Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays

Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume XX, Numbe 0xx, XXXXXX 20xx pp. – SOME NEW REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR 2D NAVIER-STOKES EQUATIONS WITH DELAYS Julia Ga c´ ıa-Luengo, Ped o Ma ´ ın-Rubio & Jos´ e Real∗ Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico Uni e sidad de Se illa Apdo. de Co eos 1160, 41080–Se illa, Spain (Communica ed by XXXXX) Abs ac . In his pape we s eng hen some esul s on he exis ence and p ope ies o pullback a ac o s o a 2D Na ie -S okes model wi h ini e delay o mula ed in [Ca aballo and Real, J. Di e en ial Equa ions 205 (2004), 271– 297]. Ac ually, we p o e ha unde sui able assump ions, pullback a ac o s no only o ixed bounded se s bu also o a se o empe ed uni e ses do exis . Mo eo e , hanks o egula i y esul s, he a ac ion om di e en phase spaces also happens in C([−h, 0]; V). Finally, om compa ison esul s o a ac o s, and unde an addi ional hypo hesis, we es ablish ha all hese amilies o a ac o s a e in ac he same objec . 1. In oduc ion and s a emen o he p oblem. Le Ω ⊂R2be an open bounded se wi h smoo h enough bounda y ∂Ω, and conside an a bi a y ini ial ime τ∈R, and he ollowing unc ional Na ie -S okes p oblem:                    ∂u ∂ −ν∆u+ (u· ∇)u+∇p= ( ) + g( , u ) in Ω ×(τ, ∞), di u= 0 in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ) = uτ(x), x ∈Ω, u(x, τ +s) = φ(x, s), x ∈Ω, s ∈(−h, 0), (1) whe e ν > 0 is he kinema ic iscosi y, u= (u1, u2) is he eloci y ield o he luid, pis he p essu e, is a non-delayed ex e nal o ce ield, gis ano he ex e nal o ce wi h some he edi a y cha ac e is ics, and uτand φ(x, s −τ) a e he ini ial da a in τand (τ−h, τ) espec i ely, whe e h > 0 is he ime o memo y e ec . Fo each ≥τ, we deno e by u he unc ion de ined a.e. on (−h, 0) by he ela ion u (s) = u( +s), a.e. s∈(−h, 0). 2010 Ma hema ics Subjec Classi ica ion. P ima y: 35B41, 35Q30, 37L30. Key wo ds and ph ases. 2D Na ie -S okes equa ions; delay e ms; pullback a ac o s. ∗Deceased on Janua y 27 h, 2012. The i s wo coau ho s dedica e his pape o he memo y o hei coau ho Jos´e Real. 1 2 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL The impo ance o physical models o luid mechanic p oblems including delay e ms is ela ed, o ins ance, o eal applica ions whe e de ices o con ol p ope - ies o luids ( empe a u e, eloci y, e c.) a e inse ed in domains and make a local in luence on he beha iou o he sys em (e.g., c . [13] o a wind- unnel model). The s udy o Na ie -S okes models including delay e ms –exis ence, unique- ness, s a iona y solu ions, exponen ial decay, and o he asymp o ic p ope ies such as he exis ence o a ac o s– was ini ia ed in he e e ences [3,4,5], and a e ha , many di e en ques ions, as dealing wi h unbounded domains, and models ( o ins ance in h ee dimensions o modi ied e ms) ha e been add essed (e.g., c . [10,17,21,19,14,20,11,15,16] among o he s). In he ecen pape [9], we ha e ea ed a elaxa ion on he assump ions o he delay ope a o in ol ed, emo ing condi ions ela ed o he con ol o he L2no m o he delay e ms (see assump ions (IV) and (V) below). Al hough his implies o es ic he phase space o con inuous unc ions ins ead o squa e in eg able in ime, he delay unc ions d i ing he delayed ime wi hin his heo y can be aken jus measu able, wi hou any addi ional assump ion as con inui y no C1wi h bounded de i a i e, as usual in he li e a u e. Mo eo e , in [9] we we e also able o es ablish a ac ion in a highe no m (namely, H1ins ead o L2) making a sha p use o egula iza ion o he equa ions in dimension wo and by ene gy me hods. Rela ionships among a ac o s in di e en me ics was success ully ca ied ou he e, oo. Ou goal in his pape is o keep all usual condi ions o he delay ope a o (in- cluding (IV) and (V)) and o compa e bo h kind o a ac o s, o bo h possibili ies o phase spaces (con inuous in ime, o jus squa e in eg