Navier-Stokes equations with delays on unbounded domains

Na ie -S okes equa ions wi h delays on unbounded domains By Ma ´ ıa Jos´ e Ga ido-A ienza, & Ped o Ma ´ ın-Rubio † 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 Some esul s on he exis ence and uniqueness o solu ions o Na ie -S okes equa- ions when he domain is unbounded and he ex e nal o ce con ains some he edi- a y cha ac e is ics a e p o ed o bo h he e olu iona y and he s a iona y cases. Exponen ial s abili y o he s a iona y solu ion is also es ablished in dimension wo. Keywo ds: Na ie -S okes equa ions, a iable and dis ibu ed delays, unbounded domains. 1. In oduc ion and s a emen o he p oblem The Na ie -S okes equa ions go e n he mo ion o usual luids like wa e , ai , oil, e c. These equa ions ha e been he objec o nume ous wo ks since he i s pape o Le ay was published in 1933 (see Cons an in & Foias 1988; Lions 1969; Temam 1979, and he e e ences he ein), e en wi h unbounded domains, allowing he possibili y o channel and mul i-channel lows among o he a ia ions (see o ins ance Rosa 1998; Temam 1979). On o he hand, delay e ec s ha e been p o ed o be use ul in many physical and biological si ua ions. These si ua ions may appea when we wan o con ol he sys em (in a ce ain sense) by applying a o ce which akes in o accoun no only he p esen s a e o he sys em bu he his o y o he solu ion. To ou knowledge, his has been a ely ea ed in he con ex o Na ie -S okes equa ions (c . Ca aballo & Real 2001, 2003, 2004). To da e, we ha e no ound in he li e a u e any wo k ha conside s he combina ion o delay e ms and unbounded domains. The aim o he pape is wo- old: i s ly we conside se e al si ua ions in which he ex e nal o ce con ains some he edi a y ea u es and he domain is no bounded, and p o e exis ence ( o dimension N= 2 and 3) and uniqueness (N= 2) o solu ions. In a second pa o he pape he exis ence and uniqueness o a s a iona y solu ion a e es ablished in dimensions 2 and 3,and, in he case N= 2 exponen ial s abili y o he solu ion unde an addi ional assump ion. Le Ω ⊂RN(N= 2 o 3) be an open se wi h bounda y Γ ha is no necessa ily bounded bu sa is ies a Poinca ´e inequali y: The e exis s λ1>0 such ha ZΩ|φ|2dx≤1 λ1ZΩ|∇φ|2dx, ∀φ∈H1 0(Ω).(1.1) †E-mails: [email p o ec ed] ; [email p o ec ed] A icle submi ed o Nonlinea Analysis TMA T EX Pape 2M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio Conside he ollowing unc ional Na ie -S okes p oblem ( o u he de ails and no a ions see Lions 1969 and Temam 1979):                ∂u ∂ −ν∆u+PN i=1 ui ∂u ∂xi = ( )−∇p+g( , u ) in (0, T)×Ω, di u= 0 in (0, T)×Ω, u= 0 on (0, T)×Γ, u(0, x) = u0(x), x ∈Ω, u( , x) = φ( , x), ∈(−h, 0) x∈Ω, whe e we assume ha T > 0 is gi en, ν > 0 is he kinema ic iscosi y, uis he eloci y ield o he luid, p he p essu e, u0 he ini ial eloci y ield, a non- delayed ex e nal o ce ield, gano he ex e nal o ce con aining some he edi a y cha ac e is ic and φ he ini ial da um in he in e al o ime (−h, 0),whe e his a posi i e ixed numbe . To s a , we conside he