Three Dimensional System of Globally Modified Navier-Stokes Equations with Infinite Delays

Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume 00, Numbe 0, Xxxx XXXX pp. 000–000 THREE DIMENSIONAL SYSTEM OF GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS WITH INFINITE DELAYS P. Ma ´ ın-Rubio, A. M. M´ a quez-Du ´ an & J. 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 Aim Sciences) Abs ac . Exis ence and uniqueness o solu ion o a globally modi ied e - sion o Na ie -S okes equa ions con aining in ini e delay e ms a e es ablished. Mo eo e , we also analyze he s a iona y p oblem and, unde sui able addi- ional condi ions, we ob ain global exponen ial decay o he solu ions o he e olu iona y p oblem o he s a iona y solu ion. Keywo ds: Globally Modi ied Na ie -S okes Equa ions; in ini e delays. Ma hema ics Subjec Classi ica ions (2000): 35K55, 35Q30, 34D23, 34K20 1. In oduc ion and s a emen o he p oblem. Le Ω ⊂R3be an open bounded se wi h egula bounda y Γ, and le N∈(0,+∞) be ixed. Le us de ine FN: [0,+∞)→(0,1] by FN( ) := min 1,N , ∈[0,+∞), and conside he ollowing sys em o globally modi ied Na ie -S okes equa ions on Ω wi h homogeneous Di ichle bounda y condi ion                ∂u ∂ −ν∆u+FN(kuk) [(u· ∇)u] + ∇p= ( ) in (τ, T∗)×Ω, ∇ · u= 0 in (τ, T∗)×Ω, u= 0 on (τ, T∗)×Γ, u(τ, x) = u0(x), x ∈Ω, (1) whe e ν > 0 is he kinema ic iscosi y, u he eloci y ield o he luid, p he p es- su e, τ∈Ran ini ial ime, u0 he ini ial eloci y ield, ( ) a gi en ex e nal o ce ield, and T∗∈(τ, +∞] a gi en inal ime. The sys em (1) is indeed a globally modi ied e sion o he Na ie -S okes sys- em – he modi ying ac o FN(kuk) depends on he no m kuk=k∇uk(L2(Ω))3×3, 2000 Ma hema ics Subjec Classi ica ion. P ima y: 35K55; 35Q30; 34D23; 34K20. Key wo ds and ph ases. Globally Modi ied Na ie -S okes Equa ions; in ini e delays. Pa ially suppo ed by Minis e io de Ciencia e Inno aci´on (Spain), p ojec MTM2008-00088 and Jun a de Andaluc´ıa, g an P07-FQM-02468. 1 2 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL which in u n depends on ∇uo e he whole domain Ω and no jus a o nea he poin x∈Ω unde conside a ion. Essen ially, i p e en s la ge g adien s domina ing he dynamics and leading o explosions. I iola es he basic laws o mechanics, bu ma hema ically he sys em (1) is a well de ined sys em o equa ions, jus like he modi ied e sions o he Na ie -S okes equa ions o Le ay and o he s wi h o he molli ica ions o he nonlinea e m, see he e iew pape [8]. I is wo h men ioning ha