able in ime). Obse e ha in he au onomous amewo k his issue would be almos immedia e since one inclusion is clea by con inuous embedding, and he o he is ob ained a e an elapsed ime as long as he memo y e ec . Howe e , in he non-au onomous case ( ha we a e dealing wi h) his is no he case a all. Using he heo y o a ac- ion o uni e ses (c . [1,2,18]) we deal wi h di e en amilies and unde di e en me ics. Namely, we conside uni e ses o ixed (in ime) bounded se s and also ime-dependen amilies gi en by a empe ed condi ion when ime goes o −∞. Mo eo e , we also imp o e some esul s p e iously ob ained in he li e a u e (c . [5]) since we can deal wi h he phase space V×L2(−h, 0; V) and no only H×L2(−h, 0; H).Finally, om compa ison esul s o a ac o s and unde an ad- di ional assump ion, we es ablish ha all hese amilies o a ac o s a e in ac he same objec . The s uc u e o he pape is he ollowing. We con inue his sec ion wi h he abs ac se ing o he p oblem, gene al de ini ions and some well-known esul s on exis ence o weak and s ong solu ions and egula i y p ope ies. In Sec ion 2 we ecall he basic heo y o pullback a ac o s o non-au onomous dynamical sys ems wi hin he amewo k o uni e ses, and compa ison esul s, when di e en me ics a e in ol ed, a e also gi en. Sec ion 3 is de o ed o es ablish all possible a ac o s o di e en phase-spaces bu aking in o accoun he L2no m in space. Ou main esul s, es ablished in he highe no m H1(in space), a e gi en in Sec ion REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 3 4. In hese wo las sec ions, ene gy me hods (in oduced in his con ex by Rosa in [23]) a e used o p o e asymp o ic compac ness in he espec i e uni e ses. As said be o e, ela ionships among all hese objec s a e ob ained. To se ou p oblem in he abs ac amewo k, we conside he ollowing usual unc ion spaces: V=u∈(C∞ 0(Ω))2: di u= 0, H= he closu e o Vin (L2(Ω))2wi h he no m |·|, and inne p oduc (·,·), whe e o u, ∈(L2(Ω))2, (u, ) = 2 X j=1 ZΩ uj(x) j(x)dx, V= he closu e o Vin (H1 0(Ω))2wi h he no m k·k associa ed o he inne p oduc ((·,·)), whe e o u, ∈(H1 0(Ω))2, ((u, )) = 2 X i,j=1 ZΩ ∂uj ∂xi ∂ j ∂xi dx. We will use k · k∗ o he no m in V0and h·,·i o he duali y be ween V0and V. We conside e e y elemen h∈Has an elemen o V0, gi en by he equali y hh, i= (h, ) o all ∈V. I ollows ha V⊂H⊂V0, whe e he injec ions a e dense and con inuous, and, in ac , compac . Now, we de ine he ope a o A:V→V0as hAu, i= ((u, )) ∀u, ∈V. Le us deno e D(A) = {u∈V:Au ∈H}. By he egula i y o ∂Ω, one has ha D(A)=(H2(Ω))2∩V, and Au =−P∆u o all u∈D(A) is he S okes ope a o (Pis he o ho-p ojec o om (L2(Ω))2on o H). On D(A) we conside he no m |·|D(A)de ined by |u|D(A)=|Au|. Obse e ha on D(A) he no ms k·k(H2(Ω))2and |·|D(A)a e equi alen (see [6] o [25]), and D(A) is compac ly and densely injec ed in V. Le us de ine b(u, , w) = 2 X i,j=1 ZΩ ui ∂ j ∂xi wjdx, o e e y unc ions u, , w : Ω →R2 o which he igh -hand side is well de ined. In pa icula , bhas sense o all u, , w ∈V, and is a con inuous ilinea o m on V×V×V. Some use ul p ope ies conce ning b ha we will use in he nex sec ions a e he ollowing (see [22] o [24]): b(u, , w) = −b(u, w, ) o all u, ,w∈V, which also implies ha b(u, , ) = 0 o all u, ∈V. Mo eo e , he e exis s a cons an C1>0, only dependen on Ω, such ha ( ecall ha we a e in dimension wo) |b(u, , w)| ≤ C1|u|1/2|Au|1/2k k|w| ∀ u∈D(A), ∈V, w ∈H. (2) Now, we es ablish some sui able spaces in o de o deal wi h he delay e m, and some app op ia e assump ions on he e m in (1) con aining he delay. Le us deno e CH=C([−h, 0]; H), wi h he no m |ϕ|CH= maxs∈[−h,0] |ϕ(s)|, and L2 X=L2(−h, 0; X) o X=H,V. On he delay ope a o om (1), we conside ha is well de ined as g:R×CH→(L2(Ω))2, and i sa is ies he ollowing assump ions: 4 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL (I) o all ξ∈CH, he unc ion R3 7→ g( , ξ)∈(L2(Ω))2is measu able, (II) g( , 0) = 0, o all ∈R, (III) he e exis s Lg>0 such ha o