Global solution of nematic liquid crystals models

Global solu ion o nema ic liquid c ys als models F ancisco GUILL´ EN-GONZ´ ALEZ a, Ma ko ROJAS-MEDAR b Abs ac . We p o e exis ence o a global weak solu ion o a nema ic liquid c ys al p ob- lem by means o a penaliza ion me hod using a simpli ied E icksen-Leslie model and a new compac ness p ope y o he g adien o he di ec o ield. 1 In oduc ion In his No e, we es ablish he well-posedness in he la ge o a nema ic liquid c ys als model ( o mula ed o ins ance in [7]) by means o a penalisa ion a gumen using a simpli ied E icksen-Leslie model wi h he Ginzbu g-Landau app oxima ion [1, 2]. Le us conside a simpli ied e sion o he E icksen-Leslie model, in oduced by Lin in [4] and analysed by Lin and Liu [5, 6] who used a modi ied Gale kin app oach, and by Shkolle [8] who elied on a con ac ion mapping a gumen coupled wi h app op ia e ene gy es ima es. This model is a modi ied Na ie -S okes sys em ha ake in o accoun o he liquid c ys allini y, coupling wi h he Ginzbu g- Landau equa ions. A ull e sion o his E icksen-Leslie model has been ecen ly s udied by Cou and and Shkolle in [3], whe e local well-posedness (global o small enough da a) is p o ed. Now, we a e in e es ed in he asymp o ic beha iou espec o he penalisa ion pa ame e . The unknowns a e he ime-dependen di e gence- ee eloci y ield u( , x) and p essu e p( , x) o he luid and he di ec o ield d( , x) ep esen ing he o ien a ion o he liquid c ys als molecules. The luid is con ined in an open bounded domain Ω ⊂IRn(n= 2 o 3) wi h bounda y ∂Ω o C2 ype. In he penalised model one e i ies he cons ain |d| ≤ 1 as consequence o a maximum p inciple o he Ginzbu g-Landau equa ion whe e he app oxima ion ε(d) = ε−2(|d|2−1)d is conside ed (ε > 0). He e |d|=|d( , x)|deno es he punc ual Euclidean no m in IRn. This penalisa ion unc ion exhibi s po en ial s uc u e, i.e. he e exis s a po en ial unc ion Fε(d) = ε−2(|d|2−1)2 such ha ε(d) = ∇d(Fε(d)) o all d∈IRn. Acco dingly, we conside he penalised model in (0, T)×Ω as ollows |d| ≤ 1, ∂ d+u· ∇d+γ( ε(d)−∆d) = 0 (1) ∂ u+u· ∇u−ν∆u+∇p+λ∇ · (∇d¯ ∇d) = 0 (2) ∇· u= 0 (3) u|∂Ω= 0,d|∂Ω=h(4) u| =0 =u0,d| =0 =d0(5) 1 He e, u0and d0a e, espec i ely, he ini ial eloci y and di ec o ields. In o de o ob ain he dissipa i i y o he model, we shall assume (as in all p e ious wo ks) ime independen (Di ichle ) bounda y da a o he di ec o ield d, gi en by h:∂Ω→IRn. Conce ning he coe icien s, ν > 0 ep esen s he iscosi y o he luid, λ > 0 is an elas ici y cons an , ε > 0 is a penalisa ion pa ame e (wi h espec o he uni a y cons ain ), and γ > 0 is a elaxa ion- ime cons an . We ha e used he enso ial no a ion (∇d¯ ∇d)ij = n X k=1 ∂xidk∂xjdk In he ε-limi model we will ind he es ic ion |d|= 1 and he Lag ange mul iplie associa ed |∇d|2d. Indeed, when ε→0 we will ind a limi p oblem, whe e i changes (1) by |d|= 1, ∂ d+u· ∇d−γ∆d−γ|∇d|2d= 0.