a global cu o unc ion in ol ing he D(A1/4) no m o he wo dimensional s ochas ic Na ie -S okes equa ions is used in [9], and a cu -o unc ion simila o he one we will use he e was conside ed in [23]. The sys em (1) was in oduced and s udied in [1] (see also [2,14,15,16,3] and he e iew pape [13]). Howe e , he e a e si ua ions in which he model is be e desc ibed i some e ms con aining delays appea in he equa ions. These delays may appea , o ins ance, when one wan s o con ol he sys em by applying a o ce which akes in o accoun no only he p esen s a e bu he comple e his o y o he solu ions. The e o e, in his pape we a e in e es ed in he case in which e ms con aining in ini e delays appea . We conside he ollowing e sion:              ∂u ∂ −ν∆u+FN(kuk) [(u· ∇)u] + ∇p= ( ) + g( , u ) in (τ, T∗)×Ω, ∇ · u= 0 in (τ, T∗)×Ω, u= 0 on (τ, T∗)×Γ, u(τ+s, x) = φ(s, x), s ∈(−∞,0], x ∈Ω, (2) whe e gis ano he ex e nal o ce con aining some he edi a y cha ac e is ic and φ is a gi en unc ion de ined in he in e al (−∞,0]. Ou goal is o es ablish he exis ence and uniqueness o solu ion o he abo e p oblem and o s udy i s asymp o ic beha iou ( o a simila goal in Na ie -S okes models wi h ini e delay c . [5,6,7], and o ODEs and PDEs wi h unbounded delay e ms c . [4,18]). The s uc u e o he pape is he ollowing: in Sec ion 2we ecall some spaces use ul o he abs ac amewo k and some p ope ies and es ima es ela ed o he ope a o s in ol ed in he model. In Sec ion 3 he exis ence and uniqueness o solu ion o he (e olu iona y) p oblem is gi en. The s a iona y p oblem is ea ed in Sec ion 4, whe e we also p o e ha unde adequa e assump ions, any solu ion o he e olu iona y p oblem has an exponen ial decay owa d he s a iona y solu ion. 2. P elimina ies. To se ou p oblem in he abs ac amewo k, we conside he ollowing usual abs ac spaces (see [17] and [21,22]): V=nu∈(C∞ 0(Ω))3: di u= 0o, H= he closu e o Vin (L2(Ω))3wi h inne p oduc (·,·) and associa e no m |·| , whe e o u, ∈(L2(Ω))3, (u, ) = 3 X j=1 ZΩ uj(x) j(x)dx, 3D-GMNS EQUATIONS WITH INFINITE DELAYS 3 V= he closu e o Vin (H1 0(Ω))3wi h scala p oduc ((·,·)) and associa e no m k·k ,whe e o u, ∈(H1 0(Ω))3, ((u, )) = 3 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 pai ing be ween V0 and V. Finally, we will iden i y e e y u∈Hwi h he elemen u∈V0gi en by h u, i= (u, )∀ ∈V. I ollows ha V⊂H⊂V0,whe e he injec ions a e dense and compac . We conside he linea con inuous ope a o A:V→V0de ined by hAu, i= ((u, )) ∀u, ∈V. (3) Deno ing D(A) = {u∈V:Au ∈H},wi h inne p oduc (u, )D(A)= (Au, A ), hen, by he egula i y o Γ, D(A) = (H2(Ω))3∩V, and Au =−P∆u, ∀u∈D(A), is he S okes ope a o (Pis he o ho-p ojec o om (L2(Ω))3on o H). Le us deno e λ1= in ∈V {0} k k2 | |2>0, he i s eigen alue o he S okes ope a o . Now we de ine he ilinea o m bon V×V×Vby b(u, , w) = 3 X i,j=1 ZΩ ui ∂ j ∂xi wjdx, ∀u, , w ∈V, and we deno e bN(u, , w) = FN(k k)b(u, , w),∀u, , w ∈V. The o m bNis linea in uand w, bu i is nonlinea in . E iden ly we ha e bN(u, , ) = 0, o all u, ∈V. We will also make use o he ollowing inequali y (see [21] and [10]) |b(u, , w)| ≤ 2−1|u|1/4kuk3/4k k|w|1/4kwk3/4,∀u, , w ∈V. (4) In pa icula , his implies ha he e exis s a cons an C1>0 only dependen on Ω (namely, C1= (2λ1/4 1)−1) such ha |b(u, , w)| ≤ C1kukk kkwk,∀u, , w ∈V. Thus by he de ini ion o FN,i we deno e hBN(u, ), wi=bN(u, , w),∀u, , w ∈V, we ha e kBN(u, )k∗≤NC1kuk,∀u, ∈V. (5) We ecall (see [21]) ha he e exis s a cons an C2>0 depending only on Ω such ha |b(u, , w)| ≤ C2kuk1/2|Au|1/2k k|w|,(6) o all u∈D(A), ∈V, w ∈H, and |b(u, , w)| ≤ C2kukk k|w|1/2kwk1/2,(7) o all u, , w ∈V. (See [20] o he p oo o (7)). 4 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL Le Xbe a Banach space. The no a ion BX(a, ) will be used o deno e he open ball o cen e aand adius in he space X. Gi en a unc ion u: (−∞, T∗)→X, o each <T∗we deno e by u he unc ion de ined on (−∞,0) by he ela ion u (s) = u( +s), s ∈(−∞,0). One possibili y o deal wi h in ini e delays, and which we will use he e (c . [18,11,12]), is o conside , o any γ > 0, he space : Cγ(H) = ϕ∈C((−∞,0]; H) : ∃lim s→−∞ eγsϕ(s)∈H, which is a Banach space wi h he no m kϕkγ:= sup s∈(−∞,0] eγs|ϕ(s)|. In o de o s a e he p oblem in he co ec amewo k, le us i s es ablish some ini ial assump ions on some e ms in he equa ion: We will assume ha ∈L2(τ, T; (L2(Ω))3) o all T∈(τ, T∗]∩R. Fo he e m g, in which he delay is p esen , we assume ha g: (τ, T∗)× Cγ(H)→(L2(Ω))3sa is ies (g1) Fo any ξ∈Cγ(H) he mapping (τ, T∗)3 7→ g( , ξ) is measu able, (g2) g( , 0) = 0 o all ∈(τ, T∗), (g3) he e exis s a cons an Lg>0 such ha o any ∈(τ, T∗) and all ξ, η ∈ Cγ(H), |g( , ξ)−g( , η)| ≤ Lgkξ−ηkγ. Rema k 1. (i) Condi ion (g2) is no eally a es ic ion, since o he wise, i |g(·,0)| ∈ L2(τ, T ) o all T∈(τ, T∗]∩R,we could ede ine ˆ ( ) = ( ) + g( , 0) and ˆg( , ·) = g( , ·)−g( , 0).In his way he p oblem is exac ly he same and ˆ and ˆgsa is y he equi ed assump ions. (ii) Condi ions (g2) and (g3) imply ha |g( , ξ)| ≤ Lgkξkγ, so ha |g(·, ξ)| ∈ L∞(τ, T∗). We will deno e Pm he o hogonal p ojec o o Hon o he ec o space gene a ed by he i s meigen unc ions o he S okes p oblem in Ω wi h homogeneous Di ichle bounda y condi ions. An example o ope a o sa is ying assump ions (g1)-(g3) is gi en he e. Example 1. We conside he ope a o g: (τ, T∗)×Cγ(H)→(L2(Ω))3de ined as ollows: g( , ξ) := Z0 −∞ G( , s, ξ(s))ds ∀ ∈(τ, T∗),∀ξ∈Cγ(H), whe e he unc ion G: (τ, T∗)×(−∞,0) ×R3→R3sa is ies he ollowing assump- ions: (a) G( , s, 0) = 0 o all ( , s)∈(τ, T∗)×(−∞,0). (b) The e exis s a unc ion κ: (−∞,0) →(0,+∞) such ha kG( , s, u)−G( , s, )kR3≤κ(s)ku− kR3 ∀u, ∈R3,∀( , s)∈(τ, T∗),×(−∞,0), 3D-GMNS EQUATIONS WITH INFINITE DELAYS 5 (c) and he unc ion κsa is ies ha κ(·)e−(γ+)·∈L2((−∞,0)) o some  > 0. Namely, he ope a o gde ines an elemen o (L2(Ω))3in he ollowing way: g( , ξ)(x) = Z0 −∞ G( , s, ξ(s)(x))ds ∀x∈Ω. We check now ha gsa is ies he assump ion (g3), and using (a) abo e, we ob ain ha i is well de ined as a map wi h alues in (L2(Ω))3: ZΩZ0 −∞ κ(s)kξ(s)(x)−η(s)(x)kR3ds2 dx =ZΩZ0 −∞ κ(s)e−(γ+)se(γ+)skξ(s)(x)−η(s)(x)kR3ds2 dx ≤ZΩZ0 −∞ κ2(s)e−2(γ+)sdsZ0 −∞ e2(γ+)skξ(s)(x)−η(s)(x)k2 R3dsdx =Z0 −∞ κ2(s)e−2(γ+)sdsZΩZ0 −∞ e2(γ+)skξ(s)(x)−η(s)(x)k2 R3dsdx =CκZ0 −∞ ZΩ e2(γ+)skξ(s)(x)−η(s)(x)k2 R3dxds ≤Cκ"sup s∈(−∞,0] e2γs ZΩ kξ(s)(x)−η(s)(x)k2 R3dx#Z0 −∞ e2sds =Cκkξ−ηk2 γ 1 2 =Lgkξ−ηk2 γ, whe e we ha e deno ed Cκ=kκ(·)e−(γ+)·k2 L2(−∞,0) and Lg=Cκ/(2). 3. Exis ence o solu ions. In his sec ion we es ablish exis ence o solu ion o (2) by a compac ness me hod using a Faedo-Gale kin scheme. De ini ion 1. A weak solu ion o (2) is a unc ion u∈C((−∞, T]; H)∩L2(τ, T ;V) o all T∈(τ, T ∗]∩R, wi h uτ=φand such ha o all ∈V, d d (u( ), ) + ν((u( ), )) + bN(u( ), u( ), )=( ( ), )+(g( , u ), ), in he sense o D0(τ, T ∗). Rema k 2. I uis a solu ion o (2) in he sense gi en abo e, hen usa is ies an ene gy equali y, namely: |u( )|2+ 2νZ s ku( )k2d =|u(s)|2+ 2 Z s [( ( ), u( )) + (g( , u ), u( ))] d ∀s, ∈[τ, T∗]∩R. Fi s , we will p o e he uniqueness o weak solu ions o ou model in a simila way as done in [20] o he model wi hou delay. We will only include he de ailed es ima es which in ol e he delay e m. 6 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL Theo em 1. Unde he abo e assump ions, he e exis s a mos a weak solu ion u o (2). The p oo is simila o, bu a bi mo e complica ed han o he 2D-Na ie -S okes equa ions and depends on he ollowing Lemma 1. ([20]) Fo e e y u, ∈V, and each N > 0, 1. 