Global in time solution and time-periodicity for a smectic-A liquid crystal model

Global in ime solu ion and ime-pe iodici y o a smec ic-A liquid c ys al model Blanca Climen -Ezque a∗ , F ancisco Guill´ en-Gonz´ alez∗ Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Ap do. 1160, 41080 Se illa, Spain. E-mails: bclimen[email p o ec ed], [email p o ec ed] No embe 5, 2012 Abs ac In his pape some esul s a e ob ained o a smec ic-A liquid c ys al model wi h ime- dependen bounda y Di ichle da a o he so-called laye a iable ϕ( he le el se s o ϕ desc ibe he laye s uc u e o he smec ic-A liquid c ys al). Fi s , he ini ial-bounda y p oblem o a bi a y ini ial da a is conside ed, ob aining he exis ence o weak solu ions which a e bounded up o in ini y ime. Second, he exis ence o ime-pe iodic weak so- lu ions is p o ed. A e wa ds, he p oblem o he global in ime egula i y is a acked, ob aining he exis ence and uniqueness o egula solu ions (up o in ini y ime) o bo h p oblems, i.e. he ini ial- alued p oblem and he ime-pe iodic one, bu assuming a dom- inan iscosi y coe icien in he linea pa o he di usion enso . Keywo ds: solu ion bounded up o in ini y ime, ime-pe iodic solu ions, global in ime eg- ula solu ions, Na ie -S okes equa ions, Smec ic-A liquid c ys al, coupled non-linea pa abolic sys em. 1 In oduc ion In his wo k, we s udy he ime e olu ion o a smec ic-A liquid c ys al model p oposed in [E’97]. Smec ic c ys als a e in a liquid-c ys alline phase, whe e he molecules o he liquid c ys al no only ha e a ce ain o ien a ional o de (as in he nema ic case) bu also ha e a ∗This wo k has been pa ially inanced by DGI-MEC (Spain), G an MTM2006–07932 and by he p ojec P06-FQM-02373 (Jun a de Andaluc´ıa) 1 ce ain posi ional o de (laye s uc u e); he molecules a e a anged in almos incomp essible laye s o almos cons an wid h. Wi hin each laye , he smec ic-A sys em consis s o a single op ical axis npe pendicula o he laye such ha ∇ × n= 0. In his case, n=∇ϕ o a po en ial unc ion ϕ, and he le el se s o ϕwill ep esen he laye s uc u e in he sample. This s udy is mo i a ed by he ollowing p oblem in liquid c ys als. The usual nema ic molecule con igu a ion is de e mined by minimizing he Oseen F ank ene gy, which in he mo e simple case o equal cons an s de i es o Di ichle ene gy ZΩ |∇d|2. He e, he uni ec o ds ands o he o ien a ion o liquid c ys al molecules. Now, in he smec ic-A case, his o ien a ion dcoincides wi h he no mal ec o no each laye . Then, in o de o s udy he ene gy ZΩ |∇(∇ϕ)|2unde he cons ain |∇ϕ|= 1, i is na u al o in oduce he penalized ene gy ZΩ 1 2|∆ϕ|2+ (∇ϕ) whe e is he Ginzbu g-Landau penaliza ion unc ion (n) = 1 ε2(|n|2−1)n, which has he po en ial unc ion F(n) = 1 4ε2(|n|2−1)2 e i ying (n) = ∇nF(n) o each n∈IRN. As ε→0, one can hope ha he minimize o he penalized ene gy, o he solu ion o he co esponding Eule -Lag ange equa ion ∆2ϕ−∇· (∇ϕ) = 0, will be con e gen o he minimize o he ene gy RΩ 1 2|∆ϕ|2wi h he non-con ex cons ain |∇ϕ|= 1 (c . [Kinde leh e ,Liu’96] and [E’97]). Thus, i is impo an o s udy he asymp o ic beha io as ε→0 (c . [Guill´en,Rojas’02] o nema ic c ys al models). Howe e , e y li le is known abou his. We assume he smec ic-A liquid c ys al con ined in an open bounded domain Ω ⊂IRN (N= 2 o 3) wi h egula bounda y ∂Ω. We conside he ollowing PDE sys em in Ω × (0,+∞):    ρ(∂ u+ (u· ∇)u)−∇·(σd+λ σe) + ∇p= 0,∇ · u= 0, ∂ ϕ+u· ∇ϕ+γ(∆2ϕ−∇· (∇ϕ)) = 0, (1) whe e u: Ω ×[0,+∞)7→ RNis he low eloci y, p: Ω ×[0,+∞)7→ Rdesc ibes he luid p essu e and ϕ: Ω×[0,+∞)7→ Ris he laye a iable. The cons an s ρ,λ, and γa e posi i e, ep esen ing espec i ely, he densi y o he luid, he a io be ween he kine ic ene gy and 2 he elas ic one, and he elas ic elaxa ion ime. Mo eo e , we conside he same cons i u i e laws o he dissipa i e s ess enso σdand he elas ic s ess