all ∈R, and o all ξ,η∈CH, |g( , ξ)−g( , η)| ≤ Lg|ξ−η|CH, (IV) he e exis s Cg>0 such ha o all τ≤ , and o all u, ∈C([τ−h, ]; H), Z τ |g(s, us)−g(s, s)|2ds ≤C2 gZ τ−h |u(s)− (s)|2ds. Examples o ixed, a iable, and dis ibu ed delay ope a o s can be ound, o in- s ance, in [3, Sec ion 3], [5, Sec ions 3.5 and 3.6], and [10, Sec ion 3], and we omi hem he e jus o he sake o b e i y. Obse e ha (I)−(III) imply ha gi en T > τ and u∈C([τ−h, T]; H), he unc ion gu: [τ, T]→(L2(Ω))2de ined by gu( ) = g( , u ) o all ∈[τ, T ], is measu able and, in ac , belongs o L∞(τ, T; (L2(Ω))2). Then, hanks o (IV), he mapping G:u∈C([τ−h, T]; H)→gu∈L2(τ, T; (L2(Ω))2) has a unique ex ension o a mapping e Gwhich is uni o mly con inuous om L2(τ− h, T;H) in o L2(τ, T; (L2(Ω))2). F om now on, we will deno e g( , u ) = e G(u)( ) o each u∈L2(τ−h, T ;H), and hus p ope y (IV) will also hold o all u, ∈L2(τ−h, T;H). Assume ha uτ∈H,φ∈L2 H, and ∈L2 loc(R;V0). De ini ion 1. A weak solu ion o (1) is a unc ion u ha belongs o L2(τ−h, T;H) ∩L2(τ, T;V)∩L∞(τ, T;H) o all T > τ, wi h u(τ) = uτand u( ) = φ( −τ) a.e. ∈(τ−h, τ), and such ha o all ∈V, d d (u( ), ) + νhAu( ), i+b(u( ), u( ), ) = h ( ), i+ (g( , u ), ),(3) whe e he equa ion mus be unde s ood in he sense o D0(τ, ∞). Rema k 1. I uis a weak solu ion o (1), hen om (3) we deduce ha o any T > τ, one has u0∈L2(τ, T;V0), and so u∈C([τ, ∞); H), whence he ini ial da um u(τ) = uτhas ull sense. Mo eo e , in his case he ollowing ene gy equali y holds: |u( )|2+2νZ s ku( )k2d =|u(s)|2+2Z sh ( ), u( )i+(g( , u ), u( ))d ∀τ≤s≤ . A no ion o mo e egula solu ion is also sui able o p oblem (1). De ini ion 2. A s ong solu ion o (1) is a weak solu ion uo (1) such ha u∈ L2(τ, T;D(A)) ∩L∞(τ, T;V) o all T > τ. Rema k 2. I ∈L2 loc(R; (L2(Ω))2) and uis a s ong solu ion o (1), hen u0∈ L2(τ, T;H) o all T > τ, and so u∈C([τ, ∞); V). In his case he ollowing ene gy equali y holds: ku( )k2+ 2νZ s |Au( )|2d + 2 Z s b(u( ), u( ), Au( )) d =ku(s)k2+ 2 Z s ( ( ) + g( , u ), Au( )) d ∀τ≤s≤ . (4) REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 5 Conce ning he exis ence and uniqueness o weak and s ong solu ions o (1), we ha e he ollowing esul which can be p o ed simila ly as [3, Theo em 2.1] o [4, Theo em 2.5] (see also [10, Theo em 2.3] o a mo e gene al case). Theo em 1. Le us conside uτ∈H,φ∈L2 H, ∈L2 loc(R;V0), and g:R×CH→ (L2(Ω))2sa is ying (I)–(IV). Then, o each τ∈R, he e exis s a unique weak solu ion u=u(·;τ, uτ, φ)o (1). Mo eo e , i ∈L2 loc(R; (L2(Ω))2), hen (a) u∈C([τ+ε, T]; V)∩L2(τ+ε, T;D(A)) o all T > τ +ε > τ. (b) I uτ∈V,uis in ac a s ong solu ion o (1). Be o e es ablishing he o iginal esul s abou he egula i y o pullback a ac o s, we ecall he main exis ence esul s s udied in [5,17,19]. Fi s ly, in o de o do ha , we emembe b ie ly he abs ac heo y on pullback a ac o s in he nex sec ion. 2. Abs ac esul s on minimal pullback a ac o s. Now, we p esen a sum- ma y o some esul s om [8] abou he exis ence o minimal pullback a ac o s (see also [1,2,18]). In pa icula , we assume ha he p ocess Uis closed (see De ini ion 3below). Conside gi en a me ic space (X, dX), and le us deno e R2 d={( , τ)∈R2:τ≤ }. A p ocess Uon Xis a mapping R2 d×X3( , τ, x)7→ U( , τ)x∈Xsuch ha U(τ, τ)x=x o any (τ, x)∈R×X, and U( , )(U( , τ)x) = U( , τ)x o any τ≤ ≤ and all x∈X. De ini ion 3. Le Ube a p ocess on X. (a) Uis said o be con inuous i o any pai τ≤ , he mapping U( , τ) : X→X is con inuous. (b) Uis said o be closed i o any τ≤ , and any sequence {xn} ⊂ X, i xn→x∈Xand U( , τ)xn→y∈X, hen U( , τ)x=y. Rema k 3. I is clea ha e e y con inuous p ocess is closed. Le us deno e by P(X) he amily o all nonemp y subse s o X, and conside a amily o nonemp y se s b D0={D0( ) : ∈R}⊂P(X). De ini ion 4. We say ha a p ocess Uon Xis pullback b D0-asymp o ically compac i o any ∈Rand any sequences {τn} ⊂ (−∞, ] and {xn} ⊂ Xsa is ying τn→ −∞ and xn∈D0(τn) o all n, he sequence {U( , τn)xn}is ela i ely compac in X. Deno e Λ( b D0, ) = s≤ [ τ≤s U( , τ)D0(τ) X ∀ ∈R, whe e {· · · }Xis he closu e in X. Gi en wo subse s o X,O1and O2, we deno e by dis X(O1,O2) he Hausdo semi-dis ance in Xbe ween hem, de ined as dis X(O1,O2) = sup x∈O1 in y∈O2 dX(x, y). Le be gi en Da nonemp y class o amilies pa ame e ized in ime b D={D( ) : ∈R}⊂P(X). The class Dwill be called a uni e se in P(X). 