0 ≤ kukFN(kuk)≤N, 2. |FN(kuk)−FN(k k)| ≤ 1 NFN(kuk)FN(k k)ku− k. P oo o Theo em 1:Le u, be wo weak solu ions wi h he same ini ial condi ions and se w= −u. Then, using he ene gy equali y, we ob ain 1 2 d d |w|2+νkwk2+hNL(u, ), wi= (g( , )−g( , u ), w), ∈(τ, T∗), whe e we ha e se hNL(u, ), wi=FN(kuk)b(u, u, w)−FN(k k)b( , , w).F om he p ope ies o he ilinea o m bi easily ollows ha hNL(u, ), wi=FN(kuk)b(w, u, w)+(FN(kuk)−FN(k k))b( , u, w) +FN(k k)b( , w, w). Now using Lemma 1, o mula (7) and Young’s inequali y (see [20] o he de ails) he e exis s a cons an C3>0,which depends on C2and ν, such ha , | hNL(u, ), wi | ≤ νkwk2+C3N4|w|2. Consequen ly, aking (g3) in o accoun , we ob ain d d |w( )|2≤2C3N4|w( )|2+ 2Lgkw kγ|w( )|, ∈(τ, T∗). Obse e ha w(s) = 0 i s≤τ. The e o e, o ∈(τ, T∗) : kw kγ= sup θ≤0 eγθ|w( +θ)| ≤ sup θ∈[τ− ,0] |w( +θ)|. Thus we ob ain |w( )|2≤2C3N4Z τ |w(s)|2ds + 2LgZ τ sup ∈[τ,s] |w( )||w(s)|ds ≤(2C3N4+ 2Lg)Z τ sup ∈[τ,s] |w( )|2ds, o any ∈[τ, T∗). Now we deduce ha sup ∈[τ, ] |w( )|2≤(2C3N4+ 2Lg)Z τ sup ∈[τ,s] |w( )|2ds, o any ∈[τ, T∗),whence he G onwall lemma inishes he p oo . Ou main esul is he ollowing Theo em 2. Suppose ha ∈L2(τ, T; (L2(Ω))3) o all T∈(τ, T∗]∩R, g : (τ, T∗)×Cγ(H)→(L2(Ω))3sa is ying he assump ions (g1)–(g3), and φ∈Cγ(H) a e gi en, and ha 2γ > νλ1.Then, he e exis s a unique weak solu ion uo (2), which in ac is a s ong solu ion in he sense ha u∈C((τ, T ]; V)∩L2(τ+ε, T;D(A)), 3D-GMNS EQUATIONS WITH INFINITE DELAYS 7 o all 0< ε < T∗−τand any T∈(τ+ε, T∗]∩R. Mo eo e , i φ(0) ∈V, hen usa is ies u∈C([τ, T ]; V)∩L2(τ, T;D(A)), o all T∈(τ, T∗]∩R. P oo . We spli he p oo in se e al s eps. S ep 1: A Gale kin scheme. Le us conside { j} ⊂ V, he o hono mal basis o Ho all he eigen unc ions o he S okes ope a o . Deno e Vm= span[ 1, . . . , m] and conside he p ojec o Pmu=Pm j=1(u, j) j. De ine also um( ) = m X j=1 αm,j( ) j whe e he uppe sc ip mwill be used ins ead o (m) o sho since no con usion is possible wi h powe s o u, and whe e he coe icien s αm,j a e equi ed o sa is y he ollowing sys em: d d (um( ), j) + ν((um( ), j)) + bN(um( ), um( ), j) = ( ( ), j)+(g( , um ), j),1≤j≤m, (8) whe e he equa ions a e unde s ood in he sense o D0(τ, T∗),and he ini ial condi- ions is um(τ+s) = Pmφ(s) o s∈(−∞,0]. The abo e sys em o o dina y unc ional di e en ial equa ions wi h in ini e delay ul ills he condi ions o exis ence and uniqueness o local solu ion o [12, Th.1.1, p.36]. Nex , we will deduce a p io i es ima es ha assu e ha he solu ions do exis o all ime ∈[τ, T∗]∩R. S ep 2: A p io i es ima