enso σeas in [Liu’00]: σd=µ1(n D(u)n)n⊗n+µ4D(u) + µ5(D(u)n⊗n+n⊗D(u)n), σe=− (n)⊗n+∇(∇ · n)⊗n−(∇ · n)∇n whe e µ1≥0, µ4>0, µ5≥0 a e dissipa i e cons an coe icien s, n=∇ϕand D(u) deno es he symme ic enso o he eloci y g adien : D(u) = 1 2(∇u+∇ u). The p oblem (1) is comple ed wi h he (Di ichle ) bounda y condi ions u|∂Ω= 0, ϕ|∂Ω=ϕ1, ∂nϕ|∂Ω=ϕ2(2) (assuming ime-depending bounda y da a ϕ1, ϕ2:∂Ω×(0,+∞)7→ IRN) and one o he ollowing condi ions: •ei he he ini ial condi ions u(0) = u0ϕ(0) = ϕ0in Ω (3) •o he ime-pe iodic condi ions: u(0) = u(T), ϕ(0) = ϕ(T) in Ω,(4) whe e T > 0 is a gi en inal ime. In he i s case, he compa ibili y condi ion ϕ0|∂Ω=ϕ1(0) mus be assumed. In his las case, one assumes ϕ1(0) = ϕ1(T) and ϕ2(0) = ϕ2(T). By spli ing he symme ic dissipa i e enso in o he linea and nonlinea pa σd=µ4D(u) + σd nl(D(u),∇ϕ), whe e σd nl := µ1(n D(u)n)n⊗n+µ5(D(u)n⊗n+n⊗D(u)n), no ice ha −∇ · σd=−µ4∇ · D(u)−∇·σd nl =−µ4 2∆u−∇·σd nl since ∇ · u= 0. By decomposing he e m due o he penaliza ion o he o he e ms in he elas ic enso as ollows σe=− (n)⊗n+σe np(n), whe e σe np := ∇(∇ · n)⊗n−(∇ · n)∇nis he non-penalized enso , and aking in o accoun ha ∇ · ( (n)⊗n)=(∇ · (∇ϕ))∇ϕ+ i(∇ϕ)∂i∇ϕ= (∇ · (∇ϕ))∇ϕ+∇F(∇ϕ) 3 and (∇ · σe np)j= (∇ · (∇(∇ · n)⊗n−(∇ · n)∇n))j= (∇ · (∇(∆ϕ)⊗ ∇ϕ−∆ϕ∇2ϕ))j =∂i(∂i(∆ϕ)∂jϕ−∆ϕ∂ijϕ)=∆2ϕ∂jϕ+∂i(∆ϕ)∂i∂jϕ−∂i(∆ϕ)∂ijϕ−∆ϕ∂i∂ijϕ = ∆2ϕ∂jϕ−∆ϕ∂j∆ϕ= ∆2ϕ∂jϕ−1 2∂j(|∆ϕ|2), we ha e −∇ · σe= (∇ · (∇ϕ))∇ϕ+∇F(∇ϕ)−∆2ϕ∇ϕ+∇|∆ϕ|2 2. Then, joining oge he all he g adien e ms, he momen um sys em o (1) can be w i en as: ρ(∂ u+ (u· ∇)u)−µ4 2∆u−∇·σd nl −λ(∆2ϕ−∇· (∇ϕ))∇ϕ+∇q= 0 (5) whe e qis he po en ial unc ion q=p+λF(∇ϕ) + λ|∆ϕ|2 2. One obse es ha he liquid c ys al model (1)-(2) lacks o maximum o compa ison p in- ciples ( o ∇ϕ) so one looses one o he s onges ools in analyzing nonlinea pdes. Assuming ime-independen bounda y da a ϕ1, ϕ2, an impo an ac o he model (1)-(2) is i s dissipa i e cha ac e , because his sys em admi s (a leas o mally) he ollowing ene gy equali y: d d ZΩ1 2|u|2+λ1 2|∆ϕ|2+F(∇ϕ) +ZΩµ4 2|∇u|2+σd nl :D(u) + λγ|∆2ϕ−∇· (∇ϕ)|2= 0. (6) This equali y is ob ained mul iplying he ϕ-equa ion by −λ(∆2ϕ− ∇ · (∇ϕ)), he u-sys em (5) by uand in eg a ing by pa s, because all he nonlinea con ec i e and elas ic e ms cancel and he bounda y e ms anish by using ha u|∂Ω= 0, ∂ ϕ|∂Ω= 0 and ∂ ∂nϕ|∂Ω= 0 (see (20) below o an ene gy equali y ela ed o a sys em wi h ime-dependen bounda y da a). In pa icula , since RΩσd nl :D(u)≥0 (see (22) below), his equali y implies ha he o al ene gy ( ha is, he kine ic ene gy 1 2RΩ|u|2plus he elas ic ene gy λRΩ 1 2|∆ϕ|2+F(∇ϕ)) dec eases espec o he ime. Now, since ime-dependen bounda y da a ϕ1, ϕ2will be conside ed, (6) mus be modi ied wi h a igh hand side depending on ime de i a i es o ϕ1, ϕ2which ac as o ce e ms, see (20). I we conside again ime-independen bounda y da a, i is impo an o ema k ha he ollowing (s a ic) c i ical poin s a e pa icula solu ions o he ime-pe iodic p oblem: u= 0, ϕ: any solu ion o he p oblem: ∆2ϕ−∇· (∇ϕ) = 0 in Ω, ϕ=ϕ1,∂nϕ=ϕ2on ∂Ω, p=−λF(∇ϕ)−λ|∆ϕ|2 2. 4 The e o e, in o de o conside a non i ial ime-pe iodic p oblem, i will be essen ial o assume ime-dependan bounda y da a o ϕ. De ini ion 1 We say ha (u, ϕ)is a weak solu ion o (1)-(3) in [0, T),0< T < +∞i ∇ · u= 0 in Q, u|Σ= 0, ϕ|Σ=ϕ1, ∂nϕ|Σ=ϕ2 u∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)), ϕ ∈L∞(0, T;H2(Ω)) ∩L2(0, T;H4(Ω)),(7) e i ying h∂ u, i+ ((u· ∇)u, ) + ((µ4/2)∇u+σd nl,∇ )−λ((∆2ϕ−∇· (∇ϕ))∇ϕ, ) = 0 ∀ ∈V, ∂ ϕ+ (u· ∇)ϕ+γ(∆2ϕ−∇· (∇ϕ)) = 0,a.e. in Q u(0) = u0, ϕ(0) = ϕ0in Ω. De ini ion 2 We say ha a weak solu ion (u, ϕ)is a s ong solu ion o (1)-(3) in [0, T)i u∈L∞(0, T;H1(Ω)) ∩L2(0, T;H2(Ω)), ϕ ∈L∞(0, T;H4(Ω)) ∩L2(0, T;H6(Ω)),(8) e i ying poin -wise he ully di e en ial sys em (1). In [Liu’00], conside ing a p oblem like (1)-(3), wi h a iable densi