchthat limτ→−∞ dist(U(t, τ)B(τ), C(t)) = 0, then A(t) ⊂ C(t).4PROOF. (Sketch) The proof follows from the properties of the omega limit sets Λ(D̂, t). Indeed, itfollows that Λ(D̂, t) is a compact nonempty set for all D̂ ∈ D and all t ∈ R. Moreover, the family{Λ(D̂, t), t ∈ R} pullback attracts D̂, is invariant in the sense of Definition 3, and satisfies that Λ(B̂, t) =⋃D̂∈D Λ(D̂, t). The minimality property follows immediately (see [3] for more details).Remark 6 The family Â is said to be forward attracting iflimt→+∞ dist(U(t, τ)D(τ), A(t)) = 0 for all D̂ ∈ D, and all τ ∈ R.It is interesting to investigate when a pullback attractor also attracts in the forward sense. In this respect,if the pullback attraction is uniform, i.e. if it holdslims→+∞ supt∈Rdist(U(t, t− s)D(t− s), A(t)) = 0 for all D̂ ∈ D,then, equivalently, it also holds the uniform forward attraction propertylimt→+∞ supτ∈Rdist(U(t+ τ, τ)D(τ), A(t+ τ)) = 0 for all D̂ ∈ D.As far as we know, apart from the result by Chepyzhov and Vishik [4], which ensures existence of a uniformforward (and so pullback) attractor when the universe D is formed by fixed bounded sets and there existsa fixed compact set uniformly forward absorbing, this is still an open problem in the non-uniform case.3. Application to non-autonomous 2D-Navier-Stokes equations in unbounded domainsLet us now apply the theory to our model. To construct the associated evolutionary process to (2), letus consider the unique solution, u(·; τ, u0) to (2).DefineU(t, τ)u0 = u(t; τ, u0) τ ≤ t, u0 ∈ H. (4)From the uniqueness of solution to problem (2), it follows thatU(t, τ)u0 = U(t, r)U(r, τ)u0, for all τ ≤ r ≤ t, u0 ∈ H. (5)Also, it is a standard task to prove that for all τ ≤ t, the mapping U(τ, t) : H → H defined by (4), iscontinuous. Consequently, the family {U(t, τ), τ ≤ t} defined by (4) is a process U(·, ·) in H.From now on, we denote σ = νλ1. Let Rσ be the set of all functions r : R→ (0,+∞) such thatlimt→−∞ eσtr2(t) = 0, (6)and denote by Dσ the class of all families D̂ = {D(t); t ∈ R} ⊂ P(H) such that D(t) ⊂ B(0, rD̂(t)), forsome rD̂∈ Rσ, where B(0, rD̂(t)) denotes the closed ball in H centered at zero with radius rD̂(t).Now, we haveTheorem 7 Suppose that f ∈ L2loc(R;V ′) is such that∫ t−∞ eσξ‖f(ξ)‖2∗ dξ < +∞ for all t ∈ R. Then,there exists a unique global pullback Dσ-attractor belonging to Dσ for the process U defined by (4).PROOF. (Sketch) Let τ ∈ R and u0 ∈ H be fixed, and denote u(t) = u(t; τ, u0) = U(t, τ)u0, for allt ≥ τ. It easily followsddt(eσt|u(t)|2)+ 2νeσt‖u(t)‖2 = σeσt|u(t)|2 + 2eσt〈f(t), u(t)〉, (7)5andeσt|u(t)|2 ≤ eστ |u0|2 + 1νt∫τeσξ‖f(ξ)‖2∗ dξ for all τ ≤ t. (8)Let D̂ ∈ Dσ be