Regularity and time-periodicity for a nematic liquid crystal model

Regula i y and ime-pe iodici y o a nema ic liquid c ys al model Blanca Climen -Ezque a∗ , F ancisco Guill´ en-Gonz´ alez∗, M. Jesus Mo eno-I abe e 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], [email p o ec ed] Abs ac In his pape wo main esul s a e ob ained o a nema ic liquid c ys al model wi h ime- dependen bounda y Di ichle da a o he o ien a ion o he c ys al molecules. Fi s , he ini ial-bounda y p oblem is conside ed, ob aining he exis ence o global in ime (up o in ini y ime) weak solu ion, he exis ence o global egula solu ion o iscosi y coe icien big enough, and he weak/s ong uniqueness. Second, using hese p e ious esul s and he exis ence o ime-pe iodic weak solu ions p o ed in [2], he egula i y o any ime-pe iodic weak solu ion is deduced o iscosi y coe icien big enough. Keywo ds: solu ion up o in ini y ime, ime-pe iodic solu ions, uniqueness, Na ie -S okes equa- ions, Nema ic liquid c ys al models, coupled non-linea pa abolic sys em. 1 In oduc ion In his wo k, a simpli ied E icksen-Leslie e sion o a nema ic liquid c ys al model is consid- e ed; see o ins ance [8] o a o mula ion o a mo e comple e liquid c ys al p oblem. This model can be seen as a a ian o he Na ie -S okes p oblem ( espec o he eloci y- p essu e unknowns (u, p)) coupled wi h a con ec ion-di usion sys em o a new a iable d, which is a uni ec o ial unc ion modelling he o ien a ion o he c ys al molecules. On he o he hand, i is usual o conside an app oxima ion by Ginzbu g-Landau penaliza ion ([1]) o he cons ain |d|= 1 (|d|=|d( , x)|deno es he poin -wise euclidean no m). This penalized model (whe e he cons ain |d|= 1 is elaxed by |d| ≤ 1) was in oduced by Lin in [6] and s udied ( om a ma hema ical poin o iew) by Lin and Liu in [7, 8]. Cou and and Shkolle in [4] also s udied his simpli ied model bu including s e ching e ec s. We assume a (new onian) luid con ined in an open bounded domain Ω ⊂IRN(N= 2 o 3) wi h egula bounda y ∂Ω. In he penalized model he cons ain |d|= 1 is pa ially conse ed o ∗Fi s and second au ho s ha e been pa ially inanced by he p oje s P06-FQM-02373 and MTM2006–07932. 