6 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL De ini ion 5. A p ocess Uon Xis said o be pullback D-asymp o ically compac i i is pullback b D-asymp o ically compac o any b D∈ D. I is said ha b D0={D0( ) : ∈R}⊂P(X) is pullback D-abso bing o he p ocess Uon Xi o any ∈Rand any b D∈ D, he e exis s a τ0( , b D)≤ such ha U( , τ)D(τ)⊂D0( )∀τ≤τ0( , b D). Wi h he abo e de ini ions, we may es ablish he main esul o his sec ion (c . [8, Theo em 3.11]). Theo em 2. Conside a closed p ocess U:R2 d×X→X, a uni e se Din P(X), and a amily b D0={D0( ) : ∈R}⊂P(X)which is pullback D-abso bing o U, and assume also ha Uis pullback b D0-asymp o ically compac . Then, he amily AD={AD( ) : ∈R}de ined by AD( ) = Sb D∈D Λ( b D, ) X , has he ollowing p ope ies: (a) o any ∈R, he se AD( )is a nonemp y compac subse o X, and AD( )⊂ Λ( b D0, ), (b) ADis pullback D-a ac ing, i.e., limτ→−∞ dis X(U( , τ)D(τ),AD( )) = 0 o all b D∈ D, and any ∈R, (c) ADis in a ian , i.e., U( , τ)AD(τ) = AD( ) o all ( , τ)∈R2 d, (d) i b D0∈ D, hen AD( ) = Λ( b D0, )⊂D0( )X o all ∈R. The amily ADis minimal in he sense ha i b C={C( ) : ∈R} ⊂ P(X)is a am- ily o closed se s such ha o any b D={D( ) : ∈R}∈D,lim τ→−∞ dis X(U( , τ)D(τ), C( )) = 0, hen AD( )⊂C( ). Rema k 4. Unde he assump ions o Theo em 2, he amily ADis called he minimal pullback D-a ac o o he p ocess U. I AD∈ D, hen i is he unique amily o closed subse s in D ha sa is ies (b)–(c). A su icien condi ion o AD∈ D is o ha e ha b D0∈ D, he se D0( ) is closed o all ∈R, and he amily Dis inclusion-closed (i.e., i b D∈ D, and b D0={D0( ) : ∈R}⊂P(X) wi h D0( )⊂D( ) o all , hen b D0∈ D). We will deno e by DF(X) he uni e se o ixed nonemp y bounded subse s o X, i.e., he class o all amilies b Do he o m b D={D( ) = D: ∈R}wi h Da ixed nonemp y bounded subse o X. Now, i is easy o conclude he ollowing esul . Co olla y 1. Unde he assump ions o Theo em 2, i he uni e se Dcon ains he uni e se DF(X), hen bo h a ac o s, ADF(X)and AD, exis , and ADF(X)( )⊂ AD( ) o all ∈R. Rema k 5. I can be p o ed (see [18]) ha , unde he assump ions o he p eceding co olla y, i o some T∈R, he se ∪ ≤TD0( ) is a bounded subse o X, hen ADF(X)( ) = AD( ) o all ≤T. Now, and since i will be use ul below, we es ablish an abs ac esul (c . [8, Theo em 3.15]) ha allows us o compa e wo a ac o s o a p ocess unde app o- p ia e assump ions. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 7 Theo em 3. Le {(Xi, dXi)}i=1,2be wo me ic spaces such ha X1⊂X2wi h con inuous injec ion, and o i= 1,2, le Dibe a uni e se in P(Xi), wi h D1⊂ D2. Assume ha we ha e a map U ha ac s as a p ocess in bo h cases, i.e., U:R2 d×Xi→ Xi o i= 1,2is a p ocess. Fo each ∈R, le us deno e Ai( ) = [ b Di∈Di Λi(b Di, ) Xi i= 1,2, whe e he subsc ip iin he symbol o he omega-limi se Λiis used o deno e he dependence o he espec i e opology. Then, A1( )⊂ A2( ) o all ∈R. Suppose mo eo e ha he wo ollowing condi ions a e sa is ied: (i) A1( )is a compac subse o X1 o all ∈R, (ii) o any b D2∈ D2and any ∈R, he e exis a amily b D1∈ D1and a ∗ b D1≤ (bo h possibly depending on and b D2), such ha Uis pullback b D1- asymp o ically compac , and o any s≤ ∗ b D1 he e exis s a τs≤ssuch ha U(s, τ)D2(τ)⊂D1(s) o all τ≤τs. Then, unde all he condi ions abo e, A1( ) = A2( ) o all ∈R. Rema k 6. In he p eceding heo em, i ins ead o assump ion (ii) we conside he ollowing condi ion: (ii’) o any b D2∈ D2and any sequence τn→ −∞, he e exis ano he amily b D1∈ D1and ano he sequence τ0 n→ −∞ wi h τ0 n≥τn o all n, such ha U is pullback b D1-asymp o ically compac , and U(τ0 