given. From (8), we obtain for all u0 ∈ D(τ), and all t ≥ τ that|U(t, τ)u0|2 ≤ e−σ(t−τ)r2D̂(τ) +e−σtνt∫−∞eσξ‖f(ξ)‖2∗ dξ, (9)For each t ∈ R, denote by Rσ(t) the constant given by (Rσ(t))2 = 2e−σtν∫ t−∞ eσξ‖f(ξ)‖2∗ dξ, and considerthe family B̂σ of closed balls in H defined by Bσ(t) = {v ∈ H; |v| ≤ Rσ(t)}. It is straightforward tocheck that B̂σ ∈ Dσ, and that B̂σ is pullback Dσ-absorbing for the process U(·, ·).Let D̂ ∈ Dσ, a sequence τn → −∞, a sequence u0n ∈ D(τn) and t ∈ R, be fixed. To prove that fromthe sequence {U(t, τn)u0n} we can extract a subsequence that converges in H we follow a scheme similarto that one in [9] but with necessary changes.References[1] J.M. Ball, Global attractors for damped semilinear wave equations, Discrete and Continuous Dynamical Systems10(2004), no. 1-2, 31-52.[2] T. Caraballo & J.A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamicalsystems, Dynamics of Continuous Discrete and Impulsive Systems A, 10(2003), 491-514.[3] T. Caraballo, G. ÃLukaszewicz & J. Real, Pullback attractors for asymptotically compact non-autonomous dynamicalsystems, manuscript, Technical Report no. 141 of the Institute of Applied Mathematics and Mechanics, University ofWarsaw.[4] V.V. Chepyzhov & M.I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium publications fol. 49,American Mathematical Society, Providence, Rhode Island, (2002).[5] H. Crauel, A. Debussche & F. Flandoli, Random attractors, J. Dyn. Diff. Eq. 9(1995), No. 2, 307-341.[6] P.E. Kloeden & B. Schmalfuss, Asymptotic behaviour of nonautonomous difference inclusions. Systems Control Lett.33(1998), no. 4, 275–280.[7] J.A. Langa & B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partialdifferential equations, Stochastics and Dynamics 4(2004), no. 3, 385-404.[8] J.L. Lions, Quelques me´thodes de re´solution des proble`mes aux limites non line´aires, Paris; Dunod, Gauthier-Villars,(1969).[9] G. ÃLukaszewicz & W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains,Z. angew. Math. Phys. 55(2004), 1-11.[10] I. Moise, R. Rosa & X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete andContinuous Dynamical Systems 10(2004), No. 1 & 2, 473-496.[11] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis, TMA32(1998), No. 1, 71-85.[12] B. Schmalfuss, Attractors for non-autonomous dynamical systems, in Proc. Equadiff 99, Berlin, Eds. B. Fiedler, K.Gro¨ger and J. Sprekels (World Scientific, 2000), pp. 684-689.[13] G.R. Sell, Non-autonomous differential equations and topological dynamics, I, II Trans. Amer. Math. Soc. 127(1967),241-262, 263-283.[14] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed., North Holland, Amsterdam, (1979).6