ollowing usual abs ac spaces: V=nu∈(C∞ 0(Ω))N: di u= 0o, H= he closu e o Vin (L2(Ω))Nwi h he no m |·|,and inne p oduc (·,·) whe e o u, ∈(L2(Ω))N, (u, ) = N X j=1 ZΩ uj(x) j(x)dx, V= he closu e o Vin (H1 0(Ω))Nwi h he no m [ hanks o (1.1)] k·k associa ed o he inne p oduc ((·,·)),whe e o u, ∈(H1 0(Ω))N, ((u, )) = N X i,j=1 ZΩ ∂uj ∂xi ∂ j ∂xi dx. I ollows ha V⊂H≡H0⊂V0,whe e he injec ions a e dense and con inuous. We will use k·k∗ o he no m in V0and h·,·i o he duali y hV0, V i.Now we deno e a(u, ) = ((u, )), and de ine he ilinea o m bon V×V×Vby b(u, , w) = N X i,j=1 ZΩ ui ∂ j ∂xi wjdx∀u, , w ∈V. Le Xbe a Banach space. Gi en a unc ion u: (−h, T)→X, o each ∈(0, T) we deno e by u he unc ion de ined on (−h, 0) by he ela ion u (s) = u( +s), s ∈ (−h, 0). In o de o s a e he p oblem in he co ec amewo k, le us i s es ablish sui able assump ions on he e m in which he delay is p esen . In a gene al way, le Xand Ybe wo sepa able Banach spaces, and g: [0, T ]× C0([−h, 0]; X)→Ysuch ha (I) o all ξ∈C0([−h, 0]; X), he mapping ∈[0, T]→g( , ξ)∈Yis measu - able, A icle submi ed o Nonlinea Analysis TMA Na ie -S okes equa ions wi h delays on unbounded domains 3 (II) o each ∈[0, T], g( , 0) = 0, (III) he e exis s Lg>0 such ha ∀ ∈[0, T],∀ξ, η ∈C0([−h, 0]; X) kg( , ξ)−g( , η)kY≤Lgkξ−ηkC0([−h,0];X), (IV) he e exis s Cg>0 such ha ∀ ∈[0, T],∀u, ∈C0([−h, T]; X) Z 0kg(s, us)−g(s, s)k2 Yds≤CgZ −hku(s)− (s)k2 Xds. Obse e ha (I)-(III) imply ha gi en u∈C0([−h, T]; X), he unc ion gu: ∈ [0, T]→Yde ined by gu( ) = g( , u )∀ ∈[0, T], is measu able (see Bensoussan e al. 1992) and, in ac , belongs o L∞(0, T;Y). Then, hanks o (IV), he mapping G:u∈C0([−h, T]; X)→gu∈L2(0, T;Y) has a unique ex ension o a mapping e Gwhich is uni o mly con inuous om L2(−h, T;X) in o L2(0, T;Y). F om now on, we will deno e g( , u ) = e G(u)( ) o each u∈ L2(−h, T;X), and hus, ∀ ∈[0, T],∀u, ∈L2(−h, T;X),we will ha e Z 0kg(s, us)−g(s, s)k2 Yds≤CgZ −hku(s)− (s)k2 Xds. Wi h he con en ion abo e, assume ha u0∈H,φ∈L2(−h, 0; V) , ∈L2(0, T ;V0), and he ollowing delay ope a o s: g1: [0, T]×C0([−h, 0]; V)→(L2(Ω))N sa is ying hypo heses (I)-(IV) wi h X=V,Y= (L2(Ω))N,Lg1=L1and Cg1=C1, and g2: [0, T]×C0([−h, 0]; V)→V0 sa is ying hypo heses (I)-(IV) wi h X=V,Y=V0,Lg2=L2and Cg2=C2. We a e in e es ed in he ollowing p oblem:          To ind u∈L2(−h, T;V)∩L∞(0, T;H) such ha , o all ∈V, d d (u( ), ) + νa(u( ), ) + b(u( ), u( ), ) = h ( ), i+ (g1( , u ), ) +hg2( , u ), i, u(0) = u0, u( ) = φ( ), ∈(−h, 0), (1.2) whe e he equa ion in (1.2) mus be unde s ood in he sense o D0(0, T ). Rema k 1.1. Obse e ha he e ms in (1.2) a e well de ined. In pa icula , by hypo heses (I)-(IV), i u∈L2(−h, T;V) he e m g1( , u )de ines a unc ion in L2(0, T; (L2(Ω))N), and he e m g2( , u )de ines a unc ion in L2(0, T;V0). Thus (see Lions 1969), i u∈L2(−h, T;V)∩L∞(0, T;H)sa is ies he equa ion in (1.2),u is weakly con inuous om [0, T ]in o H, and he e o e he ini ial condi ion u(0) = u0 makes sense. O cou se, o N= 2, i he e exis s a solu ion u o he p oblem (1.2), i hen belongs