y and ime-independen bounda y condi ions o ϕ, au ho p o es he exis ence o weak solu ions using a semi-Gale kin p ocedu e keeping he anspo equa ion o densi y and he ϕ-equa ion a in ini y dimension. Mo eo e , he global egula i y o weak solu ions ( o big enough µ4i N= 3) is deduced in [Liu’00], and a p elimina y analysis abou he asymp o ic beha io in ime is made (see also [Lai,Liu’06] o o he asymp o ic beha io s udy o a ela ed model). The main esul s o p esen pape a e he ollowing, always o bounda y da a ϕ1and ϕ2 depending on he ime: 1. he uniqueness o weak/s ong solu ions o he ini ial- alue p oblem (1)-(3), 2. he exis ence o global weak solu ions o p oblem (1)-(3), which is bounded up o in ini y ime (wi h an exponen ial weigh ed no m o he L2(0,+∞)-no m, see (31)), 3. he exis ence o weak ime-pe iodic solu ions, 4. he exis ence o egula solu ions o bo h p e ious cases, he ini ial- alued p oblem and he ime-pe iodic one, bu assuming a dominan iscosi y coe icien µ4in he linea pa o he di usion enso . 5 The esul s ob ained in his pape a e in a ce ain sense simila o he esul s p esen ed in [Climen e al.’06] and [Climen e al.] o he weak solu ions and he egula solu ions o a nema ic liquid c ys al model, espec i ely. The pape is o ganized as ollows. In Sec ion 2, some di e en ial inequali ies a e deduced, which will be used in he es o he pape . In Sec ion 3, he uniqueness o weak/s ong solu ions o he ini ial- alue p oblem (1)-(3) is analyzed. In Sec ion 4, he global in ime solu ion o he ini ial alued p oblem is s udied a in ini y ime and he exis ence o weak ime-pe iodic solu ions is ob ained in Sec ion 5. Finally, unde he cons ain s o iscosi y coe icien µ4big enough, he exis ence and uniqueness o global egula solu ions o he ini ial alued p oblem is p o ed in Sec ion 6 and he exis ence o egula ime-pe iodic solu ions is deduced in Sec ion 7. Fo simplici y we ix he cons an s excep ing he iscosi y µ4, aking ρ=λ=γ=µ1=µ5= 1, ν =µ4/2. No a ion •We deno e Q= (0,+∞)×Ω, QT= (0, T)×Ω, Σ = (0,+∞)×∂Ω and ΣT= (0, T)×∂Ω. •In gene al, he no a ion will be ab idged. We se Lp=Lp(Ω), p≥1, H1 0=H1 0(Ω), e c. I X=X(Ω) is a space o unc ions de ined in he open se Ω, we deno e by Lp(X) he Banach space Lp(0, T ;X). Also, bold ace le e s will be used o ec o ial spaces, o ins ance L2=L2(Ω)N. •The Lpno m is deno ed by | · |p, 1 ≤p≤ ∞, he Hmno m by k · km(in pa icula |·|2=k·k0) and he p oduc no m in Hn×Hmby k·kn×m. The inne p oduc o L2(Ω) is deno ed by (·,·). •We se V he space o med by all ields u∈C∞ 0(Ω)Nsa is ying ∇ · u= 0. We deno e H ( espec i ely V) he closu e o Vin L2( espec i ely H1). Hand Va e Hilbe spaces o he no ms |·|2and k·k1, espec i ely. Fu he mo e, H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω},V={u∈H1;∇ · u= 0,u= 0 on ∂Ω} •In he sequel, C, C1, C2>0 will deno e di e en cons an s, depending only on he ixed da a o he p oblem, as Ω, ϕ1, ϕ2,ε(and u0, ϕ0 o he ini ial- alue p oblem). 6 2 P elimina ies 2.1 A li ing unc ion We de ine eϕ=eϕ( ) as he weak solu ion o he p oblem    −∆2eϕ= 0 in Ω, eϕ=ϕ1( )∂neϕ=ϕ2( ) on ∂Ω. (9) In he ime-pe iodic case, since by hypo hesis ϕ1(0) = ϕ1(T) and ϕ2(0) = ϕ2(T) on ∂Ω, hen eϕ(0) = eϕ(T) in Ω. The e o e, i we de ine bϕ( ) = ϕ( )−eϕ( ), hen ∆2bϕ= ∆2ϕin Qand bϕ=∇bϕ= 0 on Σ. In he ime-pe iodic case, one has ϕ(0) = ϕ(T) i and only i bϕ(0) = bϕ(T). Then, we can ew i e he p oblem (1)-(2) espec o he a iables (u,bϕ) (wi h bϕ( ) = ϕ( )−eϕ( )) as ollows ( ecall ha all coe icien s ha e been aken equal o one, excep ing iscosi y ν=µ4/2):                  ∂ u+ (u· ∇)u−ν∆u−∇·σd nl −(∆2bϕ−∇· (∇ϕ))∇ϕ+∇q= 0 in QT, ∇ · u= 0 in QT, ∂ bϕ+u· ∇ϕ+ ∆2bϕ−∇· (∇ϕ) = ∂ eϕin QT, u= 0,bϕ= 0, ∂nbϕ= 0 on ΣT (10) join ly wi h ei he he ini ial condi ions u(0) = u0,bϕ(0) = ϕ0−eϕ(0) o he ime-pe iodic condi ions u(0) = u(T), bϕ(0) = bϕ(T). Since u∈H1 0and bϕ∈H2 0, he ollowing no ms a e equi alen s: kuk1≈ |∇u|2,kbϕk2≈ |∆bϕ|2kbϕk4≈ |∆2bϕ|2. 