Comptes Rendus Mathematique

Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains

Caraballo, Tomás

Łukaszewicz, Grzegorz

Real, José

Numérisation de Documents Anciens Mathématiques

1/existencempactnesstonomousthe non-é.’existencepullback etback pourtant donnéetion, auxeC. R. Acad. Sci. Paris, Ser. I 342 (2006) 263–268http://france.elsevier.com/direct/CRASSDynamical SystemsPullback attractors for non-autonomous 2D-Navier–Stokesequations in some unbounded domainsTomás Caraballoa,1, Grzegorz Łukaszewiczb,2, José Reala,1a Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spainb University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, PolandReceived 28 February 2005; accepted after revision 7 December 2005Available online 5 January 2006Presented by Haïm BrezisAbstractIn this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring theof a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic coand the existence of a pullback absorbing family of sets. Finally, we prove the existence of a pullback attractor for a non-au2D Navier–Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work sinceautonomous term is quite general.To cite this article: T. Caraballo et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméUn attracteur pullback pour un système de Navier–Stokes bidimensionel non autonome dans un domaine non bornDans cette Note, on présente d’abord la notion de compacité asymptotique pullback. On établit ensuite un résultat dd’un attracteur pullback pour un système dynamique non autonome, sous les hypothèses de compacité asymptotiqued’existence d’une famille d’ensembles absorbants au sens pullback. On prouve finalement l’existence d’un attracteur pullun système de Navier–Stokes bidimensionel non autonome dans un domaine non borné, une situation dans laquelle, éla généralité du terme non autonome, la théorie des attracteurs uniformes ne peut pas être appliquée.Pour cit r cet article : T. Ca-raballo et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeLa numérotation, les notations et la terminologie que nous utilisons se réfèrent directement à la numérotanotations et à la terminologie de la version anglaise.On appelle processus sur un espace métriqueX toute famille{U(t, τ ); −∞ < τ  t < +∞} d’applications conti-nuesU(t, τ ) :X → X, telle queU(τ, τ )x = x, et (3) est satisfaite. On suppose donnée une famille non vidDE-mail addresses:caraball@us.es (T. Caraballo), glukasz@mimuw.edu.pl (G. Łukaszewicz), jreal@us.es (J. Real).1 Supported by Ministerio de Ciencia y Tecnología (Spain) and FEDER (European Community) grant BFM2002-03068.2 Supported by Polish Government grant KBN 2 P303A 030 22.1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.12.015264 T. Caraballo et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 263–268idese fa-estneéd’ensembleŝD = {D(t); t ∈ R} ⊂ P(X), où on dénote parP(X) la classe de tous les sous-ensembles non vdeX. On introduit les notions de processusD-asymptotiquement compact au sens pullback (Definition 2.1), dmille D-absorbante pullback (Definition 2.2) et deD-attracteur pullback (Definition 2.3). Notre résultat principalle suivant :Théorème 0.1.Supposons que le processusU(·, ·) estD-asymptotiquement compact au sens pullback, et queB̂ ∈ Dest une familleD-absorbante pullback. Alors, la famillêA = {A(t); t ∈ R} ⊂ P(X) definie parA(t) = Λ(B̂, t), t ∈ R,où pour chaquêD ∈ DΛ(D̂, t) = ⋂st(⋃τsU(t, τ )D(τ)),est unD-attracteur pullback pourU(·, ·), et satisfait queA(t) = ⋃D̂∈D Λ(D̂, t), pour tout t ∈ R. De plus,Â estminimal au sens que sîC = {C(t); t ∈ R} ⊂ P(X) est une famille d’ensembles fermés telle quelimτ→−∞ dist(U(t, τ )B(τ),C(t)) = 0,alorsA(t) ⊂ C(t).On applique le Théorème 0.1 au système de Navier–Stokes bidimensionel non autonome(2) dans un domaine noborné satisfaisant (1). On denoteσ = νλ1, etRσ l’ensemble de toutes les fonctionsr :R → (0,+∞) qui satisfont (6).On definitDσ comme étant la classe de toutes les famillesD̂ = {D(t); t ∈ R} ⊂ P(H) pour lesquelles il y a