es. Mul iplying (8) by umwe ob ain 1 2 d d |um( )|2+νλ1 2|um( )|2+ν 2kum( )k2≤( ( ), um( )) + (g( , um ), um( )) ≤ | ( )||um( )|+Lgkum kγ|um( )| ≤ν 4kum( )k2+| ( )|2 νλ1 +Lgkum k2 γ. Hence |um( )|2+ν 2Z τ e−νλ1( −s)kum(s)k2ds ≤e−νλ1( −τ)|u(τ)|2+ 2 Z τ e−νλ1( −s)| (s)|2 νλ1 +Lgkum sk2 γds. (9) Fu he kum k2 γ≤max (sup θ∈(−∞,τ− ] e2γθ|φ(θ+ −τ)|2,sup θ∈[τ− ,0] e2γθ−νλ1( −τ+θ)|u(τ)|2 +2e2γθ Z +θ τ e−νλ1( +θ−s)| (s)|2 νλ1 +Lgkum sk2 γds). 8 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL On he one hand sup θ∈(−∞,τ− ] eγθ|φ(θ+ −τ)|= sup θ≤0 eγ(θ−( −τ))|φ(θ)| =e−γ( −τ)kφkγ. On he o he hand, as we a e assuming ha 2γ > νλ1, sup θ∈[τ− ,0] e2γθ−νλ1( −τ+θ)|u(τ)|2≤e−νλ1( −τ)|u(τ)|2 and sup θ∈[τ− ,0] e2γθ Z +θ τ e−νλ1( +θ−s)| (s)|2 νλ1 +Lgkum sk2 γds ≤Z τ e−νλ1( −s)| (s)|2 νλ1 +Lgkum sk2 γds. Collec ing hese inequali ies we deduce kum k2 γ≤e−νλ1( −τ)kφk2 γ+ 2 Z τ e−νλ1( −s)| ( )|2 νλ1 +Lgkum sk2 γds. By he G onwall lemma we ha e kum k2 γ≤e−(νλ1−2Lg)( −τ)kφk2 γ+2 νλ1Z τ e−(λ1ν−2Lg)( −s)| (s)|2ds. Then we ob ain he ollowing es ima es: o any R > 0 and T∈(τ, T∗]∩R, he e exis s a cons an C=C(τ, T, R),depending on some cons an s o he p oblem (namely, λ1, ν, Lgand ), and on τ, T and R, such ha kum k2 γ≤C(τ, T, R)∀ ∈[τ, T],∀ kφkγ≤R, ∀m≥1.(10) In pa icula , his implies ha {um}is bounded in L∞(τ, T;H)∀T∈(τ, T∗]∩R.(11) Now, i ollows om (9) and (10) ha ν 2e−νλ1(T−τ)ZT τ kum(s)k2ds ≤ν 2ZT τ e−νλ1(T−s)kum(s)k2ds ≤ |u(τ)|2+ 2 ZT τ e−νλ1(T−s)| (s)|2 νλ1 +Lgkum sk2 γds ≤R2+ 2 ZT τ e−νλ1(T−s)| (s)|2 νλ1 +LgC(τ, T, R)ds, so ha we conclude he exis ence o ano he cons an ( elabelled he same) C(τ, T, R) such ha kumk2 L2(τ,T ;V)≤C(τ, T, R)∀ kφkγ≤R∀m≥1,∀T∈(τ, T∗]∩R.(12) Now, obse e ha (8) is equi alen o dum d =−νAum−PmBN(um, um) + Pm ( ) + Pmg( , um ).(13) 3D-GMNS EQUATIONS WITH INFINITE DELAYS 9 F om (5), (11), (12) and (13), by he choice o he basis one also deduces ha k(um)0k2 L2(τ,T ;V0)≤C(τ, T, R)∀ kφkγ≤R∀m≥1,∀T∈(τ, T∗]∩R.(14) So, his implies he exis ence o a u∈L∞(τ, T ;H)∩L2(τ, T;V) wi h u0∈L2(τ, T;V0),∀T∈(τ, T∗]∩R, and a subsequence o {um}which con e ges weak-s a o uin L∞(τ, T;H),weakly o uin L2(τ, T ;V),wi h {(um)0}con e ging weakly o u0in L2(τ, T;V0) o all T∈(τ, T∗]∩R. Obse e in pa icula ha u∈C([τ, T]; H) o all T∈(τ, T∗]∩R. By a compac ness esul (c . [17, Ch.1,Th.5.1]), one can hen deduce ha a subsequence in ac con e ges s ongly o uin L2(τ, T;H) and a.e. in (τ, T) wi h alues in Hand a.e. in (τ, T)×Ω o all T∈(τ, T∗]∩R. S ep 3: Some mo e a p io i es ima es. The es