2.2 Some inequali ies We will gi e wo inequali ies in he nex wo lemmas, ela ing he ellip ic ope a o ∆2bϕ− ∇ · (∇ϕ) and he penalized ene gy ZΩ F(∇ϕ) wi h some no ms. Lemma 3 The ollowing inequali y holds: |∆bϕ|2 2+1 2ε2|∇bϕ|4 4≤1 2|∆2bϕ−∇· (∇ϕ)|2 2+C1,(11) whe e C1>0is a cons an depending on ε,|Ω|, and k∇eϕkL∞(L4). P oo . We deno e ω= ∆2bϕ−∇· (∇ϕ). Tes ing his equali y by bϕone has: (ω, bϕ) = (∆2bϕ, bϕ)−(∇ · (∇ϕ),bϕ) = |∆bϕ|2 2+ ( (∇ϕ),∇bϕ).(12) 7 The las e m on he igh hand side o (12) can be w i en as ( (∇ϕ),∇bϕ)=( (∇ϕ)− (∇bϕ),∇bϕ)+( (∇bϕ),∇bϕ) = ( (∇ϕ)− (∇bϕ),∇bϕ) + 1 ε2|∇bϕ|4 4−1 ε2|∇bϕ|2 2 (13) F om (12) and (13), one has |∆bϕ|2 2+1 ε2|∇bϕ|4 4= (ω, bϕ) + 1 ε2|∇bϕ|2 2−( (∇ϕ)− (∇bϕ),∇bϕ).(14) The i s e m on he igh hand side o (14) can be bounded as |(ω, bϕ)|≤|ω|2|bϕ|2≤1 2|ω|2 2+C 2|∇bϕ|2 4≤1 2|ω|2 2+1 6ε2|∇bϕ|4 4+C ε2(15) whe e he cons an Cdepends on εand |Ω|. On he o he hand, he second e m on he igh hand side o (14) will be bounded as: 1 ε2|∇bϕ|2 2≤1 6ε2|∇bϕ|4 4+C ε2.(16) Now, we a e going o bound he hi d e m o (14). Taking in o accoun ha (a)− (b) = 1 ε2|a|2+|b|2+a·b−1(a−b)∀a,b∈IRN, in pa icula (∇ϕ)− (∇bϕ) = 1 ε2|∇ϕ|2+|∇bϕ|2+∇ϕ· ∇bϕ−1∇eϕ. (17) Consequen ly, by using ha ∇eϕ∈L∞(L4), he H¨olde and Young’s inequali ies, he las e m on he igh hand side o (14) can be bounded as ollows |( (∇ϕ)− (∇bϕ),∇bϕ)| ≤ 1 ε2||∇ϕ|2+|∇bϕ|2+∇ϕ· ∇bϕ−1|2|∇eϕ|4|∇bϕ|4 ≤C ε2|∇ϕ|2 4+|∇bϕ|2 4+ 1|∇bϕ|4 ≤C ε2|∇bϕ|2 4+ 1|∇bϕ|4≤1 6ε2|∇bϕ|4 4+C ε2. (18) whe e Cdepends on |Ω|and k∇eϕkL∞(L4). Finally, om (14)-(18), he inequali y (11) is deduced. Lemma 4 The ollowing inequali y holds: ZΩ F(∇ϕ)≤1 2ε2|∇bϕ|4 4+C2 ε2(19) whe e C2depends on |Ω|and k∇eϕkL∞(L4) 8 P oo . Since F(∇ϕ) = 1 4ε2|∇ϕ|2−12, one has ZΩ F(∇ϕ) = 1 4ε2ZΩ |∇ϕ|4+1 2ε2ZΩ |∇ϕ|2+1 4ε2|Ω| ≤1 2ε2ZΩ |∇ϕ|4+C ε2≤1 2ε2ZΩ |∇bϕ|4+C2 ε2 whe e Cdepends on |Ω|and C2depends, mo eo e , on k∇eϕkL4(L4). The e o e, (19) holds. 2.3 Ene gy Inequali y Lemma 5 (Ene gy equali y) I (u, ϕ)is a egula enough solu ion o (10), he ollowing ene gy equali y holds: d d 1 2|u|2 2+1 2|∆bϕ|2 2+ZΩ F(∇ϕ)+|∇ϕTD(u)∇ϕ|2 2+|D(u)∇ϕ|2 2 +ν|∇u|2 2+|ω|2 2= (∂ eϕ, ω)+(∂ ∇eϕ, (∇ϕ)). (20) whe e ω= ∆2bϕ−∇· (∇ϕ). P oo . Taking uas es unc ion in he u-sys em o (10), one has 1 2 d d |u|2 2+ν|∇u|2 2+ (σd nl,∇u)−(ω· ∇ϕ, u)=0.(21) The nonlinea dissipa i e enso σd nl e i ies: (σd nl,∇u) = (σd nl, D(u)) = |∇ϕTD(u)∇ϕ|2 2+|D(u)∇ϕ|2 2,(22) since (n⊗n) : D(u) = ninjD(u)ij =nTD(u)n and (D(u)n⊗n+n⊗D(u)n) : D(u) = 2(D(u)n⊗n) : D(u) = D(u)iknknjD(u)ij =|D(u)n|2. The e o e, om (21) we ob ain 1 2 d d |u|2 2+ν|∇u|2 2+|∇ϕTD(u)∇ϕ|2 2+|D(u)∇ϕ|2 2+|ω|2 2−(ω· ∇ϕ, u) = 0.(23) On he o he hand, by aking ωas es unc ion in he ϕ-equa ion o (10), one has 1 2 d d |∆bϕ|2 2−(∂ bϕ, ∇ · (∇ϕ)) + (u· ∇ϕ, ω) + |ω|2 2= (∂ eϕ, ω).(24) The second e m on he le hand side o (24) can be w i en as −(∂ bϕ, ∇ · (∇ϕ)) = (∂ ∇ϕ, (∇ϕ)) −(∂ ∇eϕ, (∇ϕ)) = d d ZΩ F(∇ϕ)−(∂ ∇eϕ, (∇ϕ)).(25) By adding (23) and (24) and in o accoun (25) we ob ain (20). 9 5 Weak ime-pe iodic solu ions In his sec ion, le T > 0 a ini e ixed numbe which s a es he ime pe iod. De ini ion 11 We say ha (u, ϕ)is a weak ime-pe iodic solu ion o (1),(2) and (4) i u∈L∞(0, T;H)∩L2(0, T;H1), ϕ ∈L∞(0, T;H2)∩L2(0, T;H4) sa is ying (1) and bounda y condi ions (2) as in De ini ion 9 and ime-pe iodic condi ions u(0) = u(T),ϕ(0) = ϕ(T)in he sense o spaces L2and H2 espec i ely. Theo em 12 (Exis ence o weak ime-pe iodic solu ions) Le Ω,ϕ1and ϕ2be egula enough wi h ϕ1(0) = ϕ1(T),ϕ2(0) = ϕ2(T), and such ha he li ing unc ion eϕde ined in (9) sa is ies eϕ∈L∞(0, T ;H4(Ω)), ∂ eϕ∈L∞(0, T ;W1,4(Ω)). Then, he e exis s a weak ime-pe iodic solu ion o (1),(2) and (4). P oo . In he p oo o his heo em, a ully Gale kin me hod (app oxima ing in ini e dimen- sion bo h a iables uand