unfonctionrD̂ ∈ Rσ telle queD(t) ⊂ B(0, rD̂(t)), et on démontre le résultat suivant :Théorème 0.2.Supposons quef ∈ L2loc(R;V ′) et satisfait∫ t−∞ eσξ‖f (ξ)‖2 dξ < +∞ pour toutt ∈ R. Alors, il existeun uniqueDσ -attracteur pullback, appartenant àDσ , pour le processusU(·, ·) defini par(4).1. IntroductionLet O ⊂ R2 be an open set, not necessarily bounded, with boundary∂O, and suppose thatO satisfies the Poincarinequality, i.e., there exists a constantλ1 > 0 such thatλ1∫Oφ2 dx ∫O|∇φ|2 dx for all φ ∈ H 10 (O). (1)Consider the following 2D-Navier–Stokes problem (for further details see Lions [8] and Temam [14]):∂u∂t− ν	u +2∑i=1ui∂u∂xi= f (t) − ∇p in (τ,+∞) ×O,divu = 0 in (τ,+∞) ×O,u = 0 on(τ,+∞) × ∂O,u(τ, x) = u0(x), x ∈O.This problem is set in a suitable abstract framework by considering the usual abstract spaces:V = {u ∈ (C∞0 (O))2; divu = 0},H = the closure ofV in (L2(O))2 with norm | · |, and inner product(·, ·), where(u, v) =2∑j=1∫Ouj (x)vj (x)dx, for u,v ∈(L2(O))2.V = the closure ofV in (H 10 (O))2 with norm‖ · ‖, and associated scalar product((·, ·)), where((u, v)) =2∑i,j=1∫∂uj∂xi∂vj∂xidx, for u,v ∈ (H 10 (O))2.OT. Caraballo et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 263–268 265ter validfor non-3], etc.),uss [7],s, and itenergyto obtainenamicalsituations.establishpactnessNavier–riodic norso veryrocessesIt follows thatV ⊂ H ≡ H ′ ⊂ V ′, where the injections are continuous and dense.Finally, we will use‖ · ‖∗ for the norm inV ′ and〈·, ·〉 for the duality pairing betweenV andV ′.Consider the trilinear formb onV × V × V given byb(u, v,w) =2∑i,j=1∫Oui∂vj∂xiwj dx, u, v,w ∈ V,defineB :V × V → V ′ by 〈B(u, v),w〉 = b(u, v,w), for u,v,w ∈ V , and denoteB(u) = B(u,u). Assume now thau0 ∈ H , f ∈ L2loc(R;V ′). For eachτ ∈ R we consider the problem:u ∈ L2(τ, T ;V ) ∩ L∞(τ, T ;H) for all T > τ,ddt(u(t), v) + ν((u(t), v)) + 〈B(u(t)), v〉 = 〈f (t), v〉 for all v ∈ V,in the sense of scalar distributions on(τ,+∞),u(τ ) = u0.(2)It follows [14] that problem (2) has a unique solution,u(·; τ,u0), that moreover belongs toC0([τ,+∞);H).It is known that the theory of global attractors previously developed for autonomous systems is no longto analyse the long-term behaviour of this kind of problems, and it is necessary to use some techniquesautonomous systems. Having several possible options (e.g. kernel sections [4], skew-product semiflows [1we will use the already well established theory of pullback attractors (Crauel et al. [5], Langa and SchmalfKloeden and Schmalfuss [6], Schmalfuss [12]) since it allows to handle more general non-autonomous termworks under the presence of random environments as well.In order to prove the existence of the attractor we will exploit an approach which is based on the use of theequations which are in direct connection with the concept of asymptotic compactness, and which permits usinformations on the asymptotic behaviour of our systems when the injectionV ↪→ H is not compact (as in our casof unbounded domain). This method was first used by Ball (see e.g. [1]) in the context of autonomous dysystems, and by Łukaszewicz and Sadowski in [9] (and later also in [10]) to extend to the non-autonomousthe corresponding one in the autonomous framework (see Rosa [11]), but related touniform asymptotic compactnesNow we are interested in the case withoutuniformityproperties.In Section 2 we introduce the concept of asymptotically compact non-autonomous dynamical system, anda general result ensuring the existence of a minimal pullback attractor under assumptions of asymptotic comand existence of a family of absorbing sets. Then, we prove the existence of a pullback attractor for our 2DStokes model in an unbounded domain in which the external force needs not be bounded, neither almost petranslation compact. It is enough that this term satisfies an appropriate integrability condition.2. Pullback attractors for asymptotically compact non-autonomous dynamical systemsAlthough it is possible to state all our analysis using the