n, τn)D2(τn)⊂D1(τ0 n)∀n, hen, wi h a simila p oo , one can ob ain ha he equali y A1( ) = A2( ) also holds o all ∈R. Obse e ha a su icien condi ion o (ii’) is ha he e exis s T > 0 such ha o any b D2∈ D2, he e exis s a b D1∈ D1sa is ying ha Uis pullback b D1-asymp o ically compac , and U(τ+T, τ)D2(τ)⊂D1(τ+T) o all τ∈R. 3. P e ious esul s on p ocesses and pullback a ac o s in H.In his sec ion we ecall some known esul s (c . [5,17,19]) on he exis ence o minimal pullback a ac o s in he Hno m o sui able p ocesses associa ed o p oblem (1). In o de o apply he heo y o he abo e sec ion, and ollowing [5,17,19], we may conside he Banach space CH, and he Hilbe space M2 H=H×L2 Hwi h associa ed no m k(uτ, φ)k2 M2 H=|uτ|2+Z0 −h |φ(s)|2ds o (uτ, φ)∈M2 H. We can de ine wo p ocesses o p oblem (1). P oposi ion 1. Assume ha ∈L2 loc(R;V0), and g:R×CH→(L2(Ω))2sa is ies (I)–(IV). Then, he bi-pa ame ic amilies o mappings U( , τ) : CH→CHand S( , τ) : M2 H→M2 Hgi en espec i ely by U( , τ)φ=u (·;τ, φ(0), φ) o φ∈CH, τ ≤ , (5) and S( , τ)(uτ, φ)=(u( ;τ, uτ, φ), u (·;τ, uτ, φ)) o (uτ, φ)∈M2 H, τ ≤ , (6) 8 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL whe e uis he unique weak solu ion o (1), a e well de ined con inuous p ocesses on CHand M2 H espec i ely. P oo . The esul ollows om Theo em 1abo e, and om [5, Theo em 9]. Now, in o de o es ablish asymp o ic es ima es o he solu ions o (1), we impose a i h assump ion on gand . Deno e by λ1 he i s eigen alue o he S okes ope a o A. (V) Assume ha νλ1> Cg, and ha he e exis s a alue η∈(0,2(νλ1−Cg)) such ha o e e y u∈L2(τ−h, ;H), Z τ eηs|g(s, us)|2ds ≤C2 gZ τ−h eηs|u(s)|2ds o any τ≤ , and Z0 −∞ eηsk (s)k2 ∗ds < ∞. Lemma 1. Suppose ha ∈L2 loc(R;V0), and ha and g:R×CH→(L2(Ω))2 sa is y (I)–(V). Then, o any (uτ, φ)∈M2 H, he ollowing es ima e holds o he solu ion u o (1) o all ≥τ: |u( )|2≤e−η( −τ)max{1, Cg}k(uτ, φ)k2 M2 H+β−1e−η Z τ eηsk (s)k2 ∗ds, (7) whe e β= 2ν−(η+ 2Cg)λ−1 1.(8) P oo . By he ene gy equali y (see Rema k 1), and Young’s inequali y, we ha e d d |u( )|2+ 2νku( )k2 ≤βku( )k2+β−1k ( )k2 ∗+Cg|u( )|2+C−1 g|g( , u )|2,a.e. > τ. Thus, d d eη |u( )|2+eη 2ν−β−(η+Cg)λ−1 1ku( )k2 ≤eη β−1k ( )k2 ∗+eη C−1 g|g( , u )|2,a.e. > τ, and he e o e, in eg a ing abo e and using p ope y (V), we ob ain eη |u( )|2+2ν−β−(η+Cg)λ−1 1Z τ eηsku(s)k2ds ≤eητ |uτ|2+β−1Z τ eηsk (s)k2 ∗ds +CgZ τ−h eηs|u(s)|2ds ≤eητ max{1, Cg}k(uτ, φ)k2 M2 H+β−1Z τ eηsk (s)k2 ∗ds +CgZ τ eηs|u(s)|2ds, o all ≥τ, and om his las inequali y and (8), in pa icula we deduce (7). A e he abo e esul , i u ns ou app op ia e he in oduc ion o he ollowing empe ed uni e ses. De ini ion 6. Fo any η > 0, we will deno e by Dη(CH) he class o all amilies o nonemp y subse s b D={D( ) : ∈R}⊂P(CH) such ha lim τ→−∞ eητ sup ϕ∈D(τ) |ϕ|2 CH= 0. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 9 Analogously, we will deno e by Dη(M2 H) he class o all amilies o nonemp y subse s b D={D( ) : ∈R}⊂P(M2 H) such ha lim τ→−∞ eητ sup (w,ϕ)∈D(τ) k(w, ϕ)k2 M2 H= 0. Fu he mo e, acco dingly o he no a ion in oduced in he p e ious sec ion, DF(CH) and DF(M2 H) will deno e he uni e ses o ixed bounded se s in CHand M2 H espec i ely. Rema k 7. (i) The choices o he abo e uni e ses a e igh and con enien o keep, in he sense ha , on he one hand, M2 His mo e gene al as phase space o he ini ial da a o p oblem (1). On he o he hand, he egula i y o he solu ion o (1) (c . Theo em 1) makes ha , a e an elapsed ime h, e e y solu ion is con inuous wi h alues on H. Indeed, as i was obse ed in [5], in he case o he uni e ses o ixed bounded se s, pullback a ac o s in bo h spaces do exis , and hey a e in insically ela ed h ough he canonical embedding j:CH→M2 Hde ined by j(ϕ) = (ϕ(0), ϕ) (see Theo em 4below). (ii) The uni e ses Dη(CH) and Dη(M2 H),which a e inclusion-closed, con ain e- spec i ely he uni e ses DF(CH) and DF(M2 H). Now, we ob ain pullback abso bing amilies o U:R2 d×CH→CHand S: R2 d×M2 H→M2 H. Co olla y 