ϕ) will be used. The eason is ha his ini e-dimensional Gale kin p oblem le us o ind ime-pe iodic app oxima e solu ions ia a ixed-poin a gumen o he ope a o mapping he ini ial and inal ime alues. Fi s ly, we conside he ini ial-bounda y Gale kin p oblem associa ed o any a bi a y ini e-dimensional ini ial da a. A e wa ds, he key is o ind an ini ial da a a = 0 which will be “ ep oduced” a inal ime =T. Finally, by means o a pass o he limi p ocedu e, a weak ime-pe iodic solu ion will be ound. We di ide he p oo in se e al s eps. S ep 0: Exis ence o local in ime Gale kin solu ion. Le {wi}n≥1 and {φi}n≥1 “special” basis o Vand H2 0(Ω), espec i ely, o med by eigen unc ions o he S okes p oblem (∇wi,∇ ) = λi(wi, )∀ ∈V,wi∈V,con kwikL2= 1, λi%+∞ and o he bilaplacian p oblem (∆φi,∆e) = µi(φi, e)∀e∈H2 0, φi∈H2 0,con kφikL2= 1, µi%+∞. Le Vmand Wmbe he ini e-dimensional subspaces spanned by {w1,w2,...,wn}and {φ1, φ2, . . . , φn} espec i ely. Gi en u0∈Hand ϕ0∈H2 0, o each m≥1, we seek an app oxima e solu ion (um, ϕm), wi h um: [0, T]7→ Vmand ϕm=bϕm+eϕ, wi h bϕm: [0, T ]7→ Wm, e i ying he ollowing a ia ional o mula ion a.e. ∈(0, T): 16                          (∂ um( ), m) + ((um( )· ∇)um( ), m) + ν(∇um( ),∇ m)+(σd,m nl ( ), D m) −(Qm∆2bϕm( )−∇· (∇ϕm( ))∇ϕm( ), m)=0 ∀ m∈Vm, (∂ bϕm( ), em) + ((um( )· ∇)ϕm( ), em) + (∆2bϕm( )−∇· (∇ϕm( )), em) = (∂ eϕ( ), em),∀em∈Wm, um(0) = u0m=Pm(u0), ϕm(0) = ϕ0m=Pm(ϕ0) in Ω. (42) He e, Pm:H7→ Vmdeno es he usual o hogonal p ojec o om Hon o Vm, and Qm: L27→ Wm he o hogonal p ojec o om L2on o Wm. In pa icula , u0m→u0in L2and ϕ0m→ϕ0in H2(as m→0). I we w i e um( ) = m X j=1 ξi,m( )φiand bϕm( ) = m X j=1 ζi,m( )ϕi, (42) can be ew i en as a i s o de o dina y di e en ial sys em (in no mal o m) associa ed o he unknowns (ξi,m( ), ζi,m( )). Then, one has exis ence o a maximal solu ion (de ined in some in e al [0, τm)⊂[0, T]) o he ela ed Cauchy p oblem. Mo eo e , om a p io i es ima es (independen on m) which will be ob ained below, in pa icula one has ha τm=T. S ep 1: Ene gy es ima es. By aking in (42) m=um∈Vmand em=Qm(∆2bϕm−∇· (ϕm( ))) ∈Wmas es unc ions in (42), one can a i es a a simila inequali y o (26) changing (u,bϕ) by (um,bϕm) and |∆2bϕm−∇· (ϕm( ))|2 2by |Qm(∆2bϕm−∇· (ϕm( )))|2 2. Tha is, one has d d |um|2 2+|∆bϕm|2 2+ 2 ZΩ F(∇ϕm)+ 2ν|∇um|2 2+|Qm(∆2bϕm−∇· (∇ϕm))|2 2 ≤1 4ε2|∇bϕm|4 4+C. (43) On he o he hand, he p oo o Lemma 3 can be mimic o he case o Gale kin solu ions, ob aining he ollowing inequali y (simila o (11)) |∆bϕm|2 2+1 2ε2|∇bϕm|4 4≤1 2|Qm(∆2bϕm−∇· (∇ϕm))|2 2+C1.(44) Following he same a gumen o he p oo o Theo em 10, om (43) and (44) one has (35) and (36). Since now he inal ime T > 0 is ini e, in pa icula , he ollowing es ima es holds: umis bounded in L∞(0, T;H)∩L2(0, T;V), 17 ϕmis bounded in L∞(0, T;H2) and ωm:= Qm(∆2bϕm−∇· (∇ϕm)) is bounded in L2(0, T ;L2). S ep 2: ϕmis bounded in L2(0, T;H4). We ha e de ined ωm=Qm(∆2bϕm−∇· (∇ϕm)), namely, ωm∈Wm,(ωm, em) = ∆2bϕm−∇· (∇ϕm), em∀em∈Wm.(45) By aking em= ∆2bϕm∈Wmas es unc ion in (45) ( ha is possible because a spec al basis o he eigen unc ions o he bilaplacian has been conside ed), one ob ains |∆2bϕm|2 2≤ |∇ · (∇ϕm))|2|∆2bϕm|2+|ωm|2|∆2bϕm|2, hence kbϕmk4≤C|∇ · (∇ϕm)|2+|ωm|2.(46) F om (39) and by using he bound o ϕmin L∞(0,+∞;H2) and he in e pola ion inequali y kϕk3≤Ckϕk1/2 2kϕk1/2 4, one has |∇ · (∇ϕm))|2 2≤C|∇ϕm|4 6|∇∇ϕm|2 6+|∆ϕm|2 2≤Ckϕmk4 2kϕmk2 3+kϕmk2 2 ≤C(kϕmk4kϕmk2+ 1) ≤δkbϕmk2 4+δkeϕk2 4+C≤δkbϕmk2 4+C. (47) By using his las inequali y o δsmall enough in (46) we ob ain kbϕmk2 4≤C+C|ωm|2 2. As ωmis bounded in L2(Q), in eg a ing in [0, T] we ha e ha ϕmis bounded in L2(0, T;H4). S ep 3: Uniqueness o Gale kin solu ion By applying he a gumen s gi en in Theo em 8 o (um, ϕm), we can ob ain he uniqueness o Gale kin solu ion. No ice ha his is possible because ∆2ϕm∈Wm(and um∈Vm). S ep 4: Exis ence o ime-pe iodic Gale kin solu ion Gi en (um 0, ϕm 0)∈Vm×Wm, we de ine he map Lm: [0, T]7→ IRm×IRm 7→ (ξ1m( ), ..., ξmm( ), ζ1m( ), ..., ζmm( )) whe e (ξ1m( ), ..., ξmm( )) and (ζ1m( ), ..., ζmm( )) a e he coe icien s o um( ) and bϕm( ) espec o Vmand Wm espec i ely, being (um( ),bϕm( )) he (unique) app oxima e solu ion o (42) co esponding o he ini ial da a (um 0, ϕm 0). 