cocycle formalism (see [3]), which has proven aluseful, in particular, in the case of random dynamical systems, we prefer to use the language of evolutionary psince it seems to be more appropriate for our situation.Let us consider a process (also called a two-parameter semigroup)U on a metric spaceX, i.e., a family{U(t, τ ); −∞ < τ  t < +∞} of continuous mappingsU(t, τ ) :X → X, such thatU(τ, τ )x = x, andU(t, τ ) = U(t, r)U(r, τ ) for all τ  r  t. (3)SupposeD is a nonempty class of parameterised setsD̂ = {D(t); t ∈ R} ⊂ P(X), whereP(X) denotes the familyof all nonempty subsets ofX.Definition 2.1. The processU(·, ·) is said to be pullbackD-asymptotically compact if for anyt ∈ R, anyD̂ ∈ D, anysequenceτn → −∞, and any sequencexn ∈ D(τn), the sequence{U(t, τn)xn} is relatively compact inX.Definition 2.2. It is said thatB̂ ∈ D is pullbackD-absorbing for the processU(·, ·) if for any t ∈ R and anyD̂ ∈ D,there exists aτ0(t, D̂)  t such thatU(t, τ )D(τ) ⊂ B(t) for all τ  τ0(t, D̂).266 T. Caraballo et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 263–268loonditionsr,generaltepullbackforwardct setDefinition 2.3. The familyÂ = {A(t); t ∈ R} ⊂ P(X) is said to be a pullbackD-attractor forU(·, ·) if(i) A(t) is compact for allt ∈ R,(ii) Â is pullbackD-attracting, i.e.,limτ→−∞ dist(U(t, τ )D(τ),A(t)) = 0 for all D̂ ∈D, and allt ∈ R,(iii) Â is invariant, i.e.,U(t, τ )A(τ) = A(t), for − ∞ < τ  t < +∞.Remark 1. Observe that Definition 2.3 does not guarantee the uniqueness of pullbackD- ttractors (see Carabaland Langa [2] for a discussion on this point). In order to ensure uniqueness one needs to impose additional cas, for instance, the condition that the attractor belongs to the same familyD, or some kind of minimality. Howevewe do not include this stronger assumption in the definition since, as we will show in Theorem 2.4, under veryhypotheses, it is possible to ensure the existence of a global pullbackD-attractor which is minimal in an appropriasense.We have the following resultTheorem 2.4. Suppose that the processU(·, ·) is pullbackD-asymptotically compact and that̂B ∈ D is a family ofpullbackD-absorbing sets forU(·, ·).Then, the familyÂ = {A(t); t ∈ R} ⊂ P(X) defined byA(t) = Λ(B̂, t), t ∈ R, where for eacĥD ∈ DΛ(D̂, t) =⋂st(⋃τsU(t, τ )D(τ)),is a pullbackD-attractor for U(·, ·) which satisfies in addition thatA(t) = ⋃D̂∈D Λ(D̂, t), for t ∈ R. Furthermore,Â is minimal in the sense that if̂C = {C(t); t ∈ R} ⊂ P(X) is a family of closed sets such thatlimτ→−∞ dist(U(t, τ )B(τ),C(t)) = 0,thenA(t) ⊂ C(t).Proof (Sketch). The proof follows from the properties of the omega limit setsΛ(D̂, t). Indeed, it follows thatΛ(D̂, t)is a compact nonempty set for all̂D ∈ D and allt ∈ R. Moreover, the family{Λ(D̂, t), t ∈ R} pullback attractŝD, isinvariant in the sense of Definition 2.3, and satisfies thatΛ(B̂, t) = ⋃D̂∈D Λ(D̂, t). The minimality property followsimmediately (see [3] for more details).Remark 2. The familyÂ is said to be forward attracting iflimt→+∞ dist(U(t, τ )D(τ),A(t)) = 0 for all D̂ ∈D, and allτ ∈ R.It is interesting to investigate when a pullback attractor also attracts in the forward sense. In this respect, if theattraction is uniform, i.e. if it holdslims→+∞ supt∈Rdist(U(t, t − s)D(t − s),A(t)) = 0 for all D̂ ∈D,then, equivalently, it also holds the uniform forward attraction propertylimt→+∞ supτ∈Rdist(U(t + τ, τ )D(τ),A(t + τ)) = 0 for all D̂ ∈D.As far as we know, apart from the result by Chepyzhov and Vishik [4], which ensures existence of a uniform(and so pullback) attractor when the universeD is formed by fixed bounded sets and there exists a fixed compauniformly forward absorbing, this is still an open problem in the non-uniform case.T. Caraballo et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 263–268 267consider.syceths, Dynam.ript, Tech-ociety,3. Application to non-autonomous 2D-Navier–Stokes equations in unbounded domainsLet us now apply the theory to our model. To construct the associated evolutionary process