1 |d| ≤ 1 as consequence o he maximum p inciple o he Ginzbu g-Landau equa ion conside ing he unc ion (d) = 1 ε2(|d|2−1)dwhe e ε > 0 is he penaliza ion pa ame e . The e exis s a po en ial unc ion F(d) = 1 4ε2(|d|2−1)2such ha (d) = ∇dF(d) o each d∈IRN. Then, we conside he ollowing PDE sys em in (0,+∞)×Ω:    ∂ u+ (u· ∇)u−ν∆u+∇p=−∇d ∆d,∇ · u= 0, ∂ d+ (u· ∇)d= ∆d− (d),|d| ≤ 1, (1) The cons an s ν,λand γa e posi i e, ep esen ing espec i ely, he luid iscosi y, an elas ici y cons an and a elaxa ion ime ( o simplici y we conside λ=γ= 1 and ν > 0 and o he las esul , la ge enough). The p oblem (1) is comple ed wi h he (Di ichle ) bounda y condi ions u(x, )=0,d(x, ) = h(x, ) on ∂Ω×(0,+∞) (2) (assuming as in [2] a ime-depending bounda y da a o dgi en by h:∂Ω×(0,+∞)7→ IRN; in [7, 8] only a ime-independen bounda y da a is conside ed) and ei he he ini ial condi ion u(x, 0) = u0d(x, 0) = d0in Ω (3) o he ime-pe iodic condi ion: u(x, 0) = u(x, T),d(x, 0) = d(x, T) in Ω,(4) whe e T > 0 is a gi en inal ime. In his las case, we assume, mo eo e , ha h(0) = h(T). This model has, beside well known di icul ies o he Na ie -S okes p oblem (a nonlinea pa abolic sys em wi h he ee di e gence cons ain ela ed o he p essu e), o he di e en di i- cul ies which come om he s ongly nonlinea coupling be ween he o ien a ion ec o dand he eloci y-p essu e (u, p) and om he cons ain |d| ≤ 1. An essen ial cha ac e is ic o he p oblem o d(gi en u), ei he he ini ial- alue p oblem wi h (3) o he ime-pe iodic case wi h (4), is he ollowing weak maximum p inciple (see [7, 2]): Assume |h| ≤ 1 on ∂Ω×(0, T) and ei he |d0| ≤ 1 in Ω o he ini ial- alue p oblem o h(0) = h(T) on ∂Ω o he ime-pe iodic p oblem. Then, gi en u∈L2(0, T;V)∩L∞(0, T;H) (see he no a ions below o he de ini ion o spaces Vand H), any poin -wise solu ion o he d-p oblem e i ies |d(x, )| ≤ 1 a.e. in Ω ×(0, T). In [7], conside ing he ini ial- alue p oblem (1)-(3) wi h ime-independen bounda y condi ions o d, au ho s p o e exis ence o global weak solu ion (wi h u∈L∞(L2)∩L2(H1), d∈L∞(H1)∩ L2(H2)), exis ence o global egula solu ion (wi h u∈L∞(H1)∩L2(H2), d∈L∞(H2)∩L2(H3)) i νis big enough o N= 3 and uniqueness o egula solu ions. Howe e , in hese p e ious esul s o [7] he e is an impo an simpli ica ion; he bounda y da a hdoes no depend on ime. In his case, he ime-pe iodic p oblem (1),(2),(4) wi h bounda y condi ion independen o he ime (d(x, )|∂Ω×(0,T )=d0(x)), leads o a i ial p oblem (see [2]), because all “s a ic” solu ions u= 0 and d e i ying s a iona y p oblem −∆d+ (d) = 0 in Ω, d|∂Ω=d0, a e in pa icula ime-pe iodic solu ions. Respec o he non i ial case o ime-dependen bounda y condi ion, he exis ence o weak ime-pe iodic solu ions o (1),(2),(4) is p o ed in [2]. 