18 Now, a ying he ini ial da a (um 0, ϕm 0), we a e going o de ine a new map Φm: IRm×IRm7→ IRm×IRmas ollows: gi en Lm 0∈IRm×IRm, we de ine Φm(Lm 0) = Lm(T), whe e Lm( ) is ela ed o he solu ion o p oblem (42) wi h ini ial da a Lm 0(= Lm(0)). By uniqueness o app oxima e solu ion o (42), his map is well-de ined. Mo eo e , using egula i y o he co esponding o dina y di e en ial sys em (equi alen o (42)), his map is con inuous. In o de o p o e exis ence o ixed poin o Φm, we will use Le ay-Schaude ’s Theo em. Consequen ly, we ha e o p o e ha o all λ∈[0,1], solu ions Lm 0(λ) o Lm 0(λ) = λΦm(Lm 0(λ)) a e uni o mly bounded (independen o λ). Since Lm 0(0) = {0}, i is su ices o analyze λ∈(0,1] and he equa ion 1 λLm 0(λ)=Φm(Lm 0(λ)). Since we ha e conside ed he eigen unc ions o ∆2 o u nish Wmand (45), i is easy o jus i y he compu a ions o lemma 3, lemma 4 and Co olla y 6, in o de o a i e a (35). Conside ing he no m kLm( )kIRm×IRm=kum( )k2 L2+k∆bϕm( )k2 L21/2in IRm×IRm, inequali y (35) yields k1 λLm 0(λ)k2 IRm×IRm≤e−C0TkLm 0(λ)k2 IRm×IRm+C(1 −eC0T). Since λ∈(0,1], one has kLm 0(λ)k2 IRm×IRm≤C(1 −eC0T) eC0T−1 which is a bound independen o λ(and m). Consequen ly, Le ay-Schaude Theo em implies he exis ence o ixed poin o Φm, and he e o e he exis ence o ime-pe iodic Gale kin solu ions. Mo eo e , o each ime-pe iodic Gale kin solu ion (um, ϕm), hei co esponding ini ial- end da a (um(0), ϕm(0)) = (um(T), ϕm(T)) is bounded in he L2×H2-no m, i.e k(um,bϕm)(0)kL2×H2≤C(Cindependen o m). S ep 5: Pass o he limi in ime-pe iodic Gale kin solu ions The pass o he limi in a ia ional o mula ion (42) can be done using es ima ions (inde- penden s o m) and compac ness ob ained in o de o con ol nonlinea e ms. Consequen ly, he e we will only w i e he pass o he limi in ime-pe iodic condi ions. F om es ima ions o (ϕm) in L∞(H2) and (∂ ϕm) in L2(L3/2) and using he iple o spaces H2,→H1,→L3/2, one has ha (ϕm) is ela i ely compac in C([0, T]; H1), hence 19 ϕm(T)→ϕm(T) and ϕm(0) →ϕ(0) in H1(Ω). Since ϕm(T) = ϕm(0), hen ϕ(T) = ϕ(0) in H1(Ω). Mo eo e , i is easy o see ha ϕ∈Cw([0, T]; H2) (i.e. ϕis con inuous om [0, T] on o H2, espec o he weak opology in H2), he e o e ϕ(T) = ϕ(0) in H2(Ω). The a gumen o uis simila . Consequen ly, we ha e ound a weak ime-pe iodic solu ion o p oblem (1)-(2), (4) and he p oo o Theo em 12 is inished. 6 Regula i y o he ini ial- alue p oblem The idea now is o ob ain egula i y o he weak solu ions o he ini ial- alue p oblem (1)-(3) (see [Lin,Liu’95], [Lin,Liu’00] o a nema ic liquid c ys al case and [Liu’00] o he smec ic-A case, imposing ime-independen bounda y da a in all hese p e ious cases). In his sense, we will see ha a global egula i y esul hold bu only o he case o dominan iscosi y, ha is o νbig enough. In ou opinion, he global egula i y imposing cons ain s o ini ial da a nea o special equilib ium solu ions is an in e es ing p oblem, which up o ou knowledge emains as an open p oblem. De ini ion 13 We say ha a weak solu ion (u, ϕ)o (1)-(3) is a s ong solu ion i k(u( ), ϕ( ))k1×4≤C3∀ ≥0,(48) ∀γ > 0, e−γ Z 0 eγsk(u(s), ϕ(s))k2 2×6ds ≤C4,∀ ≥0 (49) and e i ying poin -wise he ully di e en ial sys em (1). Theo em 14 In he condi ions o heo em 10, i mo eo e (u0, ϕ0)∈H1×H4wi h ku0k1≤ R1,kϕ0k4≤R2, ∂ eϕ∈L∞(0,+∞;W1,4(Ω)) and ∂ eϕ∈L∞(0,+∞;L2(Ω)), hen o each ν≥ν0, wi h ν0=ν0(R1, R2, ∂ eϕ, ∂ eϕ), he e exis s a unique s ong solu ion o (1)-(3) in [0,+∞), which e i ies (48) and (49) wi h cons an s C3and C4depending on ν0 (bu independen o ν). P oo . We de ine bω=−∂ bϕ−(u· ∇)bϕ. (50) By owing o bϕ|Σ= 0, ∇bϕ|Σ= 0 and u|Σ= 0, we ha e bω|Σ= 0,∇bω|Σ= 0.