to (2), let usthe unique solution,u(·; τ,u0) to (2).DefineU(t, τ )u0 = u(t; τ,u0), τ  t, u0 ∈ H. (4)From the uniqueness of solution to problem (2), it follows thatU(t, τ )u0 = U(t, r)U(r, τ )u0, for all τ  r  t, u0 ∈ H. (5)Also, it is a standard task to prove that for allτ  t , the mappingU(τ, t) :H → H defined by (4), is continuousConsequently, the family{U(t, τ ), τ  t} defined by (4) is a processU(·, ·) in H .From now on, we denoteσ = νλ1. LetRσ be the set of all functionsr :R → (0,+∞) such thatlimt→−∞ eσ t r2(t) = 0, (6)and denote byDσ the class of all familieŝD = {D(t); t ∈ R} ⊂ P(H) such thatD(t) ⊂ B(0, rD̂(t)), for somerD̂ ∈Rσ , whereB(0, rD̂(t)) denotes the closed ball inH centred at zero with radiusrD̂(t).Now, we haveTheorem 3.1. Suppose thatf ∈ L2loc(R;V ′) is such that∫ t−∞ eσξ‖f (ξ)‖2∗ dξ < +∞ for all t ∈ R. Then, there exista unique global pullbackDσ -attractor belonging toDσ for the processU defined by (4).Proof (Sketch). Letτ ∈ R andu0 ∈ H be fixed, and denoteu(t) = u(t; τ,u0) = U(t, τ )u0, for all t  τ . It easilyfollowsddt(eσ t∣∣u(t)∣∣2) + 2ν eσ t∥∥u(t)∥∥2 = σ eσ t ∣∣u(t)∣∣2 + 2 eσ t 〈f (t), u(t)〉, (7)andeσ t∣∣u(t)∣∣2  eστ |u0|2 + 1νt∫τeσξ∥∥f (ξ)∥∥2∗ dξ for allτ  t. (8)Let D̂ ∈ Dσ be given. From (8), we obtain for allu0 ∈ D(τ), and allt  τ that∣∣U(t, τ )u0∣∣2  e−σ(t−τ)r2D̂(τ ) + e−σ tνt∫−∞eσξ∥∥f (ξ)∥∥2∗ dξ. (9)For eacht ∈ R, denote byRσ (t) the constant given by(Rσ (t))2 = 2e−σ tν∫ t−∞ eσξ‖f (ξ)‖2∗ dξ , and consider the familB̂σ of closed balls inH defined byBσ (t) = {v ∈ H ; |v|  Rσ (t)}. It is straightforward to check that̂Bσ ∈ Dσ , andthatB̂σ is pullbackDσ -absorbing for the processU(·, ·).Let D̂ ∈ Dσ , a sequenceτn → −∞, a sequenceu0n ∈ D(τn) andt ∈ R, be fixed. To prove that from the sequen{U(t, τn)u0n} we can extract a subsequence that converges inH we follow a scheme similar to that one in [9] but winecessary changes.References[1] J.M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dynam. Systems 10 (1–2) (2004) 31–52.[2] T. Caraballo, J.A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systemContin. Discrete Impuls. Systems A 10 (2003) 491–514.[3] T. Caraballo, G. Łukaszewicz, J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, manuscnical Report no. 141 of the Institute of Applied Mathematics and Mechanics, University of Warsaw.[4] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, Colloq. Publ., vol. 49, American Mathematical SProvidence, RI, 2002.268 T. Caraballo et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 263–26875–280.quations,. Math.am. Sys-9, Berlin,3–283.[5] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dynam. Differential Equations 9 (2) (1995) 307–341.[6] P.E. Kloeden, B. Schmalfuss, Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett. 33 (4) (1998) 2[7] J.A. Langa, B. Schmalfuss, Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential eStochastics and Dynamics 4 (3) (2004) 385–404.[8] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.[9] G. Łukaszewicz, W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. AngewPhys. 55 (2004) 1–11.[10] I. Moise, R. Rosa, X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyntems 10 (1 & 2) (2004) 473–496.[11] R. Rosa, The global attractor for the 2D Navier–Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1) (1998) 71–85.[12] B. Schmalfuss, Attractors for non-autonomous dynamical systems, in: B. Fiedler, K. Gröger, J. Sprekels (Eds.), Proc. Equadiff 9World Scientific, 2000, pp. 684–689.[13] G.R. Sell, Non-autonomous differential equations and topological dynamics, I, II, Trans. Amer. Math. Soc. 127 (1967) 241–262, 26[14] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, second ed., North-Holland, Amsterdam, 1979.

Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

Tomás Caraballo

Grzegorz Łukaszewicz

José Real

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2005.12.015.pdf

Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains

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