(6) Le us in oduce he ollowing space o unc ions H1 h={d∈H1(Ω)n/d=hon ∂Ω} H={u∈L2(Ω)n/∇ · u= 0,u·n= 0 on ∂Ω} V={u∈H1 0(Ω)n/∇ · u= 0}. Fo simplici y, le us deno e L2,H1ins ead o L2(Ω)n, H1(Ω)ne c. Ou main esul is he ollowing Theo em 1.1 Le T > 0and Ω⊂IRnbe an open, bounded and C2domain. Le us assume u0∈H, d0∈H1 hand h∈H2such ha |d0|= 1 in Ωand |h|= 1 on ∂Ω. Then, he e exis s a global weak solu ion u∈L2(0, T;V)∩L∞(0, T;H),d∈L∞(0, T;H1 h)o he limi p oblem (2)-(6) ob ained as a limi o “semi-s ong” solu ions uε∈L2(0, T;V)∩L∞(0, T ;H),dε∈L2(0, T;H2∩H1 h)∩L∞(0, T;H1 h)o he coupled Na ie -S okes and Ginzbu g-Landau model (1)-(5) as εgoes o ze o. Rema k 1 Up o ou known, his heo em is he i s esul o exis ence o a global in ime solu ion (wi hou es ic ions on he da a) o he limi p oblem (2)-(6). In [7], P ohl p o es exis ence (and uniqueness) o a local in ime s ong solu ion o (2)-(6). On he o he hand, Lin and Liu s udied in [6] he asymp o ic beha iou o (1)-(5) when εgoes o ze o, bu he limi o he (s ongly) nonlinea e ms ∇dε¯ ∇dεis only ob ained owa ds a measu e alued enso M. The main con ibu ion in his no e is o iden i y M wi h he limi enso ∇d¯ ∇d. Rema k 2 No ice ha in he ε-limi p oblem (2)-(6) one loses he H2- egula i y o he di ec o ield d, hence al hough (uε,dε) e i ies (1) poin -wise a.e. ( , x), hei limi (u,d) e i ies (6) only in a dis ibu ional sense. 2ε-app oxima e solu ions and dissipa i i y Fo each ε > 0, le us conside a “semi-s ong” solu ion (uε,dε) o he ε-app oxima e p oblem (1)-(5), ha is uε∈L2(0, T;V)∩L∞(0, T;H),dε∈L2(0, T;H2∩H1 h)∩L∞(0, T;H1 h),(7) 2 e i ying he u-sys em (2) in he dis ibu ional sense and he d-sys em (1) poin -wise a.e. The bounda y condi ions a e e i ied in he ace sense. Finally, he ini ial condi ions ha e classical sense; indeed uε and dεa e ime-con inuous unc ions, as consequence o he addi ional egula i y ∂ uε∈Lp(0, T;V0) and ∂ dε∈Lp(0, T;L2) (p= 2 i n= 2 o p= 4/3 i n= 3) ha can be ob ained applying he p e ious egula i y (7) o he equa ions (1) and (2). The exis ence o (uε,dε) can be p o ed ([5]) by means o h ee main a gumen s: a semi-gale kin me hod (space-disc e iza ion o he u-sys em (2), emaining he d-sys em (1) in he con inuous sense), a maximum p inciple o he d-sys em in o de o ob ain he cons ain |dε| ≤ 1 and he ollowing ene gy inequali y, d d µ1 2kuεk2+λ 2k∇dεk2+λZΩ Fε(dε)dx¶+νk∇uεk2+λγk ε(dε)−∆dεk2≤0,(8) ob ained aking espec