2. Unde he assump ions o Lemma 1, he amily b D1,η ={D1,η( ) : ∈ R} ⊂ P(CH)de ined by D1,η( ) = BCH(0, η( )), he closed ball in CHo cen e ze o and adius η( ), whe e 2 η( ) = 1 + β−1e−η( −h)Z −∞ eηsk (s)k2 ∗ds, wi h βgi en by (8), is pullback Dη(CH)-abso bing o he p ocess Uon CHde ined by (5) (and he e o e pullback DF(CH)-abso bing oo), and b D1,η belongs o Dη(CH). Besides, he amily b D2,η ={D2,η( ) : ∈R} ⊂ P(M2 H)de ined by D2,η( ) = BM2 H(0, Rη( )), he closed ball in M2 Ho cen e ze o and adius Rη( ), wi h R2 η( ) = 1 + β−1(1 + heηh)e−η Z −∞ eηsk (s)k2 ∗ds, is pullback Dη(M2 H)-abso bing o he p ocess Son M2 Hgi en by (6) (and hus also pullback DF(M2 H)-abso bing), and b D2,η belongs o Dη(M2 H). Since i will be use ul in o de o compa e he pullback a ac o s de ined in he spaces CHand M2 H, we conside he bi-pa ame ic amily o mappings e U( , τ) : M2 H→L2 Hde ined as e U( , τ)(uτ, φ) = u (·;τ, uτ, φ) o (uτ, φ)∈M2 H, τ ≤ . Rema k 8. Obse e ha e U( , τ) maps M2 Hin o CHi ≥τ+h, and he e o e we can w i e S( , τ)(uτ, φ) = j(e U( , τ)(uτ, φ)) o (uτ, φ)∈M2 H, ≥τ+h, whe e S(·,·) is gi en by (6). Mo eo e , i is clea ha U( , τ)φ=e U( , τ)j(φ) o φ∈CH, ≥τ, 16 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL Then, since all Jna e non-inc easing, we deduce ha o all n≥n(kδ) Jn( n)−J( ∗)≤Jn(˜ kδ)−J( ∗) ≤ |Jn(˜ kδ)−J( ∗)| ≤ |Jn(˜ kδ)−J(˜ kδ)|+|J(˜ kδ)−J( ∗)|< δ. The e o e, as δ > 0 is a bi a y, we ob ain ha lim sup n→∞ Jn( n)≤J( ∗), and consequen ly, again by (15) and (16), lim sup n→∞ kun( n)k≤ku( ∗)k, which combined wi h (20) and (17) allows us o claim ha un( n)→u( ∗) s ongly in V, in con adic ion wi h (19). Thus, (18) is p o ed as desi ed. As an immedia e consequence o he p e ious lemma, we ha e he ollowing esul . Co olla y 3. Unde he assump ions o Lemma 4, i holds: (a) Fo any ˜ h∈[0, h], he p ocess U:R2 d×C˜ h,V H→C˜ h,V His pullback D˜ h,V η(CH)- asymp o ically compac . (b) The p ocess S:R2 d×M2 V→M2 Vis pullback DV η(M2 H)-asymp o ically compac . We es ablish now he ollowing esul abou he exis ence o minimal pullback a ac o s o he p ocess Uon C˜ h,V H, which can be p o ed in a same way as [9, Theo em 5.1]. Theo em 5. Assume ha ∈L2 loc(R; (L2(Ω))2), and ha and g:R×CH→ (L2(Ω))2sa is y (I)–(V). Then, o any ˜ h∈[0, h], he p ocess Uon C˜ h,V Hpossesses a minimal pullback D˜ h,V η(CH)-a ac o AD˜ h,V η(CH), a minimal pullback D˜ h,V F(CH)- a ac o AD˜ h,V F(CH), and a minimal pullback DF(C˜ h,V H)-a ac o ADF(C˜ h,V H). Be- sides, he ollowing ela ions hold: ADF(C˜ h,V H)( )⊂ AD˜ h,V F(CH)( ) ⊂ ADF(CH)( ) ⊂ AD˜ h,V η(CH)( ) = ADη(CH)( ) ⊂CV∀ ∈R,(21) and o any amily b D∈ Dη(CH), lim τ→−∞ dis CV(U( , τ)D(τ),ADη(CH)( )) = 0 ∀ ∈R. Finally, i mo eo e sa is ies sup s≤0e−ηs Zs −∞ eηθ| (θ)|2dθ<∞,(22) hen all a ac o s in (21) coincide, and his amily is empe ed in CV, in he sense ha lim →−∞ eη sup ∈ADη(CH)( ) k k2 CV= 0, whe e k kCV= maxs∈[−h,0] k (s)k o any ∈CV. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 17 Rema k 13. Obse e ha , unde he assump ions o Theo em 5, one has ha AD˜ h,V η(CH)≡ ADh,V η(CH) o any ˜ h∈[0, h], i.e., he pullback a ac o AD˜ h,V η(CH)is independen o ˜ h. Ac ually, i also sa is ies (22), hen AD˜ h,V F(CH)≡ ADh,V F(CH), and ADF(C˜ h,V H)≡ ADF(Ch,V H). Rema k 14. Unde he assump ions o Theo em 5, since b D1,η,h belongs o Dh,V η(CH), and he se D1,η,h( ) is closed in Ch,V H o all ∈R, om Rema ks 4and 10, and he equali y in (21), we deduce ha ADη(CH)belongs o Dh,V η(CH). In ac , i in addi ion sa is ies (22), hen, o each T∈R, he se {ADη(CH)( ) : ≤T}is bounded in Ch,V H. We a e also able o ob ain he exis ence o minimal pullback a ac o s o he p ocess Son M2 V. Theo em 6. Suppose ha ∈L2 loc(R; (L2(Ω))2), and ha and g:R×CH→ (L2(Ω))2sa is y (I)–(V). Then, he e exis he minimal pullback DF(M2 V)-a ac o ADF(M2 V), and he minimal pullback DV η(M2 H)-a ac o ADV η(M2 H) o he p ocess S on M2 V, and he ollowing ela ions hold: ADF(M2 V)( )⊂ ADF(M2 H)( )⊂ ADη(M2 H)( ) = ADV η(M2 H)( )∀ ∈R.