2 The main esul s o he p esen a icle a e he ollowing: always o bounda y da a hdepending on he ime, we p o e exis ence o global weak solu ion (de ined in [0,+∞)) o he ini ial alue p oblem (1)-(3), exis ence o global s ong solu ions unde he cons ain o iscosi y coe icien νbig enough and uniqueness o s ong/weak solu ions, ha is any weak solu ion coincides wi h he s ong solu ion (i his s ong solu ion exis s). Mo eo e , we p o e exis ence o egula ime- pe iodic solu ions unde he same ype o cons ain . A exis ence esul o egula ime-pe iodic solu ions o a gene alized Boussinesq model can be seen in [3]. The pape is o ganized as ollows. In Sec ion 2, some di e en ial inequali ies in weak no ms a e deduced, whe eas Sec ion 3 is de o ed o ob ain di e en ial inequali ies in s ong no ms. In Sec ion 4, he global in ime solu ion o he ini ial- alue p oblem is s udied (a in ini y ime), and inally he exis ence o s ong ime-pe iodic solu ion is ob ained in Sec ion 5, using esul s o Sec ion 4 and he exis ence o weak ime-pe iodic solu ion o [2]. No a ions •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·km×n. 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, D > 0 will deno e di e en cons an s, depending only on he ixed da a o he p oblem, as Ω, λ, γ. 2 Di e en ial inequali ies in weak no ms 2.1 A li ing unc ion We de ine e d( ) as he weak solu ion o he Laplace-Di ichle p oblem    −∆e d= 0 in Ω, e d=h( ) on ∂Ω. (5) 3 In he ime-pe iodic case, since by hypo hesis h(0) = h(T) on ∂Ω, hen e d(0) = e d(T) in Ω. The e o e, i we de ine b d( ) = d( )−e d( ), hen ∆b d= ∆din Ω×(0, T) and b d= 0 on ∂Ω×(0, T). In he ime-pe iodic case, d(0) = d(T) i and only i b d(0) = b d(T). Then, we can ew i e he p oblem (1)-(2) in he a iables (u,b d) (wi h d=b d+e d) as ollows:        ∂ u+ (u· ∇)u−ν∆u+∇p+∇d ∆b d= 0,∇ · u= 0 in QT, ∂ b d+ (u· ∇)d−∆b d+ (d) = −∂ e din QT, u= 0,b d= 0 on ∂Ω×(0, T ), (6) join ly wi h ei he he ini ial condi ion u(0) = u0,b d(0) = d0−e d(0) o he ime-pe iodic condi ions u(0) = u(T), b d(0) = b d(T). Rema k: The choice o his ype o li ing unc ion allows us o ob ain es ima es up o in ini y ime, which is no possible wi h he li ing unc ion ha we will conside in Sec ion 3. 2.2 Di e en ial inequali ies We will gi e wo di e en di e en ial equali ies in he nex wo lemmas. Lemma 1 I uand da e egula enough, he ollowing di e en ial inequali y holds: d d |u|2 2+|∇b d|2 2+ 2ν|∇u|2 2+|∆b d|2 2≤2| (d)|2 2+|∂ e d|2 2,(7) P oo : Taking uand −∆b das es unc ions in (6), adding up, aking in o accoun ha ((u· ∇)u,u) = 0 and (∇d ∆b d,u)−((u· ∇)d,∆b d)=0,(8) one a i es (a leas o mally) a he ollowing ene gy equali y: 1 2 d d |u|2 2+|∇b d|2 2+ν|∇u|2 2+|∆b d|2 2= ( (d),∆b d)+(∂ e d,∆b d). Consequen