We prove some regularity results for the pullback attractors of a non-autonomous 2D Navier–Stokes model in a bounded domain Ω of R2. We establish a general result about (H2(Ω))2∩V-boundedness of invariant sets for the associate evolution process. Then, as a consequence, we deduce that, under adequate assumptions, the pullback attractors of the non-autonomous 2D Navier–Stokes equations are bounded not only in V but also in (H2(Ω))2

García Luengo, Julia María

Marín Rubio, Pedro

Real Anguas, José

Julia García-Luengo

Pedro Marín-Rubio

José Real

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Nonlinear Analysis

-boundedness of the pullback attractors for non-autonomous 2D Navier–Stokes equations in bounded domains

idUS. Depósito de Investigación Universidad de Sevilla

H2-boundedness of the pullback attractors fornon-autonomous 2D-Navier-Stokes equationsin bounded domainsJulia Garc´ıa-Luengo, Pedro Mar´ın-Rubio, Jose´ RealDpto. de Ecuaciones Diferenciales y Ana´lisis Nume´rico, Universidad de Sevilla,Apdo. de Correos 1160, 41080 Sevilla, SpainAbstractWe prove some regularity results for the pullback attractors of a non-autonomous2D-Navier-Stokes model in a bounded domain Ω of R2. We establish a generalresult about (H2(Ω))2 ∩V -boundedness of invariant sets for the associate evolutionprocess. Then, as a consequence, we deduce that, under adequate assumptions, thepullback attractors of the non-autonomous 2D-Navier-Stokes equations are boundednot only in V but also in (H2 (Ω))2.Key words: 2D-Navier-Stokes equations, pullback attractors, invariant sets,H2-regularity.Mathematics Subject Classifications (2010): 35B41, 35B65, 35Q30? Corresponding author: Jose´ RealThis work has been partially supported by Ministerio de Ciencia e Innovacio´n(Spain) under project MTM2008-00088, and Junta de Andaluc´ıa grant P07-FQM-02468. J.G.-L. is a fellow of Programa de FPU del Ministerio de Educacio´n.Email addresses: luengo@us.es (Julia Garc´ıa-Luengo), pmr@us.es(Pedro Mar´ın-Rubio), jreal@us.es (Jose´ Real).Preprint submitted to Nonlinear Analysis 4 March 20111 Introduction and setting of the problemLet us consider the following problem for a non-autonomous 2D-Navier-Stokessystem: ∂u∂t− ν 4 u+ (u · ∇)u+∇p = f(t) in Ω× (τ,+∞) ,∇ · u = 0 in Ω× (τ,+∞) ,u = 0 on ∂Ω× (τ,+∞) ,u(x, τ) = uτ (x), x ∈ Ω,(1)where Ω ⊂ R2 is a bounded open set, with regular boundary ∂Ω, the numberν > 0 is the kinematic viscosity, u is the velocity field of the fluid, p the pres-sure, τ ∈ R is a given initial time, uτ is the initial velocity field, and f(t) agiven external force field.To set our problem in the abstract framework, we consider the following usualabstract spaces (see [1] and [2–4]):V ={u ∈ (C∞0 (Ω))2 : div u = 0},H = the closure of V in (L2(Ω))2 with inner product (·, ·) and associate norm|·| , where for u, v ∈ (L2(Ω))2,(u, v) =2∑j=1∫Ωuj(x)vj(x)dx,V = the closure of V in (H10 (Ω))2 with scalar product ((·, ·)) and associatenorm ‖·‖ , where for u, v ∈ (H10 (Ω))2,((u, v)) =2∑i,j=1∫Ω∂uj∂xi∂vj∂xidx.We also consider the operator A : V → V ′ defined by 〈Au, v〉 = ((u, v)).Denoting D(A) = (H2(Ω))2 ∩ V, then Au = −P∆u,∀u ∈ D(A), is the Stokesoperator (P is the ortho-projector from (L2(Ω))2 onto H).Now we define the continuous trilinear form b on V × V × V byb(u, v, w) =2∑i,j=1∫Ωui∂vj∂xiwj dx, ∀u, v, w ∈ V.2It is well known thatb(u, v, v) = 0 for all u, v ∈ V. (2)We remember (see [2] or [3]) that there exists a constant C1 > 0 only dependenton Ω such that|b(u, v, w)| ≤ C1|u|1/2‖u‖1/2‖v‖|w|1/2‖w‖1/2, ∀u, v, w ∈ V, (3)|b(u, v, w)| ≤ C1|Au|‖v‖|w|, ∀u ∈ D(A), v ∈ V, w ∈ H, (4)and|b(u, v, w)| ≤ C1|u|1/2|Au|1/2‖v‖|w|, ∀u ∈ D(A), v ∈ V, w ∈ H. (5)Assume that uτ ∈ H and f ∈ L2loc(R;H).Definition 1.1 A solution of (1) is a function u ∈ C([τ, T ];H) ∩ L2(τ, T ;V )for all T > τ, with u(τ) = uτ , such that for all v ∈ V,ddt(u(t), v) + ν((u(t), v)) + b(u(t), u(t), v) = (f(t), v),where the equation must be understood in the sense of D′(τ,+∞).Under the conditions above (e.g. cf. [2] or [3]), there exists a unique solutionu(·) = u(·; τ, uτ ) of (1). Moreover, this solution u satisfies that u ∈ C([τ +ε, T ];V ) ∩ L2(τ + ε, T ; (H2 (Ω))2) for every ε > 0 and T > τ + ε. In fact, ifuτ ∈ V , then u ∈ C([τ, T ];V ) ∩ L2(τ, T ; (H2 (Ω))2) for every T > τ.Therefore, we can define a process U = {U(t, τ), τ ≤ t} in H asU(t, τ)uτ = u(t; τ, uτ ) ∀uτ ∈ H, ∀τ ≤ t, (6)and the restriction of this process to V is a process in V .A pullback attractor for the process U defined by (6) (cf. [5–7]) is a familyAˆ = {A(t) : t ∈ R} of compact subsets of H such thata) (invariance) U(t, τ)A(τ) = A(t) for all τ ≤ t,b) (pullback attraction) limτ→−∞ supuτ∈Binfv∈A(t)|U(t, τ)uτ − v| = 0, for all t ∈ R, forany bounded subset B ⊂ H.3It can be proved (see [9]) that, under the above conditions, if in addition fsatisfies ∫ 0−∞eµr |f(r)|2 dr < +∞,for some 0 < µ < 2νλ1, where λ1 denotes the first eigenvalue of the Stokesoperator A, then there exists a pullback attractor for the process U definedby (6).Several studies on this model have already been published (cf. [5], [8,9]). How-ever, as far as we know, no one refers to the H2-regularity we will consider inthis paper.In the next section we prove some results which, in particular, imply that,under suitable assumptions, any pullback attractor Aˆ for U satisfies that A(t)is a bounded subset of (H2(Ω))2 ∩ V , for every t ∈ R (for similar results forreaction-diffusion equations see [10], and for related results for Navier-Stokesequations see [11]).2 H2-boundedness of invariant setsIn this section we prove that, under suitable assumptions, any family ofbounded subsets of H which is invariant for the process U , is in fact boundedin (H2 (Ω))2 ∩ V .First, we recall a result (cf. [2]) which will be used below.Lemma 2.1 Let X, Y be Banach spaces such that X is reflexive, and theinclusion X ⊂ Y is continuous. Assume that {un} is a bounded sequence inL∞(t0, T ;X) such that un ⇀ u weakly in Lq(t0, T ;X) for some q ∈ [1,+∞)and u ∈ C0([t0, T ];Y ).Then, u(t) ∈ X and ‖u(t)‖X ≤ lim infn≥1 ‖un‖L∞(t0,T ;X) , for all t ∈ [t0, T ].For each integer n ≥ 1, we denote by un(t) = un(t; τ, uτ ) the Galerkin approx-imation of the solution u(t; τ, uτ ) of (1), which is given byun(t) =n∑j=1γnj(t)wj,and is the solution ofddt(un(t), wj) + ν((un(t), wj)) + b(un(t), un(t), wj) = (f(t), wj) ,(un(τ), wj) = (uτ , wj) j = 1, ..., n,(7)4where {wj : j ≥ 1} ⊂ V is the Hilbert basis of H formed by the eigenvectorsof the Stokes operator A. Observe that by the regularity of Ω, all the wj belongto (H2(Ω))2.We first prove the following result.Proposition 2.2 Assume that f ∈ L2loc(R;H). Then, for any bounded setB ⊂ H, any τ ∈ R, any ε > 0 and any t > τ+ε, the following three propertiesare satisfied:i) The set {un(r; τ, uτ ) : r ∈ [τ + ε, t], uτ ∈ B, n ≥ 1}, is a bounded subsetof V.ii) The set of functions {un(·; τ, uτ ) : uτ ∈ B, n ≥ 1}, is a bounded subsetof L2(τ + ε, t;D(A)).iii) The set of time derivatives functions {u′n(·; τ, uτ ) : uτ ∈ B, n ≥ 1}, is abounded subset of L2(τ + ε, t;H).Proof.Let us fix a bounded set B ⊂ H, τ ∈ R, ε > 0, t > τ + ε, and uτ ∈ B.Multiplying by γnj(t) in (7), summing from j = 1 to n, and using (2), weobtain12ddθ|un(θ)|2 + ν ‖un(θ)‖2 = (f(θ), un(θ)) , a.e. θ > τ. (8)Observing that|(f(θ), un(θ))| ≤ 12νλ1|f(θ)|2 + νλ12|un(θ)|2≤ 12νλ1|f(θ)|2 + ν2‖un(θ)‖2 ,from (8) we deduceddθ|un(θ)|2 + ν ‖un(θ)‖2 ≤ 1νλ1|f(θ)|2 ,and integrating between τ and r,|un(r)|2 + ν∫ rτ‖un(θ)‖2 dθ (9)≤ |uτ |2 + 1νλ1∫ tτ|f(θ)|2 dθ, ∀r ∈ [τ, t], ∀n ≥ 1.Now, multiplying in (7) by λjγnj(t), where λj is the eigenvalue associated tothe eigenvector wj, and summing from j = 1 to n, we obtain512ddθ‖un(θ)‖2 + ν |Aun(θ)|2 + b(un(θ), un(θ), Aun(θ)) = (f(θ), Aun(θ)) , (10)a.e. θ > τ. Observe that| (f(θ), Aun(θ)) | ≤ 1ν|f(θ)|2 + ν4|Aun(θ)|2 ,and by (5) and Young’s inequality,|b(un(θ), un(θ), Aun(θ))| ≤ C1|un(θ)|1/2‖un(θ)‖|Aun(θ)|3/2 (11)≤ ν4|Aun(θ)|2 + C(ν)|un(θ)|2‖un(θ)‖4,where C(ν) = 27C41(4ν3)−1.Thus, from (10) we deduceddθ‖un(θ)‖2 + ν |Aun(θ)|2 ≤ 2ν|f(θ)|2 + 2C(ν)|un(θ)|2‖un(θ)‖4, (12)a.e. θ > τ.From this inequality, in particular we deduce‖un(r)‖2 ≤ ‖un(s)‖2 + 2ν∫ tτ|f(θ)|2 dθ+ 2C(ν)∫ rs|un(θ)|2‖un(θ)‖4 dθfor all τ ≤ s ≤ r ≤ t, and therefore, by Gronwall’s lemma,‖un(r)‖2 ≤(‖un(s)‖2 + 2ν∫ tτ|f(θ)|2 dθ)exp(2C(ν)∫ tτ|un(θ)|2‖un(θ)‖2 dθ)for all τ ≤ s ≤ r ≤ t.Integrating this last inequality for s between τ + ε/2 and r, we obtain(r − τ − ε2) ‖un(r)‖2≤(∫ tτ‖un(s)‖2 ds+ 2(t− τ)ν∫ tτ|f(θ)|2 dθ)× exp(2C(ν)∫ tτ|un(θ)|2‖un(θ)‖2 dθ)for all τ + ε/2 ≤ r ≤ t, and in particular,6‖un(r)‖2≤ 2ε(∫ tτ‖un(s)‖2 ds+ 2(t− τ)ν∫ tτ|f(θ)|2 dθ)(13)× exp(2C(ν)∫ tτ|un(θ)|2‖un(θ)‖2 dθ)for all τ + ε ≤ r ≤ t, for any n ≥ 1.From (9) and (13), the assertion in i) holds. Moreover, by (12),ν∫ tτ+ε|Aun(θ)|2 dθ≤‖un(τ + ε)‖2 + 2ν∫ tτ|f(θ)|2 dθ+2C(ν)∫ tτ+ε|un(θ)|2‖un(θ)‖4 dθ,and therefore, by i), the assertion in ii) holds.On the other hand, multiplying by the derivative γ′nj(t) in (7), and summingfrom j = 1 till n, we obtain|u′n(θ)|2 +ν2ddθ‖un(θ)‖2 + b(un(θ), un(θ), u′n(θ)) = (f(θ), u′n(θ)) , (14)a.e. θ > τ.Observing that| (f(θ), u′n(θ)) | ≤14|u′n(θ)|2 + |f(θ)|2,and by (4)|b(un(θ), un(θ), u′n(θ))| ≤ C1|Aun(θ)|‖un(θ)‖|u′n(θ)|≤ 14|u′n(θ)|2 + C21 |Aun(θ)|2‖un(θ)‖2,we obtain from (14)|u′n(θ)|2 + νddθ‖un(θ)‖2 ≤ 2|f(θ)|2 + 2C21 |Aun(θ)|2‖un(θ)‖2.Integrating this last inequality, we deduce that∫ tτ+ε|u′n(θ)|2 dθ≤ ν‖un(τ + ε)‖2 + 2∫ tτ|f(θ)|2 dθ+2C21 supθ∈[τ+ε,t]‖un(θ)‖2∫ tτ+ε|Aun(θ)|2 dθ,and therefore iii) follows from i) and ii).7Corollary 2.3 Assume that f ∈ L2loc(R;H). Then, for any bounded set B ⊂H, any τ ∈ R, any ε > 0, and any t > τ + ε, the set ⋃r∈[τ+ε,t]U(r, τ)B is abounded subset of V.Proof. This is a straightforward consequence of Lemma 2.1, assertion i) inProposition 2.2, and the well known fact (e.g. cf. [1–4]) that for all uτ ∈B the Galerkin approximations un(·; τ, uτ ) converge weakly to u(·; τ, uτ ) inL2(τ, t;V ), and u(·; τ, uτ ) ∈ C([τ, t];H).Assuming additional regularity for the time derivative of f , we can improvethe above results.Proposition 2.4 Assume that f ∈ W 1,2loc (R;H) . Then, for any bounded setB ⊂ H, any τ ∈ R, any ε > 0, and any t > τ + ε, the following two propertiesare satisfied:iv) The set of time derivatives {u′n(r; τ, uτ ) : r ∈ [τ + ε, t], uτ ∈ B, n ≥ 1},is a bounded subset of H.v) The set {un(r; τ, uτ ) : r ∈ [τ + ε, t], uτ ∈ B, n ≥ 1} is a bounded subsetof D(A).Proof. Let us fix a bounded set B ⊂ H, τ ∈ R, ε > 0, t > τ + ε, and uτ ∈ B.As we are assuming that f ∈ W 1,2loc (R;H), we can differentiate with respect totime in (7), and then, multiplying by γ′nj(t), and summing from j = 1 to n,we obtain12ddθ|u′n(θ)|2 + ν ‖u′n(θ)‖2 + b(u′n(θ), un(θ), u′n(θ)) = (f ′(θ), u′n(θ))a.e. θ > τ.From this inequality, taking into account that| (f ′(θ), u′n(θ)) | ≤ν2‖u′n(θ)‖2 +12νλ1|f ′(θ)|2,and by (3)|b(u′n(θ), un(θ), u′n(θ))| ≤ C1|u′n(θ)|‖u′n(θ)‖‖un(θ)‖≤ ν2‖u′n(θ)‖2 +C212ν|u′n(θ)|2‖un(θ)‖2,we deduceddθ|u′n(θ)|2 ≤1νλ1|f ′(θ)|2 + C21ν|u′n(θ)|2‖un(θ)‖2.8Integrating in the last inequality,|u′n(r)|2 ≤ |u′n(s)|2 +1νλ1∫ tτ|f ′(θ)|2 dθ + C21ν∫ rs|u′n(θ)|2‖un(θ)‖2 dθ,for all τ ≤ s ≤ r ≤ t.Thus, by Gronwall’s inequality,|u′n(r)|2 ≤(|u′n(s)|2 +1νλ1∫ tτ|f ′(θ)|2 dθ)exp(C21ν∫ tτ+ε/2‖un(θ)‖2 dθ),for all τ + ε/2 ≤ s ≤ r ≤ t.Now, integrating this inequality with respect to s between τ + ε/2 and r, weobtain(r − τ − ε/2) |u′n(r)|2≤(∫ tτ+ε/2|u′n(s)|2 ds+t− τνλ1∫ tτ|f ′(θ)|2 dθ)× exp(C21ν∫ tτ+ε/2‖un(θ)‖2 dθ),for all τ + ε/2 ≤ r ≤ t, and any n ≥ 1. In particular, thus,|u′n(r)|2≤2ε(∫ tτ+ε/2|u′n(s)|2 ds+t− τνλ1∫ tτ|f ′(θ)|2 dθ)× exp(C21ν∫ tτ+ε/2‖un(θ)‖2 dθ),for all τ + ε ≤ r ≤ t, and any n ≥ 1.From this inequality and properties i) and iii) in Proposition 2.2, we obtainiv).On the other hand, multiplying again in (7) by λjγnj(t), and summing oncemore from j = 1 to n, we obtain(u′n(r), Aun(r))+ν |Aun(r)|2+b(un(r), un(r), Aun(r)) = (f(r), Aun(r)) , (15)for all r ≥ τ. But| (u′n(r), Aun(r)) | ≤2ν|u′n(r)|2 +ν8|Aun(r)|2 ,9and| (f(r), Aun(r)) | ≤ 2ν|f(r)|2 + ν8|Aun(r)|2 .Therefore, taking into account (11), we deduce from (15) thatν2|Aun(r)|2 ≤ 2ν(|u′n(r)|2 + |f(r)|2) + C(ν)|un(r)|2‖un(r)‖4 (16)for all r ≥ τ.Thus, since in particular f ∈ C(R;H), from i) in Proposition 2.2, iv) andinequality (16), we deduce v).As a direct consequence of the above, we can now establish our main results.Theorem 2.5 Assume that f ∈ W 1,2loc (R;H) . Then, for any bounded set B ⊂H, any τ ∈ R, any ε > 0, and any t > τ + ε, the set ⋃r∈[τ+ε,t]U(r, τ)B is abounded subset of D(A) = (H2(Ω))2 ∩ V.Proof. This follows from Lemma 2.1, Proposition 2.4, and the well known factsthat un(·; τ, uτ ) converges weakly to u(·; τ, uτ ) in L2(τ, t;V ), and u(·; τ, uτ )belongs to C([τ + ε, t];V ).Theorem 2.6 Assume that f ∈ L2loc (R;H) , and Aˆ = {A(t) : t ∈ R} is afamily of bounded subsets of H, such that U(t, τ)A(τ) = A(t) for any τ ≤ t.Then:i) For any T1 < T2, the set⋃t∈[T1,T2]A(t) is a bounded subset of V.ii) If moreover f ′ ∈ L2loc (R;H) , then for any T1 < T2, the set⋃t∈[T1,T2]A(t) isa bounded subset of (H2(Ω))2 ∩ V.Proof. It is enough to observe that if τ < T1 − 1 is fixed, then⋃t∈[T1,T2]A(t) ⊂ ⋃t∈[τ+1,T2]U(t, τ)A(τ).Now, apply Corollary 2.3 and Theorem 2.5.10AcknowledgmentsThe authors would like to thank the referees for their careful reading on aprevious version of the paper.References[1] J. L. Lions, Quelques me´thodes de re´solution des proble`mes aux limites nonline´aires, Dunod, Gauthier-Villars, Paris, 1969.[2] J. C. Robinson, Infinite-dimensional dynamical systems, Cambridge UniversityPress, 2001.[3] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,Springer, New York, Second Edition, 1997.[4] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed.,North Holland, Amsterdam, 1979.[5] T. Caraballo, G.  Lukaszewicz, and J. Real, Pullback attractors for asymptoticallycompact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006), 484-498.[6] T. Caraballo, G.  Lukaszewicz, and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains, C. R. Math.Acad. Sci. Paris, 342 (2006), 263–268.[7] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems,Stoch. Dyn. 3 (2003), 101–112.[8] P. Mar´ın-Rubio and J. Real, On the relation between two different concepts ofpullback attractors for non-autonomous dynamical systems, Nonlinear Anal. 71(2009), 3956–3963.[9] J. Garc´ıa-Luengo, P. Mar´ın-Rubio, and J. Real, Pullback attractors in V fornon-autonomous 2D-Navier-Stokes equations and their tempered behaviour,submitted.[10] M. Anguiano, T. Caraballo, and J. Real, H2-boundedness of the pullbackattractor for a non-autonomous reaction-diffusion equation, Nonlinear Anal.72 (2010), 876–880.[11] C. Guillope´, Comportement a` l’infini des solutions des e´quations de Navier-Stokes et proprie´te´ des ensembles fonctionnels invariants (ou attracteurs), Ann.Inst. Fourier (Grenoble), 32 (1982), 1–37.11

H^2-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains

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H^2-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains

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