i ely λ( ε(dε)−∆dε) and uεas es unc ions in (1)-(2) and using he equali y ∇ · (∇d¯ ∇d) = ∇(|∇d|2/2) + ∇d ∆d. In (8), k·kdeno es he L2(Ω)-no m. 3ε-independen es ima es Fo simplici y, le us deno e L2(H1) ins ead o L2(0, T;H1), e c. F om he maximum p inciple in (2) |dε| ≤ 1 a.e. ( , x). On he o he hand, om he ene gy inequali y (8), one has he ollowing (ε-independen ) es ima es: uεis bounded in L∞(H)∩L2(V),(9) dεis bounded in L∞(0, T;H1 h),(10) wε:= ε(dε)−∆dεis bounded in L2(L2),(11) Fε(dε) is bounded in L∞(L1).(12) Mo eo e , applying es ima es (9)-(11) in equa ions (1)-(2), ∂ uεis bounded in L2(V0) + L∞((W1, ∩V)0), > n, (13) ∂ dεis bounded in L2(L2) + Lq(L4/3), q = 4 i n= 2 o q= 8/3 i n= 3. (14) Finally, (10) implies in pa icula Mε:= ∇dε¯ ∇dεis bounded in L∞(L1).(15) 3 4ε-con e gence F om p e ious es ima es (9)-(15) and compac ness esul s o Aubin-Lions ype [9], he e exis s subse- quences (equally deno ed) dε,uε,wε, Mεand hei espec i e limi unc ions d,u,w, M, such ha uε→uin L∞(H) weak?, L2(V) weak, L2(H) s ong dε→din L∞(H1) weak?, C(L2) s ong, ∂ uε→∂ uin L2(V0) + L∞((W1, ∩V)0) weak, ∂ dε→∂ din L2(L2) + Lp(L4/3) weak, wε→win L2(L2) weak, ∇dε¯ ∇dε→Min he measu e sense. Consequen ly, aking limi s as ε→0 and using De Rham’s lemma [10], we a i e a ∂ d+u· ∇d+γw= 0 (16) ∂ u+u· ∇u−ν∆u+∇p+λ∇ · M= 0 (17) ∇· u= 0 (18) u|∂Ω= 0,d|∂Ω=h(19) u| =0 =u0,d| =0 =d0(20) On he o he hand, since ε−1(|dε|2−1) is bounded in L∞(L2) (using (12)) and dε→dpoin -wise a.e. ( , x), one ge he uni y cons ain |d( , x)|= 1 a.e. ( , x). The e o e, in o de o inish he p oo o Theo em 1.1, we ha e o iden i y wwi h −∆d− |∇d|2dand Mwi h ∇d¯ ∇d. 5 Iden i ica ion o w =−∆d− |∇d|2d. The a gumen o his sec ion is based in he known li e a u e on ha monic unc ions wi h alues in he uni s e ic su ace (see o ins ance [1, 2] and e e ences he ein ci ed). In pa icula , we will use he ollowing esul , which is a sligh ly modi ica ion (in oducing he con ec ion e ms) o Lemma 2.2 in [2] (see also Lemma 7.1 in [6]): Lemma 5.1 The ollowing wo sys ems a e equi alen : ∂ d+u· ∇d−γ∆d=γ|∇d|2d and |d|= 1,(∂ d+u· ∇d)∧d−γ∇ · (∇d∧d) = 0.(21) Since we al eady ha e ha |d|= 1, i su ices o e i y he equa ion o (21). Indeed, making he ec o ial p oduc o equa ion (2) by dε, aking in o accoun ha ε(dε)∧dε= 0 and −∆dε∧dε=−∇·(∇dε∧dε), one has (∂ dε+uε· ∇dε)∧dε−γ∇ · (∇dε∧dε) = 0.