ly, applying Young inequali y, one has (7). Lemma 2 I uand da e egula enough, he ollowing di e en ial inequali y holds: d d |u|2 2+|∇b d|2 2+ 2 ZΩ F(d)+ 2ν|∇u|2 2+|∆b d− (d)|2 2≤ |∂ e d|2 2(9) P oo : Taking uand −∆b d+ (d) as es unc ions in (6), adding up and aking in o accoun (8), ∂ d· (d) = ∂ F(d) and ((u· ∇)d, (d)) = 0,one ob ains 1 2 d d |u|2 2+|∇b d|2 2+ 2 ZΩ F(d)+ν|∇u|2 2+|∆b d− (d)|2 2= (∂ e d,∆b d− (d)). ****************** By ew i ing he second e m as (∂ e d,∆b d)=(∂ e d,∆b d− (d)) + (∂ e d, (d)), by using | (d)| ≤ 1 ε2owing o he maximum p inciple |d| ≤ 1 one has |(∂ e d,∆b d)| ≤ 1 2|∂ e d|2 2+1 2|∆b d− (d)|2 2+1 ε2|∂ e d|1. The e o e, one a i es (a leas o mally) a (9). 4 3 Di e en ial inequali ies in egula no ms 3.1 A li ing unc ion We will conside ano he sui able li ing unc ion e d o he bounda y da a h( ha we deno e equal) in such a way ha we could made es ima es o H3- ype o he homogeneous a iable ela ed o d(see [5]). Conc e ely, we de ine e das he solu ion o he p oblem: (∂ e d−∆e d= 0 in QT, e d=hon ∂Ω×(0, T),(10) join ly e d(0) = d0in Ω o he ini ial alued p oblem o e d(0) = e d(T) in Ω o he ime-pe iodic case. Then, he d−p oblem o (1) can be ew i en as ollows:    ∂ b d+ (u· ∇)d−∆b d+ (d) = 0 in QT, b d= 0 on ∂Ω×(0, T ) (11) and b d(0) = 0 in Ω o he ini ial- alued p oblem o b d(0) = b d(T) in Ω o he ime-pe iodic case. As consequence o he maximum p inciple |d| ≤ 1 and |e d| ≤ 1. Al hough we do no know i |b d| ≤ 1, we ha e kb dkL∞(L∞)≤ kdkL∞(L∞)+ke dkL∞(L∞)≤2. We a e going o conside he ollowing equi alen s no ms: kuk1≈ |∇u|2,kb dk1≈ |∇b d|2in H1 0, kuk2≈ |∆u|2,kb dk2≈ |∆b d|2in H1 0∩H2 kb dk3≈ |∇(∆b d)|2+|∆b d|2=k∆b dk1in H1 0∩H3 Rema k: Owing o he li ing unc ion conside ed in his sec ion, we ha e ha ∆b d− (d)|∂Ω= 0 because he es o he e ms in (11) anish on he bounda y. Tha is no ue wi h he li ing unc ion o he Sec ion 2. 3.2 Di e en ial inequali ies In he p e ious condi ions, he ollowing egula i y esul will be equen ly used. Lemma 3 Assume d=b d+e d, wi h |d| ≤ 1, a) i ∆b d− (d)∈L2(Ω) and e d∈H2(Ω), hen d∈H2(Ω) and kdk2≤ ke dk2+C(1 + |∆b d− (d)|2), b) i ∆b d− (d)∈H1(Ω) and e d∈H3(Ω), hen d∈H3(Ω) and kdk3≤ ke dk3+C(|∇d|2+|∇(∆b d− (d))|2). 