o he space C0([0, T ]; H). In Sec ion 2 we shall p o e exis ence o solu ions o (1.2) and he uniqueness o solu ion o he p oblem in he case N= 2.In Sec ion 3, gene al si ua ions con aining delayed e ms – a iable and dis ibu ed– a e conside ed. We inish wi h he s udy o exis ence and uniqueness o a s a iona y solu ion and i s exponen ial s abili y ( o N= 2) in Sec ion 4. A icle submi ed o Nonlinea Analysis TMA 4M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio 2. Exis ence o solu ions In his sec ion we will p o e a gene al heo em on he exis ence o solu ions when N= 2 o 3, and uniqueness i N= 2. In he p oo o exis ence we will need he ollowing wo esul s: Theo em 2.1. (c . [Ca aballo & Real 2001, Theo em A.1]) Le u0∈Rm,φ∈ L2(−h, 0; Rm), k ∈L2(0, T ;Rm),g: [0, T]×C0([−h, 0]; Rm)→Rmsa is ying hypo heses (I)-(IV) wi h X=Y=Rm, and : [0, T]×Rm→Rma con inuous unc ion such ha ( , 0) = 0 and o all n > 0 he e exis s Ln>0such ha | ( , u)− ( , )|Rm≤Ln|u− |Rm,∀|u|Rm≤n, | |Rm≤n, ∀ ∈[0, T ]. Then: a) Fo each ∗∈(0, T ] he e exis s a mos one solu ion o he p oblem        To ind u∈L2(−h, ∗;Rm)∩C0([0, ∗]; Rm)such ha u( ) = φ( ), ∈(−h, 0), u( ) = u0+Z 0 (s, u(s)) ds+Z 0 g(s, us) ds+Z 0 k(s) ds∀ ∈[0, ∗]. (2.1) b) The e exis s ∗∈(0, T]such ha he e exis s one (and only one)solu ion o he p oblem (2.1). c) Suppose ha he e exis s a cons an C > 0such ha i ∗∈(0, T ]is such ha he e is a solu ion uo (2.1), hen max ∈[0, ∗]|u( )|Rm≤C. Then, unde his addi ional assump ion, he e exis s a solu ion o p oblem (2.1) wi h ∗=T. Theo em 2.2. (c . [Simon 2003, Co olla y 2.34]) Le Θbe a bounded open se o Rd,and X⊂EBanach spaces wi h compac injec ion. Conside 1≤ < q ≤ ∞. Suppose F⊂L (Θ; E)sa is ies (i) ∀ω⊂⊂ Θ,sup ∈Fkτh − kL (ω;E)→0when h→0 [whe e τh is he ansla ion: (τh )(x) = (x+h)], (ii) Fis bounded in Lq(Θ; E)∩L1(Θ; X). Then Fis p ecompac in L (Θ; E). The main esul in his sec ion needs ex a no a ion o an addi ional condi ion (V), which will be discussed in some de ail below in Rema k 2.5. Deno e V(O) he same space as Vbu wi h an open se Oins ead o Ω,and analogously de ine V(O) he closu e o V(O) in (H1 0(Ω))N. Theo em 2.3. Le u0∈H,φ∈L2(−h, 0; V), ∈L2(0, T;V0), and assume ha g1: [0, T]×C0([−h, 0]; V)→(L2(Ω))Nand g2: [0, T]×C0([−h, 0]; V)→V0sa is y hypo heses (I)-(IV) in hei co esponding spaces. Then: a) I N= 2 and ν2> C2, he e exis s a mos one solu ion o p oblem (1.2). b) I N∈ {2,3}and ν2> C2, he e exis s a solu ion o (1.2) i , in addi ion, he ollowing assump ion (V) holds: (V) I mcon e ges weakly o in L2(−h, T;V),weakly-s a in L∞(0, T;H), and s ongly in L2(−h, T; (L2(O))N) o a bounded open se O ⊂ Ωwi h smoo h bounda y, hen gi(·, m ·)con e ges weakly o gi(·, ·)in L2(0, T ;V(O)0) o i= 1,2. A icle submi ed o Nonlinea Analysis TMA Na ie -S okes equa ions wi h delays on unbounded domains 5 P oo . a) Uniqueness o N= 2 ollows as he case wi h bounded domain gi en in Theo em 2.1 in [Ca aballo & Real 2001] and i is ep oduced he e o he sake o comple eness. I ν2> C2, le u, be wo solu ions o (1.2) and se w=u− . Then, om