ima es ob ained abo e a e no enough o pass o he limi and deduce ha uis a solu ion o (2). Namely, we ha e wo main di icul ies. On he one hand, we need o pass o he limi in g(um), his will be done in S ep 4, p o ing ha ac ually um →u in Cγ(H).On o he hand, he weak con e gence in L2(τ, T;V) is no enough o ensu e ha kum( )k→ku( )k o a leas FN(kum( )k)→FN(ku( )k) o a.a. , which is needed o manage he nonlinea e m BN(um, um). In o de o so ou his las ouble, we need o ind a s onge es ima e. We p oceed now wi h ha . Take he inne p oduc o he Gale kin ODE (8) wi h Aum and ob ain 1 2 d d kum( )k2+ν|Aum( )|2+bN(um( ), um( ), Aum( )) = ( ( ), Aum( )) + (g( , um ), Aum( )).(15) Ob iously, ( ( ), Aum( )) ≤ | ( )||Aum( )| ≤ ν 8|Aum( )|+2 ν| ( )|2 and |(g( , um ), Aum( ))| ≤ ν 8|Aum( )|2+2 ν|g( , um )|2. By (6), Lemma 1and Young’s inequali y, i ollows |bN(um( ), um( ), Aum( ))| ≤ N kum( )kC2kum( )k3/2|Aum( )|3/2 =NC2kum( )k1/2|Aum( )|3/2 ≤ν 4|Aum( )|2+CNkum( )k2, wi h CN=27(NC2)4 4ν3. Thus (15) simpli ies o d d kum( )k2+ν|Aum( )|2≤4 ν| ( )|2+4 ν|g( , um )|2+ 2CNkum( )k2.(16) 16 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL Thus, i we ake β=λ−1/2 1| | ν−Lgλ−1 1 , we ob ain ((Rmu, u)) ≥0∀u∈Vmsuch ha kuk=β. Consequen ly by a co olla y o he B ouwe ’s ixed poin heo em (see [17, p.53]), o each m≥1 he e exis um∈Vmsuch ha Rm(um)=0,wi h kumk ≤ β. Obse e mo eo e ha Aum∈Vm,and he e o e ν|Aum|2=−hBN(um, um), Aumi+ ( , Aum)+(g(um), Aum) (40) ≤ν 2|Aum|2− hBN(um, um), Aumi+| |2 ν+L2 gβ2 νλ1 . Mo eo e , by (6) and Young’s inequali y, |hBN(um, um), Aumi| ≤ ν 4|Aum|2+CNkumk2(41) ≤ν 4|Aum|2+CNβ2, wi h CN=27(NC2)4 4ν3. F om (40) and (41), we deduce ha he sequence {um}is bounded in D(A), and consequen ly, by he compac injec ion o D(A) in V, we can ex ac a subsequence {um0} ⊂ {um}, ha con e ges weakly in D(A) and s ongly in V o an elemen u∗∈D(A). I is now s anda d o ake limi s in (38) and o ob ain ha u∗is an s a iona y solu ion. In o de o p o e he inal egula i y ema k, i is enough o ake in o accoun ha e e y s a iona y solu ion u∗ o (34) is also a solu ion o (2), bu wi h ini ial da a φ( ) = u∗ o ∈(−∞,0], and o cing e m +g(u∗). Thus, we can apply Theo em 2. Uniqueness Le us suppose ha u∗and eu∗a e wo s a iona y solu ions o (34). Then, νhAu∗−Aeu∗, i+hBN(u∗, u∗)−BN(eu∗,eu∗), i = (g(u∗)−g(eu∗), ),∀ ∈V. (42) Taking =u∗−eu∗and p oceeding as in (39) we ob ain om (42) νku∗−eu∗k2≤ |FN(ku∗k)b(u∗−eu∗, u∗, u∗−eu∗)| +|(FN(ku∗k)−FN(keu∗k))b(eu∗, u∗, u∗−eu∗)| +λ−1 1Lgku∗−eu∗k2.