5 P oo : Fo he p oo , i is undamen al ha ∆b d− (d) = 0 on ∂Ω. We ha e kdk2≤ kb dk2+ke dk2 and kdk3≤ kb dk3+ke dk3. Then, by adding and sub ac ing (d) in o he no ms kb dk2≈ |∆b d|2and kb dk3≈ k∆b dk1, we ob ain he i s and second inequali y, espec i ely, using ha | (d)| ≤ Cand |∇ (d)| ≤ C|∇d|. Lemma 4 Assume e d∈L∞(0,+∞;H3(Ω)),u∈L∞(0,+∞;L2(Ω)) and d∈L∞(0,+∞;H1(Ω)). Then, i uand da e egula enough, he ollowing di e en ial inequali y holds: d d kuk2 1+|∆b d− (d)|2 2+νkuk2 2+ 2|∇(∆b d− (d))|2 2 ≤D(1 + |∆b d− (d)|2 2) + E νkuk1kuk2 2+ (1 + |∆b d− (d)|2 2)|∇(∆b d− (d))|2 2, (12) whe e D, E > 0a e cons an s independen o ν. P oo : Taking Auas es unc ions in he u-sys em o (1) (Abeing he S okes ope a o ) and applying adequa ely H¨olde and Young’s inequali ies, one ob ains: d d kuk2 1+4 3νkuk2 2≤C ν|(u· ∇)u|2 2+|∇ d∆d|2 2 ≤C ν|u|2 3|∇u|2 6+|∇ d|2 3|∆d|2 6≤C ν|u|2kuk1kuk2 2+kdk1kdk2kdk2 3 Hence, owing o Lemma 3, weak es ima es (|u( )|2≤C,kd( )k1≤Ca.e. ∈(0,+∞)) and s ong egula i y o e d∈L∞(0,+∞;H3(Ω)) (in pa icula , inequali ies o Lemma 3 de i e in he simples inequali ies: kdk2≤C(1 + |∆b d− (d)|2) and kdk3≤C(1 + |∇(∆b d− (d))|2)), we ha e: d d kuk2 1+4 3νkuk2 2≤C νkuk1kuk2 2+ (1 + |∆b d− (d)|2)|∇(∆b d− (d))|2 2 +C(1 + |∆b d− (d)|2). (13) In he las e m we ha e conside ed ha C/ν is uni o mly bounded espec o ν, as ν≥ν0. By aking g adien in he b d-sys em (11), mul iplying by −∇(∆b d− (d)), in eg a ing by pa s in he ∂ b d- e m (whe e all he bounda y e ms anish owing o he choice o he li ing unc ion d ha implies (∆b d− (d))|∂Ω= 0) and adding bo h sides he e m −(∂ (d),∆b d− (d)), we ind: 1 2 d d |∆b d− (d)|2 2+|∇(∆b d− (d))|2 2=−(∂ (d),∆b d− (d)) −((∇u· ∇)d,∇(∆b d− (d))) + (u· ∇∇d,∇(∆b d− (d))) (14) By using he b d-sys em we ha e ha −∂ (d) = −∇d (d)∂ d=∇d (d)(u· ∇)d−∆b d+ (d)−∆e d hence, he i s e m on he igh hand side o (14) can be w i en as (∇d (d)(u· ∇)d,∆b d− (d)) −(∇d (d)(∆b d− (d)),∆b d− (d)) −(∇d (d)∆e d,∆b d− (d)). Taking in o accoun ha k∇d (d)kL∞(L∞)≤C, weak es ima es (|u|2≤C,kdk1≤C) and he s ong egula i y o e d∈L∞(0,+∞;H3(Ω)), we can bound hese e ms by: C(|u|∞|∇d|2|∆b d− (d)|+|∆b d− (d)|2 2+|∆e d|2|∆b d− (d)|2) ≤C(kuk1/2 1kuk1/2 2kdk1|∆b d− (d)|2+|∆b d− (d)|2 2+ 1) ≤ν 18kuk2 2+C ν|∆b d− (d)|2 2+C(|∆b d− (d)|2 2+ 1). 6 The second e m on he igh hand side o (14) is es ima ed by | − ((∇u· ∇)d,∇(∆b d− (d)))| ≤ C|∇u|6|∇d|3|∇(∆b d− (d))|2 ≤ν 18kuk2 2+C νkdk1kdk2|∇(∆b d− (d))|2 2 ≤ν 18kuk2 2+C ν(1 + |∆b d− (d)|2)|∇(∆b d− (d))|2 2. Analogously, he las e m on he igh hand side o (14) is bounded by |(u· ∇2d,∇(∆b d− (d)))| ≤ C|u|∞|∇2d|2|∇(∆b d− (d))|2 ≤ν 18kuk2 2+C νkdk2 2|∇(∆b d− (d))|2 2 ≤ν 18kuk2 2+C ν(1 + |∆b d− (d)|2 2)|∇(∆b d− (d))|2 2. Consequen ly, applying p e ious es ima es in (14) and conside ing ha C/ν is uni o mly bounded espec o νas ν≥ν0, we a i e a d d |∆b d− (d)|2 2+ 2|∇(∆b d− (d))|2 2≤ν 3kuk2 2+C(1 + |∆b d− (d)|2 2) +C ν(1 + |∆b d− (d)|2 2)|∇(∆b d− (d))|2 2. (15) F om (13) and (15) we ob ain (12). 