he ene gy equali y, and he bound o he ilinea o m (see Ladyzhenskaya 1992), i ollows ha o all ∈(0, T) |w( )|2+ 2νZ 0kw(s)k2ds=−2Z 0 b(w(s), u(s), w(s)) ds +2 Z 0 (g1(s, us)−g1(s, s), w(s)) ds +2 Z 0hg2(s, us)−g2(s, s), w(s)ids ≤21/2Z 0|w(s)|kw(s)kku(s)kds +2 Z 0|g1(s, us)−g1(s, s)||w(s)|ds +2 Z 0kg2(s, us)−g2(s, s)k∗||w(s)|| ds. Then, om assump ion (IV), aking in o accoun ha w(s) = 0 o s∈(−h, 0), and deno ing 2ε=ν−√C2>0, we ha e o all ∈(0, T ) |w( )|2+ 2νZ 0kw(s)k2ds≤1 2εZ 0|w(s)|2ku(s)k2ds+εZ 0kw(s)k2ds +C1 εZ 0|w(s)|2ds+εZ 0kw(s)k2ds +2pC2Z 0kw(s)k2ds, and so, |w( )|2+ 2εZ 0kw(s)k2ds≤1 2εZ 0|w(s)|2ku(s)k2ds+C1 εZ 0|w(s)|2ds, om which uniqueness ollows hanks o he G onwall lemma: indeed, deno ing C= max(2−1, C1)/ε, we ha e ha d d µ|w( )|2exp ½−CZ 0 (ku(s)k2+ 1)ds¾¶≤0. b) Now o he exis ence, we assume N∈ {2,3},ν2> C2and ha condi ion (V) holds. We will de elop a p oo based on he unbounded case wi hou delays (see o ins ance Temam 1979), and on he case wi h delays on bounded domains (Ca aballo & Real 2001), bu wi h bo h di icul ies ea ed join ly. A icle submi ed o Nonlinea Analysis TMA 6M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio Conside an o hono mal basis B={w1, . . . , wn, . . .} ⊂ V o Hsuch ha linea combina ions o elemen s o Ba e dense in V.†[No ice one usually akes a special basis, using he S okes ope a o . This is no alid he e since compac ness is los , and i will ha e in luence on he way one can ob ain a con e gen subsequence, because s anda d es ima es on he de i a i es o he eloci y ields a e no alid nei he .] Le us deno e Vm=span[w1, . . . , wm], PH Vm:H→Vm he p ojec o gi en by PH Vmu=Pm j=1(u, wj)wj.We will also deno e PV Vm:V→Vm he p ojec o gi en by PV Vmu=Pm j=1(( , ˜wj)) ˜wj(whe e he sequence {˜w1, . . . , ˜wn}comes om he G am-Schmid o hono maliza ion p ocess in V; his will be use ul since he ini ial con e gence in (−h, 0) mus hold in V). Finally, de ine um( ) = Pm j=1 γmj( )wj, whe e            um∈L2(−h, T;Vm)∩C0([0, T]; Vm) d d (um( ), wj) + νa(um( ), wj) + b(um( ), um( ), wj) = h ( ), wji+ + (g1( , um ), wj) + hg2( , um ), wjiin D0(0, T),1≤j≤m, um(0) = PH Vmu0, um( ) = PV Vmφ( ), ∈(−h, 0). (2.2) The p eceding is a sys em o o dina y unc ional di e en ial equa ions in he unknown γm( ) = (γm1( ), ..., γmm( )). Exis ence and uniqueness o solu ion is ob- ained by applying Theo em 2.1 s a ed abo e. Obse e ha p oblem (2.2) has one solu ion de ined in an in e al [0, ∗] wi h 0< ∗≤T. Howe e , as usual, i can be deduced by he a p io i es ima es below, we can se ∗=T. In ac , mul iplying in (2.2) by γmj( ) and summing in j, we ge o all ∈[0, ∗] |um( )|2+ 2νZ 0kum(s)k2ds≤ |u0|2+ 2 Z 0h (s), um(s)ids +2 Z 0 (g1(s, um s), um(s)) ds +2 Z 0hg2(s, um s), um(s)ids, and a guing in a simila manne as in he p oo o uniqueness in he 2-dimensional case, we easily ge wo cons an s (depending on φ, ν, , g1, g2, h, T, bu no on m no ∗)K1and K2such ha sup ∈[0, ∗]|um( )|2≤K1,Z ∗ 0kum(s)k2ds≤K2.