(51) 20 On he o he hand, we a e going o ob ain he ollowing inequali ies: kϕk4≤C(|bω|2+|u|2+ 1),kϕk6≤C(kbωk2+kuk2+ 1).(52) Indeed, as ω= ∆2ϕ−∇· (∇ϕ) = −∂ ϕ−(u· ∇)ϕ=bω−∂ eϕ−(u· ∇)eϕ, one has ∆2bϕ= ∆2ϕ=ω+∇ · (∇ϕ) = bω−u· ∇eϕ−∂ eϕ+∇ · (∇ϕ).(53) Hence kbϕk4≤ |bω|2+|∇eϕ|∞|u|2+|∂ eϕ|2+|∇ · (∇ϕ)|2. P oceeding in he analogous way ha in (47) o bound he e m |∇ · (∇ϕ)|2and using he egula i y o eϕone a i es a he bound o kbϕk4gi en in (52). The bound o kbϕk6gi en in (52) can be ob ained in a simila way. No ice ha , 1 2 d d |bω|2 2= (bω, ∂ bω)=(bω, ∂ (ω+u· ∇eϕ+∂ eϕ)) = (bω, ∂ (∆2bϕ−∇· (∇ϕ) + u· ∇eϕ+∂ eϕ)) = (∆bω, ∂ ∆bϕ)+(∇bω, ∂ (∇ϕ)) + (bω, ∂ u· ∇eϕ+u·∂ ∇eϕ+∂ eϕ). By using (50), one has ∂ ∆bϕ=−∆bω−∆((u· ∇)bϕ) = −∆bω− ∇2u∇bϕ− ∇u∇2bϕ−(u· ∇)∆bϕ and ∂ (∇ϕ) = (3|∇ϕ|2−1)∂ ∇ϕ= (3|∇ϕ|2−1)(−∇bω− ∇((u· ∇)bϕ) + ∇∂ eϕ) = (3|∇ϕ|2−1)(−∇bω− ∇u∇bϕ−(u· ∇)∇bϕ+∇∂ eϕ),. The e o e, we ob ain ha 1 2 d d |bω|2 2+|∆bω|2 2=−(∆bω, ∇2u∇bϕ)−(∆bω, ∇u∇2bϕ)−(∆bω, (u· ∇)∆bϕ)−(∇bω, 3|∇ϕ|2∇bω) −(∇bω, 3|∇ϕ|2∇u∇bϕ)−(∇bω, 3|∇ϕ|2(u· ∇)∇bϕ)) + (∇bω, ∇bω)+(∇bω, ∇u∇bϕ) +(∇bω, (u· ∇)∇bϕ)) + (bω, ∂ u· ∇eϕ)+(bω, u·∂ ∇eϕ)+(bω, ∂ eϕ)+(∇bω, (3|∇ϕ|2−1)∂ ∇eϕ). By bounding he e ms on he igh hand side o p e ious equali y one a i es a d d |bω|2 2+kbωk2 2≤ν 2kuk2 2+1 2|∂ u|2 2+C νkbωk2 2(1 + kbϕk3+kbϕk2 3) + C|bω|2 2+C, (54) whe e C > 0 may deno e di e en cons an s, always independen o ν. 21 On he o he hand, aking Au+∂ uas es unc ions in he u-sys em (Abeing he S okes ope a o , i.e. A=−P∆ wi h P he Le ay p ojec o on o H) i is easy o ob ain d d ((ν+ 1)kuk2 1) + νkuk2 2+|∂ u|2 2≤1 2kbωk2 2+C+C νkuk1kuk2 2 +C+C ν(|bω|2 2+kuk1) + C νkbϕk4 3+kbϕk2 3+ 1kuk2 2, ( he las e m on he igh hand side o p e ious inequali y is a bound o |∇ · σd nl|2 2). Since we wan o choose νbig enough, o ins ance we assume ν0≥1. Then, o each ν > ν0≥1, we ge d d ((ν+ 1)kuk2 1) + νkuk2 2+|∂ u|2 2≤1 2kbωk2 2+Ckuk1kuk2 2+|bω|2 2+kuk1 +C νkbϕk4 3+kbϕk2 3+ 1kuk2 2. (55) Adding (54) and (55) we ha e d d ((ν+ 1)kuk2 1+|bω|2 2) + ν 2kuk2 2+1 2|∂ u|2 2+1 2kbωk2 2≤C νkbωk2 2(1 + kbϕk3+kbϕk2 3) +D(kuk1kuk2 2+|bω|2 2+kuk1) + C νkbϕk4 3+kbϕk2 3+ 1kuk2 2+E (56) whe e C,Dand Ea e cons an s independen o ν≥1. On he o he hand, using (52), he egula i y o u,bϕ,eϕand he in e pola ion inequali y kbϕk3≤Ckbϕk1/2 2kbϕk1/2 4we ge kbϕk3≤C(1 + |bω|1/2 2). Hence, om (56) one has he ollowing inequali y: d d ((ν+ 1)kuk2 1+|bω|2 2) + ν 2kuk2 2+1 2|∂ u|2 2+1 2kbωk2 2≤C νkbωk2 2(1 + |bω|1/2 2+|bω|2) +D(kuk1kuk2 2+|bω|2 2+kuk1) + C ν|bω|2 3+|bω|3+ 1kuk2 2+E. (57) I we deno e Φ1( ) = kuk2 1,Φ2( ) = |bω|2 2,Ψ1( ) = kuk2 2,Ψ2( ) = kbωk2 2, we ob ain om (57), d d ((ν+ 1)Φ1+ Φ2) + ν 2−DΦ1/2 1−C νΦ2+ Φ1/2 2+ 1Ψ1 +1 2−C ν(1 + Φ1/4 2+ Φ1/2 2)Ψ2≤D(Φ2+ Φ1/2 1) + E. (58) Le R1,R2,Mand ν0≥1 some posi i e cons an s ha we will speci y below, such ha i Φ1(0) ≤R1and Φ2(0) ≤R2, we will p o e ha (ν+ 1)Φ1( )+Φ2( )≤M∀ ∈[0,+∞),(59) 22 o any ν≥ν0. Indeed, by con adic ion, le ∗>0 he i s alue such ha (ν+ 1)Φ1( ∗) + Φ2( ∗) = M, hence (ν+ 1)Φ1( ∗)+Φ2( ∗) = Mand (ν+ 1)Φ1( )+Φ2( )< M ∀ ∈[0, ∗). Then, Φ1( )≤M ν+ 1 and Φ2( )≤M∀ ∈[0, ∗]. Assume ha he e exis s ν0big enough such ha , o each ν≥ν0 ν 2−DM ν+ 11/2 −C ν(M+M1/2+ 1) ≥ν+ 1 4 and 1 2−C ν1 + M1/4+M1/2≥1 4.