4 Global solu ion o he ini ial- alue p oblem De ini ion 5 We say ha (u,d)is a weak solu ion o (1)-(3) i ∇ · u= 0 in Q, u|Σ= 0,d|Σ=h, k(u( ),d( ))k0×1≤C1∀ ≥0i.e. (u,d)∈L∞(0,+∞;L2×H1),(16) ∀γ > 0, e−γ Z 0 eγsk(u(s),d(s))k2 1×2ds ≤C2,∀ ≥0,(17) e i ying h∂ u, i+ ((u· ∇)u, )+(∇u,∇ )+(∇d ∆d, )=0 ∀ ∈V, ∂ d+ (u· ∇)d+ (d)−∆d= 0,|d| ≤ 1a.e. in Q u(0) = u0,d(0) = d0in Ω. In he ini e ime case (T < ∞), (17) holds e en when γ= 0, i.e. (u,d)∈L2(0, T;H1×H2). Rema k: (16) and (17) imply ha (∂ u, ∂ d)∈L4/3 loc ([0,∞); V0×L2). De ini ion 6 We say ha (u, p, d)a weak solu ion o (1)-(3) is also a s ong solu ion i k(u( ),d( ))k1×2≤C3∀ ≥0,(18) 7 ∀γ > 0, e−γ Z 0 eγsk(u(s),d(s))k2 2×3ds ≤C4,∀ ≥0,(19) e i ying he ollowing sys em a.e. in Q:    ∂ u+ (u· ∇)u−ν∆u+∇p=−∇d ∆d,∇ · u= 0, ∂ d+ (u· ∇)d= ∆d− (d),|d| ≤ 1. (20) In he ini e ime case (T < ∞), again γ= 0 can be aken in (19). u Rema k: (18) and (19) imply ha o all γ > 0 and o all ≥0: e−γ Z 0 eγs|∂ u(s)|2 2ds ≤C5,(21) |∂ d( )|2≤C6, e−γ Z 0 eγsk∂ d(s)k2 1ds ≤C7,(22) and e−γ Z 0 eγs|∇p(s)|2 2ds ≤C8.(23) Theo em 7 (Exis ence and uniqueness o he ini ial- alued p oblem) (1) Le Ωbe a bounded domain in R3wi h bounda y ∂Ωo class C1,1. Assume (u0,d0)∈ H×H1wi h |d0| ≤ 1in Ω,h∈L∞(0,+∞;H3/2(∂Ω)) wi h |h| ≤ 1on Σand ∂ h∈ L∞(0,+∞;L2(∂Ω)), e i ying he compa ibili y condi ion d0|∂Ω=h(0). Then he e exis s a weak solu ion (u,d)o (1)-(3) in [0,+∞)which e i ies (16)-(17) wi h cons an s C1,C2 independen o ν o each ν≥1/2, and he ollow ene gy inequali y: |u( )|2 2+|∇b d( )|2 2+ 2 Z 0ν|∇u|2 2+|∆b d|2 2 ≤ |u0|2 2+ 2 Z 0ZΩ (d)·∆b d−(u· ∇)d·∂ e d(24) whe e he li ing unc ion e dis de ined as in Sec ion 3. (2) I mo eo e , ∂Ωis o class C2,1,(u0,d0)∈H1×H2wi h k(u0,d0)kH1×H2≤M0,h∈ L∞(0,+∞;H5/2(∂Ω)) and ∂ h∈L∞(0,+∞;H1/2(∂Ω)), o each ν≥ν0, wi h ν0=ν0(M0,h, ∂ h), he e exis s an unique s ong solu ion o (1)-(3) in [0,+∞), which e i ies (18) and (19) wi h cons an s C3,C4independen o ν. (3) I (u1,d1)is a weak solu ion o (1)-(3) which e i ies he ene gy inequali y (24) and (u2,d2) is a s ong solu ion o (1)-(3), hen bo h solu ions coincide. P oo : (1) In he p oo o his pa a semi-Gale kin me hod will be used. Le {wi}n≥1 a “special” basis o V o med by eigen unc ions o he S okes p oblem (∇wi,∇ ) = λi(wi, )∀ ∈V,wi∈V,wi h kwikL2= 1, λi%+∞. Le Vmbe he ini e-dimensional subspace spanned by {w1,w2,...,wn}. 