(2.3) So we can ake ∗=T, and ob ain ha {um}is bounded in L2(0, T;V)∩L∞(0, T;H), so he e exis s a subsequence, elabelled he same, such ha um* u in L2(0, T;V) weakly and in L∞(0, T;H) weak-s a as m→ ∞.(2.4) †This can be ob ained as ollows: Vis sepa able (since i is a subse o (H1 0(Ω))N), and by de ini ion Vis dense in Vand H, which implies ha Vis also dense in H. Thus, gi en a sequence { i}i≥1⊂Vdense in V, we may ake a sequence {wi n}i,n≥1⊂ V which accumula es o e e y poin i,and he e o e, linea combina ions o hese elemen s a e dense in Vand H. Since V⊂Ha e ec o ial subspaces o (L2(Ω))N,linea (in)dependence is equi alen conside ed in any o hem, whence Bis ob ained applying he G am-Schmid o hono maliza ion p ocess. A icle submi ed o Nonlinea Analysis TMA Na ie -S okes equa ions wi h delays on unbounded domains 7 Mo eo e , obse e ha um=PV Vmφin (−h, 0) con e ges o φin L2(−h, 0; V), and, in pa icula , hanks o (IV), g1(·, um) + g2(·, um) is bounded in L2(0, T;V0). As we no iced be o e, when i is possible o choose a special basis (in bounded do- mains i is so), s anda d es ima es on kd d umkL2(0,T ;V0)( o N= 2,and L4/3(0, T;V0) o N= 3) allow us o ob ain a compac ness esul : a subsequence umcon e ges o uin L2(0, T;H). He e we will ha e a simila esul bu no in a s aigh o wa d way, no on he whole domain Ω.Ac ually, wha holds in his case is he ollowing: Fo any bounded open se O ⊂ Ω he e exis s a subsequence (depending on Owhich we elabel) sa is ying um|O→u|Oin L2(0, T; (L2(O))N).(2.5) Fo he sake o cla i y, we pos pone he p oo (we will use Theo em 2.2) o Lemma 2.4 below. Now, le ψbe a con inuously di e en iable unc ion on [0, T ] wi h ψ(T) = 0.Conside equa ion (2.2) and a ixed elemen wjo B. Since (um(·), wj)ψ(·)∈ W1,1(0, T) (ac ually in H1(0, T) o N= 2,and W1,4/3(0, T) o N= 3) we ha e −ZT 0 (um( ), ψ0( )wj)d +νZT 0 ((um( ), wjψ( )))d +ZT 0 b(um( ), um( ), wjψ( ))d = (um(0), wj)ψ(0) + ZT 0h ( ), wjψ( )id +ZT 0 (g1( , um ), wjψ( ))d +ZT 0hg2( , um ), wjψ( )id . Taking a diagonal subsequence, deno ed again um, ha sa is ies (2.5) o a sequence o egula bounded open se s Oj⊂Ω ha con ain all suppo s o unc ions wjo he basis, we may now pass o he limi , hanks o he weak con e gence in (2.4) and condi ion (V) oo. Thus, we ob ain ( i s o any w∈ {w1, w2, . . .},and by densi y o e e y w∈V): −ZT 0 (u( ), ψ0( )w)d +νZT 0 ((u( ), wψ( )))d +ZT 0 b(u( ), u( ), wψ( ))d = (u0, w)ψ(0) + ZT 0h ( ), wψ( )id +ZT 0 (g1( , u ), wψ( ))d +ZT 0hg2( , u ), wψ( )id . (2.6) W i ing (2.6) o ψ∈ D(0, T ), u sa is ies (2.2) in he dis ibu ion sense. By Rema k 1.1 i makes sense o wonde abou he alue a ime = 0.Now, since (u( ), wj)ψ( )∈W1,4/3(0, T), o bo h N= 2 o 3,a guing as be o e, we ob- ain an analogous exp ession o (2.6) wi h (u(0), w) ins ead o (u0, w).This implies (u(0) −u0, w) = 0 o all w∈V, so u(0) = u0. A icle submi ed o Nonlinea Analysis TMA 8M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio Fo he ollowing esul , le us obse e ha he cons an s appea ing in s an- da d es ima es o he ilinea o m b– o a p oo see o ins ance Temam 1979, lemmas 3.3, 3.4, 3.5 and Theo em 3.3, pp. 