(60) Then, o each ∈[0, ∗] d d ((ν+ 1)Φ1+ Φ2) + ν+ 1 4Ψ1+1 4Ψ2≤D(Φ2+ Φ1/2 1) + E. (61) We de ine P= min{P1, P2}whe e 1/P1and 1/P2a e he Poinca ´e cons an s ha e i y Φ1≤1 P1 Ψ1and Φ2≤1 P2 Ψ2 espec i ely. The e o e, d d ((ν+ 1)Φ1+ Φ2) + P 4((ν+ 1)Φ1+ Φ2)≤D(Φ2+ Φ1/2 1) + E. (62) Mul iplying (62) by eP /4and in eg a ing in [0, ∗] we deduce (ν+ 1)Φ1( ∗)+Φ2( ∗)≤((ν+ 1)Φ1(0) + Φ2(0))e−P ∗/4 +e−P ∗/4Z ∗ 0 (D(Φ2(s)+Φ1/2 1(s)) + E)ePs/4ds. (63) By (53), bω= ∆2bϕ+u· ∇eϕ+∂ eϕ−∇· (∇ϕ), hence we ge Φ2=|bω|2 2≤C(kbϕk2 4+|u|2+1+kbϕk4)≤C(kbϕk2 4+|u|2+ 1). The e o e, aking in o accoun weak es ima es (31), he second e m on he igh hand side o (63) is bounded by a cons an Cwindependen o ν(in ac , Cwdepends on he cons an C2gi en in (31)) and (ν+ 1)Φ1( ∗)+Φ2( ∗)≤((ν+ 1)Φ1(0) + Φ2(0)) + Cw1 ν+ 1≤((ν+ 1)R1+R2)+2Cw. Hence, i we choose M > (ν+ 1)R1+R2+ 2Cw,(64) 23 hen we a i es a a con adic ion. The e o e, we could ge he es ima e (59) whe he he e exis s big enough cons an s Mand ν0such ha (60) and (64) hold, o each ν≥ν0. Indeed, i we choose M=λ ν hen (64) holds o any λ > 2R1+R2+ 2Cw. I we ix λwi h his condi ion, hen he wo condi ions gi en in (60) hold i λ1/2 ν≤εand 1 ν(1 + λ1/4ν1/4+λ1/2ν1/2)≤ε o ε > 0 small enough. Bu hese condi ions hold o each ν≥ν0wi h ν0big enough espec o λ. The e o e, we ge es ima es (59). F om (59), we ob ain u∈L∞(0,+∞;H1) and bω∈L∞(0,+∞;L2). Recalling (52) we also ob ain ϕ∈L∞(0,+∞;H4). By going back o (61), mul iplying by eγ o any γ > 0 and in eg a ing in [0, ] we deduce ∀γ > 0, e−γ Z 0 eγsk(u(s),bω(s))k2 2×2ds ≤C2,∀ ≥0. Again, by applying (52) we ge (49). 7 Regula i y o he ime-pe iodic p oblem The esul s ob ained up o now allow us o ob ain, o big enough ν, he egula i y gi en in De ini ion 13 also o he ime-pe iodic p oblem. Indeed, a guing as in [Climen e al.] o a nema ic c ys al model, o p o e ha weak ime-pe iodic solu ion is egula i su ices o use he ollowing h ee esul s: 1. he exis ence o he weak ime-pe iodic solu ion (p o ed in Sec ion 4), 2. he weak/s ong uniqueness o he ini ial- alued p oblem (p o ed in Sec ion 3), 3. he exis ence o global s ong solu ion o big enough iscosi y o he ini ial- alued p oblem (p o ed in Sec ion 6). Consequen ly, he egula i y o ime-pe iodic solu ions can be deduced. Re e ences [Climen e al.’06] B. Climen -Ezque a, F. Guill´en-Gonz´alez, M.A. Rojas-Meda . Rep oduc- i i y o a Nema ic Liquid C ys al Model, Z. Angew. Ma h. Phys., 576 (2006) no. 6, 984-998. 24 [Climen e al.] B. Climen -Ezque a, F. Guill´en-Gonz´alez, M.J. Mo eno-I abe e. Regula i y and Time-pe iodici y o a Nema ic Liquid C ys al model, accep ed in Nonlinea Analysis. h p://dx.doi.o g/10.1016/j.na.2008.10.092 [Cou and,Shkolle ’01] D. Cou and, S. Shkolle . Well posedness o he ull E icsen-Leslye model o nema ic liquid c ys als, C.R.Acad.Sci.Pa is, Se . I 333 (2001), 919-924. [Dam´azio e al.’07] P. D. Dam´azio, F. Guill´en-Gonz´alez, J. V. Gu ie ez-San ac eu, M. A. Rojas-Meda . Local and Global S ong Solu ion by he Semi-Gale kin Me hod o he Model o Mass Di usion, Ma em´a ica Con empo anea, . 32 (2007), 63-83. [E’97] W. E. Nonlinea Con inuum Theo y o Smec ic-A Liquid C ys als, A ch. Ra . Mech. Anal., 137, 2 (1997), 159-175. [Guill´en,Rojas’02] F. Guill´en-Gonz´alez, M. A. Rojas-Meda . Global solu ion o nema ic liquid c ys al models C.R.Acad.Sci.Pa is, Se . I 335 (2002) 1085-1090. [Kinde leh e ,Liu’96] D. Kinde leh e , C. Liu. Re isi ing he Focal Conic S uc u es in Smec- ic A. P oc. Symposium on Elas ici y o hono P o esso J. L. E icksen 1996, ASME Mechanics and Ma e ials Con e ence. [Lai,Liu’06] M.J. Lai, C. Liu, P. Wens on. On Two Nonlinea Biha monic E olu ion Equa- ions: Exis ence, Uniqueness and S abili y, Z. Angew. Ma h. Phys., 576 no. 6 (2006), 984-998. [Lin,Liu’95] F.H. Lin, C. Liu. Non-pa abolic dissipa i e sys ems modelling he low o liquid c ys als, Comm. Pu e Appl. Ma h., 4 (1995), 501-537. [Lin,Liu’00] F.H. Lin, C. Liu. Exis ence o solu ions o he E icksen-Leslie sys em, A ch. Ra . Mech. Anal., 154 (2000), 135-156. [Lions’96] P.L. Lions. Ma hema ical Topics in Fluids Mechanics, Cla endon P ess, Ox o d, 1996. [Liu’00] C. Liu. Dynamic Theo y o Incomp essible Smec ic Liquid C ys als: Exis ence and Regula i y, Disc e e and Con inuous Dynamical Sys ems 6, 3 (2000), 591-608. 25