(43) F om his inequali y, aking in o accoun (4), ha b(eu∗, u∗, u∗−eu∗) = 0,and he ac ha FN(ku∗k)≤1,we ob ain νku∗−eu∗k2≤λ−1/4 1ku∗k ku∗−eu∗k2+λ−1 1Lgku∗−eu∗k2, and he e o e, by he es ima e (36), νku∗−eu∗k2≤ λ−1/4 1 λ−1/2 1| | ν−λ−1 1Lg +λ−1 1Lg!ku∗−eu∗k2.(44) 3D-GMNS EQUATIONS WITH INFINITE DELAYS 17 On he o he hand, i in (43) we use (4), and Lemma 1, i.e. ku∗kFN(ku∗k)≤N and keu∗kFN(keu∗k)≤N, we ob ain νku∗−eu∗k2≤Nλ−1/4 1ku∗−eu∗k2+λ−1 1Lgku∗−eu∗k2.(45) F om (44) and (45) we deduce ha i (37) holds, hen u∗=eu∗. Theo em 4. Assume ha he assump ions in Theo em 2wi h and gindependen o ime and (37) hold. Then he e exis s a alue 0< λ < 2γsuch ha o he solu ion u(·,0, φ)o (2) wi h τ= 0,T∗= +∞and φ∈Cγ(H), he ollowing es ima es hold o all ≥0: |u( , 0, φ)−u∗|2≤e−λ |φ(0) −u∗|2+Lg 2γ−λkφ−u∗k2 γ,(46) ku (·,0, φ)−u∗k2 γ ≤max e−2γ kφ−u∗k2 γ, e−λ |φ(0) −u∗|2+Lg 2γ−λkφ−u∗k2 γ,(47) whe e u∗is he unique s a iona y solu ion o (34) gi en by Theo em 3. P oo . Fo sho deno e u( ) = u(·,0, φ).Le us also deno e w( ) = u( )−u∗. Conside ing equa ions (34) o u( ) and (35) o u∗,one has d d (w( ), ) + ν((w( ), )) + bN(u( ), u( ), )−bN(u∗, u∗, )=(g(u )−g(u∗), ), o > 0, o any ∈V. F om ene gy equali y and he Lipschi z condi ion on g, and in oducing an ex- ponen ial e m eλ wi h a posi i e alue λ o be ixed la e on, we ob ain d d (eλ |w( )|2)≤eλ λ|w( )|2−2νkw( )k2 +2|bN(u( ), u( ), w( ))−bN(u∗, u∗, w( ))|+ 2Lgkw kγ|w( )|), o > 0. Reasoning as o (44) and (45), we ha e |bN(u( ), u( ), w)−bN(u∗, u∗, w)| ≤ µku( )−u∗k2, whe e µ:= min (Nλ−1/4 1,λ−3/4 1| | ν−λ−1 1Lg). Hence, using a Young inequali y wi h δ > 0 o be ixed la e on, we conclude ha d d (eλ |w( )|2)≤eλ (−2ν+λλ−1 1+ 2µ+δλ−1 1Lg)kw( )k2+Lg δeλ kw k2 γ. The e o e, in eg a ing om 0 o , we ha e eλ |w( )|2≤ |w(0)|2+Lg δZ 0 eλskwsk2 γds +(−2ν+λλ−1 1+ 2µ+δλ−1 1Lg)Z 0 eλskw(s)k2ds. (48) 18 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL In o de o con ol he e m R 0eλskwsk2 γds, we p oceed as ollows. Z 0 eλs sup θ≤0 e2γθ|w(s+θ)|2ds =Z 0 eλs max{sup θ≤−s e2γθ|w(s+θ)|2,sup θ∈[−s,0] e2γθ|w(s+θ)|2}ds =Z 0 max{e−(2γ−λ)skφ−u∗k2 γ,sup θ∈[−s,0] e(2γ−λ)θeλ(s+θ)|w(s+θ)|2}ds. So, i λ≤2γ, using he abo e equali y in (48), we ob ain eλ |w( )|2≤ |w(0)|2+Lg δkφ−u∗k2 γZ 0 e(λ−2γ)sds +−2ν+λλ−1 1+2µ+δλ−1 1Lg+Lg(λ1δ)−1Z 0 max ∈[0,s]eλ kw( )k2ds. Obse e ha he (op imal) choice o δ= 1 makes ha δλ−1 1Lg+Lg(λ1δ)−1is minimal and he coe icien o he las in eg al is nega i e wi h a sui able choice o λ∈(0,2γ) by (37). 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