291 and o wa d– may be imp o ed (c . Ladyzhenskaya 1992 o he case N= 2). Al hough i is no essen ial o make he mos o hem o exis ence and uniqueness, since we will es ablish a s abili y esul la e , we use hem in o de o ensu e i unde he minimal condi ions. Lemma 2.4. Unde he assump ions o Theo em 2.3, he sequence umgi en in (2.2) is p ecompac in he ollowing sense: suppose a bounded open se O ⊂ Ωis gi en, hen he e exis s a subsequence depending on O,which we elabel, such ha um|O→u|Oin L2(0, T; (L2(O))N), whe e uis he weak limi gi en in (2.4). P oo . We will adap he p oo o Theo em 9.4 [c . Simon 2003] o check ou si ua- ion i s o he Theo em 2.2. Mo e exac ly, we claim i can be applied aking = 2, q= +∞,Θ = (0, T). Fo he se O ⊂ Ω le us make p ecise a echnical de ail: i O ⊂⊂ Ω one may ob ain a ini e eco e ing o balls, deno ed ˜ O ⊂ Ω,which is bounded and open, and hen X= (H1(˜ O))N⊂E= (L2(˜ O))Nwi h compac injec ion. Howe e , o a gene al O ⊂ Ω he abo e commen may no be ue since O and Ω can sha e pa o hei bounda ies. The compac injec ion om H1may no hold o lack o egula i y on he bounda y (i is no imposed o Γ), howe e i does in H1 0.One may hen use a unca ion a gumen (see o ins ance Rosa 1998): ix χ∈C1(R+) wi h χ(s) = 1 o s∈[0,1] and χ(s) = 0 o s≥4.Con- side Oas in he s a emen , le R > 0 be such ha O ⊂ B(0, R) and deno e ˜ O= Ω ∩B(0,2R),and um,R(x) = um(x)χ(|x|2/R2).Again he compac ness holds o X= (H1 0(˜ O))N⊂E= (L2(˜ O))Nwi h compac injec ion, and we conse e he o iginal unc ions umon Ω ∩B(0, R). Fo he sake o cla i y, we con inue he p oo di ec ly wi h umins ead o um,R. Since condi ion (ii) in Theo em 2.2 is ob iously sa is ied by (2.3), we concen a e on (i). Ac ually, we will p o e ha o he whole domain Ω he ollowing p ope y holds: sup m∈Nkτhum−umkL2(0,T −h;(L2(Ω))N)→0 when h→0. Conside h > 0 a bi a ily small. F om (2.2) we deduce o ( , +h)⊂(0, T) ha ZΩ (um( +h)−u( ))wjdx+νZ +h ZΩ∇um(s)·∇wjdxds+Z +h b(um(s), um(s), wj)ds =Z +h h (s), wjids+Z +h ZΩ g1(s, um s)wjdxds+Z +h hg2(s, um s), wjids. A icle submi ed o Nonlinea Analysis TMA Na ie -S okes equa ions wi h delays on unbounded domains 9 Mul iplying by γmj( +h)−γmj( ) and summing in jwe ob ain ZΩ|um( +h)−u( )|2dx=−νZ +h ZΩ∇um(s)·(∇um( +h)−∇um( ))dxds +Z +h b(um(s), um(s), um( +h)−um( ))ds +Z +h ZΩ g1(s, um s)(um( +h)−um( ))dxds +Z +h h (s) + g2(s, um s), um( +h)−um( )ids. The igh hand side may be bounded by ν|∇um( +h)−∇um( )|Z +h |∇um(s)|ds +Z +h GN(|um(s)|,kum(s)k,kum( +h)−um( )k) ds +Z +h |g1(s, um s)||um( +h)−um( )|ds +Z +h (k (s)k∗+kg2(s, um s)k∗)kum( +h)−um( )kds whe e he ilinea o m bis bounded (depending on he dimension) by he unc ion GN:R3→Rde ined as GN(x, y, z) = ½2−1/2xyz i N= 2, 2−1x1/2y3/2zi N= 3.