(23) In pa icula , o any amily b D∈ Dη(M2 H), lim τ→−∞ dis M2 V(S( , τ)D(τ),ADη(M2 H)( )) = 0 ∀ ∈R.(24) Finally, i also sa is ies (22), hen ADF(M2 V)( ) = ADF(M2 H)( ) = ADη(M2 H)( ) = ADV η(M2 H)( )∀ ∈R, and his amily is empe ed in M2 V, i.e., lim →−∞ eη sup (w,ϕ)∈ADη(M2 H)( ) k(w, ϕ)k2 M2 V= 0.(25) P oo . The exis ence o ADF(M2 V)and ADV η(M2 H)is a di ec consequence o Theo em 2, Co olla y 1, P oposi ion 2, P oposi ion 3, and Co olla y 3. In (23), he inclusions ollow om Co olla y 1, Theo em 3, and Rema k 11. The equali y holds by applying Theo em 3and Rema k 6, using Theo em 1, Lemma 2, Rema k 11, and Co olla y 3. The pullback a ac ion esul (24) comes om Rema k 8, Lemma 2, and he ac ha by he egula i y p ope y (a) in Theo em 1, o any b D∈ Dη(M2 H) and any τ < −h−1, dis M2 V(S( , τ)D(τ),ADη(M2 H)( )) = dis M2 V(S( , τ +h+ 1)(S(τ+h+ 1, τ)D(τ)),ADη(M2 H)( )) = dis M2 V(S( , τ +h+ 1)(j(e U(τ+h+ 1, τ)D(τ))),ADη(M2 H)( )) = dis M2 V(S( , τ +h+ 1)(j(D(h+1)(τ))),ADV η(M2 H)( )), since i is clea ha he amily {j(D(h+1)(τ)) : τ∈R}belongs o DV η(M2 H). I mo eo e sa is ies (22), he equali y ADF(M2 H)( ) = ADη(M2 H)( ) ollows om Rema k 5, and he equali y ADF(M2 V)( ) = ADF(M2 H)( ) is again a consequence o 18 J. GARC´ IA-LUENGO, P. MAR´ IN-RUBIO, AND J. REAL Theo em 3, by using he second es ima e in (13), Rema k 11, and Co olla y 3, since (22) is equi alen o sup s≤0Zs s−1 | (θ)|2dθ < ∞.(26) Las ly, he empe ed p ope y (25) comes om (22) (and he e o e (26)) and he empe ed cha ac e o ρ2( ) de ined in Lemma 3. Rema k 15. Unde he assump ions o Theo em 6, easoning analogously as in Rema k 14, one has ha ADη(M2 H)belongs o DV η(M2 H). To conclude, we ela e he minimal pullback a ac o s ob ained in C˜ h,V Hand M2 V h ough he canonical injec ion j. Theo em 7. Assume ha ∈L2 loc(R; (L2(Ω))2), and ha and g:R×CH→ (L2(Ω))2sa is y (I)–(V). Then, he ollowing ela ions hold: j(ADF(Ch,V H)( )) ⊂ ADF(M2 V)( )∀ ∈R,and (27) j(AD˜ h,V η(CH)( )) = ADV η(M2 H)( )∀˜ h∈[0, h], ∈R.(28) Ac ually, i also sa is ies (22), hen, o any ˜ h∈[0, h], j(ADF(C˜ h,V H)( )) = j(AD˜ h,V F(CH)( )) = ADF(M2 V)( )∀ ∈R.(29) P oo . In o de o p o e he inclusion in (27) we p oceed simila ly as in [19, Theo em 5], aking in o accoun ha he map jis con inuous om Ch,V Hin o M2 V, and ha j(DF(Ch,V H)) ⊂ DF(M2 V). The equali y in (28) is a consequence o p ope y (11) in Theo em 4, using he equali ies (21) and (23). Finally, he equali ies in (29) ollow om (28) and he known ac s ha , unde he addi ional assump ion (22), all a ac o s in (21) and (23) coincide. Acknowledgmen s. While inishing his pape among o he p ojec s, ou coau- ho P o . Jos´e Real deceased. This wo k is dedica ed o his memo y, wi h ou deepes and mos since e admi a ion, g a i ude, and lo e. He was P.M.-R. and J.G.-L.’s PhD-ad iso , cle e ma hema ician wi h a sha p iew on p oblems, gen- e ous and wonde ul colleague, and be e iend. He passed away oo soon, being only 60 yea s old. We miss him deeply, bu he will s ay o e e in ou hea s. This wo k has been pa ially suppo ed by Minis e io de Ciencia e Inno aci´on (Spain) unde p ojec MTM2011-22411. J.G.-L. is a ellow o P og ama de FPU del Minis e io de Educaci´on (Spain). The au ho s hank one o he e e ees by his/he commen s, which led o im- p o emen s in he p esen a ion o his pape . REFERENCES [1] (MR2191992) T. Ca aballo, G. Lukaszewicz, and J. Real, Pullback a ac o s o asymp o i- cally compac non-au onomous dynamical sys ems, Nonlinea Anal. 64 (2006), 484–498. [2] (MR2196010) T. Ca aballo, G. Lukaszewicz, and J. Real, Pullback a ac o s o non- au onomous 2D-Na ie -S okes equa ions in some unbounded domains, C. R. Ma h. Acad. Sci. Pa is 342 (2006), 263–268. [3] (MR1862662) T. Ca aballo and J. Real, Na ie -S okes equa ions wi h delays, R. Soc. Lond. P oc. Se . A Ma h. Phys. Eng. Sci. 457 (2001), 2441–2453. REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 19 [4] (MR2027360) T. Ca aballo and J. Real, Asymp o ic beha iou o wo-dimensional Na ie - S okes equa ions wi h delays, R. Soc. Lond. P oc. Se . A Ma h. Phys. Eng. Sci. 459 (2003), 3181–3194. [5] (MR2091818) T. Ca aballo and J. Real, A ac o s o 2D-Na ie -S okes models wi h delays, J. Di e en ial Equa ions 205 (2004), 271–297. [6] (MR0138894) L. Ca ab iga, Su un p oblema al con o no ela i o al sis ema di equazioni di S okes, Rend. Sem. Ma . Uni . Pado a 31 (1961), 308–340. [7] (MR2810726) J. Ga c´ıa-Luengo, P. Ma ´ın-Rubio, and J. Real, H2-boundedness o he pullback a ac o s o non-au onomous 2D Na ie -S okes equa ions in bounded domains, Nonlinea Anal. 74 (2011), 4882–4887. [8] (MR2881040) J. Ga c´ıa-Luengo, P. Ma ´ın-Rubio, and J. Real, Pullback a ac o s in V o non-au onomous 2D-Na ie -S okes equa ions and hei empe ed beha iou , J. Di e en ial Equa ions 252 (2012), 4333–4356. [9] J. Ga c´ıa-Luengo, P. Ma ´ın-Rubio, and J. Real, Pullback a ac o s o 2D Na ie -S okes equa ions wi h delays and hei egula i y, Ad . Nonlinea S ud. To appea . [10] (MR2196812) M. J. Ga ido-A ienza and P. Ma ´ın-Rubio, Na ie -S okes equa ions wi h delays on unbounded domains, Nonlinea Anal. 64 (2006), 1100–1118. [11] (MR2799549) S. M. Guzzo and G. Planas, On a class o h ee dimensional Na ie -S okes equa ions wi h bounded delay, Disc e e Con in. Dyn. Sys . Se . B 16 (2011), 225–238. [12] (MR0259693) J. L. Lions, Quelques M´e hodes de R´esolu ion des P obl`emes aux Limi es Non Lin´eai es, Dunod, 1969. [13] (MR0771394) A. Z. Mani ius, Feedback con olle s o a wind unnel model in ol ing a delay: analy ical design and nume ical simula ion, IEEE T ans. Au oma . Con ol 29 (1984), 1058– 1068. [14] (MR2660876) P. Ma ´ın-Rubio, A. M. M´a quez-Du ´an, and J. Real, Th ee dimensional sys em o globally modi ied Na ie -S okes equa ions wi h in ini e delays, Disc e e Con in. Dyn. Sys . Se . B 14 (2010), 655–673. [15] (MR2882145) P. Ma ´ın-Rubio, A. M. M´a quez-Du ´an, and J. Real, On he con e gence o solu ions o globally modi ied Na ie -S okes equa ions wi h delays o solu ions o Na ie - S okes equa ions wi h delays, Ad . Nonlinea S ud. 11 (2011), 917–927. [16] (MR2825639) P. Ma ´ın-Rubio, A. M. M´a quez-Du ´an, and J. Real, Pullback a ac o s o globally modi ied Na ie -S okes equa ions wi h in ini e delays, Disc e e Con in. Dyn. Sys . 31 (2011), 779–796. [17] (MR2347436) P. Ma ´ın-Rubio and J. Real, A ac o s o 2D-Na ie -S okes equa ions wi h delays on some unbounded domains, Nonlinea Anal. 67 (2007), 2784–2799. [18] (MR2536301) P. Ma ´ın-Rubio and J. Real, On he ela ion be ween wo di e en concep s o pullback a ac o s o non-au onomous dynamical sys ems, Nonlinea Anal. 71 (2009), 3956–3963. [19] (MR2600726) P. Ma ´ın-Rubio and J. Real, Pullback a ac o s o 2D-Na ie -S okes equa ions wi h delays in con inuous and sub-linea ope a o s, Disc e e Con in. Dyn. Sys . 26 (2010), 989–1006. [20] (MR2764398) P. Ma ´ın-Rubio, J. Real, and J. Vale o, Pullback a ac o s o a wo- dimensional Na ie -S okes model in an in ini e delay case, Nonlinea Anal. 74 (2011), 2012– 2030. [21] (MR2399458) G. Planas and E. He n´andez, Asymp o ic beha iou o wo-dimensional ime- delayed Na ie -S okes equa ions, Disc e e Con in. Dyn. Sys . 21 (2008), 1245–1258. [22] (MR1881888) J. C. Robinson, In ini e-Dimensional Dynamical Sys ems, Camb idge Uni e - si y P ess, 2001. [23] (MR1491614) R. Rosa, The global a ac o o he 2D Na ie -S okes low on some unbounded domains, Nonlinea Anal. 32 (1998), 71–85. [24] (MR0953967) R. Temam, In ini e-Dimensional Dynamical Sys ems in Mechanics and Physics, Sp inge , New Yo k, 1988. [25] (MR0603444) R. Temam, Na ie -S okes equa ions, Theo y and Nume ical Analysis, 2nd. ed., No h Holland, Ams e dam, 1979. Recei ed Ap il 2011; e ised July 2012. E-mail add ess, J. Ga c´ıa-Luengo: [email p o ec ed] E-mail add ess, P. Ma ´ın-Rubio: [email p o ec ed] E-mail add ess, J. Real: [email p o ec ed]