(22) Making ε→0, we can deduce (21), using he s ong con e gences o uεand dεand he weak con e gences o ∂ dεand ∇dε. 4 6 Iden i ica ion o M=∇d¯ ∇d. The key o his p oo is o ob ain L2-compac ness o ∇dε, using some ideas o he op imisa ion amewo k. Indeed, since wε=−∆dε+ ε(dε), hen dε( ) can be iewed as a solu ion o he op imisa ion (wi hou cons ain s) p oblem Jε(dε( )) = min d∈H1 h(Ω) Jε(d)·=ZΩµ1 2|∇d|2+Fε(d)−wε( )·d¶¸. On he o he hand, a.e. le us de ine ˜ d( ) as a solu ion o he op imisa ion (wi h cons ain s) p oblem J(˜ d( )) = min d∈H1 h(Ω) |d|=1 J(d)·=ZΩµ1 2|∇d|2−w( )·d¶¸. Ob iously, Jε(dε( )) ≤Jε(˜ d( )).In pa icula , ZT 0ZΩµ1 2|∇dε( )|2+Fε(dε( )) −wε( )·dε( )¶≤ZT 0ZΩµ1 2|∇˜ d( )|2−wε( )·˜ d( )¶.(23) Taking limi in as ε→0 in (23), bounding p e iously Fε(dε( )) ≥0, ZT 0 J(d( )) ≤lim in ε→0ZT 0ZΩµ1 2|∇dε( )|2−wε( )·dε( )¶ ≤lim ε→0ZT 0ZΩµ1 2|∇˜ d( )|2−wε( )·˜ d( )¶=ZT 0 J(˜ d( )), hence one has he equali y RT 0J(d( )) = RT 0J(˜ d( )) ( he opposi e inequali y is easy o deduce since |d( )|= 1). Consequen ly, all he p e ious inequali ies a e equali ies, and in pa icula ∃lim ε→0ZT 0ZΩµ1 2|∇dε( )|2−wε( )·dε( )¶=ZT 0 J(d( )) = ZT 0ZΩµ1 2|∇d( )|2−w( )·d( )¶ Since ∃lim ε→0RT 0RΩwε( )·dε( ) = RT 0RΩw( )·d( ) (using he s ong con e gence o dε), one has ∃lim ε→0k∇dεk2 L2(L2)= k∇dk2 L2(L2) he e o e, dε→din L2(H1)−s ong. In pa icula , aking in o accoun he es ima es o ∇dε, ∇dε→ ∇din Lp(Lq)−s ong,∀p, q :p < ∞, q < 2, hence we can iden i y M=∇d¯ ∇d. Acknowledgemen s. The i s au ho has been pa ially inanced by he p oje BFM2000-1317, and he second au ho by he p ojec s CNPq-B asil 300116-93-4 and Fapesp-B asil 01/07557-3 5 Re e ences [1] F. Be huel, H B ezis, F. H´ elein,Asymp o ics o he minimiza ion o a Ginzbu g-Landau unc- ional, Calc. Va . 1 (1993) 123-148. [2] Y. Chen,The weak solu ions o he e olu ion p oblems o ha monic maps, Ma h. Z. 201 (1989) 69-74. [3] D. Cou and, S. Shkolle ,Well-posedness o he ull E icksen-Leslie model o nema ic liquid c ys als, No e C.R.A.S., . 333, S`e ie I (2001) 919-924. [4] F.H. Lin,Nonlinea heo y o de ec s in nema ic liquid c ys als: phase ansi ion and low phenom- ena, Comm. Pu e Appl. Ma h. 42 (1989) 789-814. [5] 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. 48, (1995), 501-537. [6] 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. [7] A P ohl,Compu a ional Mic o-magne ism, Ad ances in Nume ical Ma hema ics, Teubne 2001. [8] S. Shkolle ,Well-posedness and global a ac o s o liquid c ys als on Riemannian mani olds, Comm. Pa ial Di e . Eq. 27, 5 & 6 (2001) 1103-1137. [9] J. Simon,Compac se s in Lp(0, T;B), Ann. Ma . Pu a Appl., 146 (1987) 65–97. [10] R. Temam,Na ie -S okes equa ions, No h-Holland Else ie , 1985. aDepa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Ap do. 1160, 41080 Se illa (Spain). E-mail:guillen@nume .us.es bDepa amen o de Ma ema ica Aplicada, IMECC-UNICAMP, C.P. 6065, 13081-970, Campinas-SP (B asil). E-mail:ma k[email p o ec ed] 6