(2.7) Thus, using (1.1) and (2.3), we ha e p o ed ha ZΩ|um( +h)−um( )|2dx≤ kum( +h)−um( )kZ +h Gm(s)ds whe e he unc ion Gm:R→Ris de ined ( ecall de ini ion gi en in (2.7)) as Gm(s)=(νkum(s)k+(2−1K1)1/2kum(s)k+k (s)k∗+kg2(s, um s)k∗+λ−1/2 1|g1(s, um s)|i N= 2, νkum(s)k+2−1K1/4 1kum(s)k3/2+k (s)k∗+kg2(s, um s)k∗+λ−1/2 1|g1(s, um s)|i N= 3. To inish he p oo , we will es ima e kτhum−umk2 L2(0,T −h;(L2(Ω))N)=ZT−h 0ZΩ|τhum−um|2dxd ≤ZT−h 0kum( +h)−um( )kZ +h Gm(s)dsd . Fo he igh hand side, he Fubini heo em yields, using he unc ion ¯s=   0 i s≤0, si 0 < s ≤T−h, T−hi s > T −h, A icle submi ed o Nonlinea Analysis TMA 16 M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio Then he e is a unique s a iona y solu ion u∗o (4.2) and e e y solu ion o (1.2) con e ges o u∗exponen ially as as →+∞, ha is, he e exis wo posi i e cons an s Cand λ, such ha o all u0∈Hand φ∈L2(−h, 0; V), he solu ion u o (1.2) wi h ( )≡ sa is ies o all ≥0 : |u( )−u∗|2≤Ce−λ ³|u0−u∗|2+kφ−u∗k2 L2(−h,0;V)´.(4.7) P oo . Conside u he solu ion o (1.2) o ( )≡ , and deno e u∗∈V he s a- iona y solu ion o (4.2), which exis ence and uniqueness is ensu ed by Theo em 4.1. Se w( ) = u( )−u∗, and obse e ha d d (w( ), )+ν((w( ), ))+b(u( ), u( ), )−b(u∗, u∗, ) = (G(u( −ρ( ))), )−(G(u∗), ). Since b(u( ), u( ), w( )) −b(u∗, u∗, w( )) =b(u∗, w( ), u∗)−b(u( ), w( ), u( )) =b(u∗, w( ), u∗)∓b(u( ), w( ), u∗)−b(u( ), w( ), u( )) =−b(w( ), w( ), u∗)−b(u( ), w( ), w( )) =−b(w( ), w( ), u∗), we can ob ain he ollowing es ima ion (he e λand δa e ixed posi i e alues o be de e mined la e on): d d (eλ |w( )|2) = λeλ |w( )|2+ eλ d d |w( )|2 =λeλ |w( )|2+ 2eλ (−νkw( )k2−b(u( ), u( ), w( )) +b(u∗, u∗, w( )) + (G(u( −ρ( ))) −G(u∗), w( ))) ≤eλ ¡λ|w( )|2−2νkw( )k2+ 2b(w( ), w( ), u∗) +2L1|w( −ρ( ))||w( )|) ≤λ−1 1eλ (λ+δL1−2νλ1)kw( )k2+ 2eλ |b(w( ), w( ), u∗)| +(δλ1)−1L1eλ kw( −ρ( ))k2.(4.8) Using again ha |b(w( ), w( ), u∗)| ≤ (2λ1)−1/2kw( )k2ku∗kand aking in o accoun he es ima e we p o ed in S ep 3 o Theo em 4.1 o he s a iona y solu ion, ku∗k ≤ k k∗/(ν−λ−1 1L1),we ha e |b(w( ), w( ), u∗)| ≤ (2λ1)−1/2k k∗ ν−λ−1 1L1kw( )k2. Subs i u ing his las inequali y in o (4.8) i ollows ha d d (eλ |w( )|2)≤λ−1 1eλ µλ+δL1−2νλ1+(2λ1)1/2k k∗ ν−λ−1 1L1¶kw( )k2 +(δλ1)−1L1eλ kw( −ρ( ))k2, A icle submi ed o Nonlinea Analysis TMA Na ie -S okes equa ions wi h delays on unbounded domains 17 and so o all ∈[0, T] eλ |w( )|2≤ |w(0)|2+ (δλ1)−1L1Z 0 eλskw(s−ρ(s))k2ds +λ−1 1µλ+δL1−2νλ1+(2λ1)1/2k k∗ ν−λ−1 1L1¶Z 0 eλskw(s)k2ds. We concen a e momen a ily in he delay e m on he igh hand side. Obse ing ha he unc ion φ( ) := −ρ( ) is s ic ly inc easing, ha ρ akes alues on [0, h], and so φ−1(η)≤η+h, we can apply he change o a iable η=s−ρ(s) = φ(s) : Z 0 eλskw(s−ρ(s))k2ds=Z −ρ( ) −ρ(0) eλτ−1(η)kw(η)k21 1−ρ0(φ−1(η)) dη ≤eλh 1−ρ∗Z −h eληkw(η)k2dη. Combining he abo e wo inequali ies we ob ain eλ |w( )|2≤ |w(0)|2+ (δλ1)−1L1 eλh 1−ρ∗Z −h eλskw(s)k2ds +λ−1 1µλ+δL1−2νλ1+(2λ1)1/2k k∗ ν−λ−1 1L1¶Z 0 eλskw(s)k2ds. Obse e he coe icien s o he in eg al R 0eλskw(s)k2ds. Le us no e ha δ∗= (1 −ρ∗)−1/2is he minimum o he map δ7→ δ+ 1/(δ(1 −ρ∗)).Then, hanks o (4.6) he e exis s λ > 0 small enough such ha λ+δ∗L1−2νλ1+(2λ1)1/2k k∗ ν−λ−1 1L1 +L1eλh δ∗(1 −ρ∗)≤0. Thus, we deduce ha eλ |u( )−u∗|2≤ |u0−u∗|2+λ−1 1L1eλh 1−ρ∗Z0 −h eληkw(η)k2dη, whence (4.7) is sa is ied wi h C= max ½1,λ−1 1L1eλh 1−ρ∗¾. Rema k 4.3. The