8 Fo each m≥1, we say ha (um,dm) is an app oxima e solu ion, i um: [0,+∞)7→ Vmand dm: [0,+∞)7→ H2wi h b dm=dm−e dand e d he li ing unc ion gi en in Sec ion 3 (in pa icula , om egula i y hypo hesis o h, one has ha e d∈L∞(H2) and e d ∈L∞(L2)), and he ollowing a ia ional o mula ion holds:                  (∂ um( ), m) + ((um( )· ∇)um( ), m) + ν(∇um( ),∇ m) +(∇d m( )∆dm( ), m)=0 ∀ m∈Vm,a.e. in , ∂ b dm( )+(um( )· ∇)dm( )=∆b dm( )− (dm( )),|dm| ≤ 1,a.e. in Q, um(0) = u0m=Pm(u0),dm(0) = d0in Ω. (25) He e, Pm:H7→ Vmdeno es he usual o hogonal p ojec o om Hon o Vm. In pa icula , u0m→u0in L2. The exis ence and uniqueness o local in ime solu ion o (25) (in QT, o small enough T) is p o ed in he Appendix. Mo eo e , one has he es ima es (independen o m): umbounded in L∞(0, T;H)∩L2(0, T;V) and dmbounded in L∞(0, T;H1)∩L2(0, T;H2). This su ices o con ol nonlinea e ms and o pass o he limi in (25). The e o e, we ge a weak solu ion o ini ial- alued p oblem (1)-(3) in [0, T]. Nex , o ex end he solu ion o whole [0,+∞) we will p o e ha he app oxima e solu ions (um( ),dm( )) a e bounded in [0,+∞). By using he li ing (5) (Sec ion 2), he app oxima e p oblem (25) can be ew i en as ollows:                  (∂ um( ), m) + ((um( )· ∇)um( ), m) + ν(∇um( ),∇ m) +(∇d m( )∆dm( ), m)=0 ∀ m∈Vm,a.e. , ∂ b dm( )+(um( )· ∇)dm( )=∆dm( )− (dm( )) −∂ e d( ) in Q, um(0) = u0m=Pm(u0),dm(0) = d0in Ω. (26) No ice ha b dmand e da e no he same unc ions in (25) and (26) espec i ely, since he li ing unc ions u nished in (5) o (10) a e di e en , bu he unc ion dmdoes no change. F om (7), one has in pa icula d d |um|2 2+|∇b dm|2 2+C0(|um|2 2+|∇b dm|2 2)≤2| (dm)|2 2+|∂ e d|2 2≤C, (27) whe e C0= min{2ν P,1 P}and Pis a Poinca ´e cons an ( o each ν≥1/2, C0= 1/P a cons an independen o ν). In he las es ima e we ha e used ha | (dm)|2 2is bounded in L∞(0,+∞) and |∂ e d|2 2∈L∞(0,+∞). Mul iplying by eC0 , d d eC0 (|um|2 2+|∇b dm|2 2)≤CeC0 and in eg a ing in [0, ] we ha e |um( )|2 2+|∇b dm( )|2 2≤e−C0 |u0m|2 2+|∇b d0|2 2+C(1 −eC0 )≤ |u0|2 2+|∇b d0|2 2+C(28) o all ≥0, wi h C > 0 a cons an independen o ν, hence (16) holds wi h a cons an C1 independen o ν o all ν≥1/2. 9 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] 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. [3] B. Climen -Ezque a, F. Guill´en-Gonz´alez, M.A. Rojas-Meda Regula ime- ep oduc i e so- lu ions o gene alized Boussinesq model wi h Neumann bounda y condi ions o empe a u e. P oc. R. Soc. A (2007) 463, 2153-2164. [4] D. Cou and, S. Shkolle , Well posedness o he ull E icsen-Leslye model o nema ic liquid c ys als, No e C.R.A.S, . 333, Se ie I (2001), 919–924. [5] F. 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