abo e esul has been gi en wi h g1as in Case 1 o Sec ion 3 o simpli y he no a ion. O he ypes o delay could be used, o example one could conside an au onomous case wi h dis ibu ed delay by emo ing in Case 2 he dependence o Gon i s i s a iable. Conclusions and inal commen s Exis ence, uniqueness and s abili y esul s ha e been es ablished unde di e en condi ions –essen ially iscosi y is asked o be la ge enough–. One may wonde abou esul s unde weake assump ions, whe e uniqueness o s abili y may no be A icle submi ed o Nonlinea Analysis TMA 18 M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio ensu ed. This leads us o conside addi ional concep s om he heo y o dynamical sys ems, namely a ac o s, bo h he classical ( o wa d) one, and he ’pullback’ de - ini ion ha is well-sui ed o non-au onomous sys ems (see Ca aballo & Real 2004 and Kloeden & Schmal uß 1997). Acknowledgemen This wo k was pa ially suppo ed by Minis e io de Ciencia y Tecnolog´ıa (Spain) and FEDER P oyec o BFM2002-03068. The au ho s hanks P o esso Jos´e Real o his ad ice on he subjec o his pape . Re e ences A ola, M. 1969 Su les pe u ba ions des ´equa ions d’e olu ion, applica ion `a des p obl`emes de e a d, Ann. Scien . Ec. No m. Sup. 4s´e ie, . 2, 137-253. Bensoussan, A., Da P a o, G., Del ou , M.C., & Mi e , S.K. 1992 Rep esen a ion and Con ol o In ini e Dimensional Sys ems, Vol. I, Bi kh¨ause , Bos on-Basel-Be lin. Ca aballo, T. & Real, J. 2001 Na ie -S okes equa ions wi h delays, P oc. R. Soc. Lond. A(2001) 457, 2441-2453. Ca aballo, T. & Real, J. 2003 Asymp o ic beha iou o 2D−Na ie -S okes equa ions wi h delays, P oc. R. Soc. Lond. A(2003) 459, 3181-3194. Ca aballo, T. & Real, J. 2004 A ac o s o 2D-Na ie -S okes models wi h delays, J. Di e en ial Equa ions 205, 271-297. Cons an in, P. & Foias, C. 1988 Na ie S okes Equa ions, The Uni e si y o Chicago P ess, Chicago. Hale, J.K. & Ve duyn Lunel, S.M. 1995 In oduc ion o Func ional Di e en ial Equa ions, Sp inge -Ve lag, New Yo k. Kloeden, P. E. & Schmal uß, B. 1997 Nonau onomous sys ems, cocycle a ac o s and a iable ime-s ep disc e iza ion. Dynamical nume ical analysis (A lan a, GA, 1995). Nume . Algo i hms 14, 141-152. Ladyzhenskaya, O. 1992 Fi s bounda y alue p oblem o he Na ie -S okes equa ions in domains wi h nonsmoo h bounda ies. C. R. Acad. Sci. Pa is S´e . I Ma h. 314, 253-258. Le ay, J. 1933 E ude de di e ses ´equa ions in ´eg ales non lin´eai es e de quelques p obl`emes que pose l’hyd odynamique, J. Ma h. Pu es Appl.12, 1-82. Lions, J.L. 1969 Quelques m´e hodes de ´esolu ion des p obl`emes aux limi es non lin´eai es, Dunod, Gau hie - Villa s, Pa is. Rosa, R. 1998 The global a ac o o he 2D Na ie -S okes low on some unbounded domains., Nonlinea Anal. 32, 71-85. Simon, J. 2003 ´ Equa ions de Na ie -S okes, Cou s de DEA 2002-2003, Uni e si ´e Blaise Pascal. h p://wwwlma.uni -bpcle mon . /∼simon/ Temam, R. 1979 Na ie -S okes equa ions, Theo y and Nume ical Analysis, 2nd. ed., No h Holland, Ams e dam. A icle submi ed o Nonlinea Analysis TMA