Stability for nematic liquid crystals with stretching terms

STABILITY FOR NEMATIC LIQUID CRYSTALS WITH STRETCHING TERMS B. CLIMENT-EZQUERRA, F. GUILL´ EN-GONZ´ ALEZ, M. A. RODR´ IGUEZ-BELLIDO Depa men 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-mail: b[email p o ec ed], guil[email p o ec ed], [email p o ec ed] Feb ua y 27, 2009 Dedica ed o he memo y o Vale y S. Melnik We s udy a nema ic c ys al model appea ing in [Liu e al.,2007] modeling s e ching e ec s depending on he di e en shape o mic oscopic molecules o he ma e ial, unde pe iodic bounda y condi ions. The aim o he p esen a icle is wo old: o ex end he esul s gi en in [Sun & Liu, 2009], o a model wi h mo e comple e s e ching e ms and o ob ain some s a- bili y and asymp o ic s abili y p ope ies o his model. Keywo ds: Nema ic Liquid C ys al sys em, asymp o ic s abili y, s abili y, s e ching e ec s, exis ence, egula i y. Ma hema ics Subjec Classi ica ions (2000): 35B65, 35Q35, 76D03 1. In oduc ion The Nema ic Liquid C ys al sys em is a Na ie - S okes ype model o incomp essible luids espec o he mac oscopic a iables, ha akes in o ac- coun he c ys allini y o he mic oscopic molecules o he ma e ial. I can be ob ained coupling Na ie - S okes equa ions wi h he Ginzbu g-Landau equa- ions, being i s unknowns he solenoidal eloci y u( , x), he p essu e o he luid p( , x), and he di- ec o ield d( , x), ha ep esen s he o ien a ion o he liquid c ys al molecules. Mo eo e , we sup- pose ha he luid is con ined in a domain Ω ⊂R3. We deal wi h an E icksen-Leslie ype o mula- ion. A simpli ied model was analyzed by F. H. Lin & C. Liu in [Lin & Liu, 1995]. In ac , his model is a penalized one depending on he Ginzbu g-Landau unc ion: (d) = 1 2|d|2−1d, whe e |d|deno es he euclidean no m in R3and  > 0 is a penaliza ion pa ame e . This penaliza- ion unc ion has a po en ial s uc u e, i. e. he e exis s he unc ion F(d) = 1 42|d|2−12such ha (d) = ∇d(F(d)) o all d∈R3. We deno e Q= (0,+∞)×Ω and Σ = (0,+∞)× ∂Ω, whe e Ω ⊂R3is a smoo h enough domain and ∂Ω i s bounda y. We conside he EDP sys em ap- pea ing in [Liu e al.,2007; sys em (1.9), p. 1187], ha eads as: (LC)     D u−ν∆u+∇p−λ∇ · σe=0in Q, ∇ · u= 0 in Q, D d+γw=0in Q, 1 2B. Climen -Ezque a e al. being D u=∂ u+ (u· ∇)u he ma e ial de i a i e o u, σe=−(∇d) ∇d−βw d −(1 + β)d w (1) he elas ic s ess enso (β∈R) and w=−∆d+ (d) he Eule -Lag ange sys em de i ed om he mini- miza ion p oblem espec o he elas ic ene gy Ee(d) = 1 2ZΩ |∇d|2+ZΩ F(d).(2) The e m D d=D d+C(d,∇u) desc ibes a gene al de i a i e con aining he ma- e ial de i a i e D d=∂ d+ (u· ∇)dand he quad a ic e m C(d,∇u) = β(∇u)d+ (1 + β)(∇u) d modeling he so-called s e ching e ec s, depending on he o m o he molecules [Liu e al., 2007]. In ac , he cons an β=−αis associa ed wi h he aspec a io o he ellipsoid pa icles. The case o αnea o 1 co esponds o od like pa icles ( hen he anspo is pu ely co a ian s e ching), he case o αnea o 0 co esponds o disc like pa icles ( hen he anspo is an i-s e ching) and he case o αnea o 1/2 co esponds o he sphe ical shape ( he anspo is he igid o a ion o he cen e o he mass). Finally, ν > 0 is he luid iscosi y, λ > 0 is he elas ici y cons an and γ > 0 is a elaxa ion in ime cons an . The heo e ical analysis o a simpli ied model wi hou s e ching e ec s, i.e o C(d,∇u) = 0 and he co esponding elas ic enso σe=−(∇d) ∇d, was made in [Lin & Liu, 1995] ob aining exis ence o global weak solu ion, i. e. u∈L∞(0, T;L2(Ω)) ∩L2(0, T ;H1(Ω)), d∈L∞(0, T;H2(Ω)) ∩L2(0, T ;H3(Ω)), o all T > 0, and he exis ence (and uniqueness) o local s ong solu ion, i. e. u∈L∞(0, T∗;H1(Ω)) ∩L2(0, T∗;H2(Ω)), d∈L∞(0, T∗;H2(Ω)) ∩L2(0, T∗;H3(Ω)), wi h T∗≤T(small enough) o T∗=T( o each T > 0) o big enough iscosi y coe icien νo o wo-dimensional domains. All hese p e ious e- sul s a e gi en o he ime-independen Di ichle bounda y da a: u=0,d=hon Σ, (h6=h( )) and o he ini ial- alue bounda y p oblem wi h ini- ial condi ion: u| =0 =u0,d| =0 =d0in Ω. (3) When ime-dependen Di ichle da a o dis con- side ed (h=h( )), he exis ence o weak ime- pe iodic solu ion, ha is solu ions ob ained by changing (3) by u(0) = u(T) and d(0) = d(T), is ob ained in [Climen -Ezque a e al.,]. The s ong egula i y up o in ini e ime o big enough iscosi y νjoin ly wi h he s ong egu- la i y o ime-pe iodic solu ions a e ob ained in [Climen -Ezque a e al.,]. The esul s co esponding o he ini ial- alue bounda y p oblem a e ex ended in [Lin & Liu, 2000] o a much mo e comple e model espec o he dissipa i e enso and consid- e ing he pa icula s e ching e ec s o he case o sphe ical molecules, i.e. aking β=−1/2 in (1). Recen ly, a liquid c ys al model wi h a s e ch- ing e m o he case o od like pa icles ( aking β=−1 in (1)) and pe iodic bounda y condi ions o bo h uand dhas been s udied in [Sun & Liu, 2009], ob aining global weak solu ion and local s ong so- lu ion (which is global o la ge enough iscosi y) The aim o he p esen a icle is wo old: o ex- end he las esul s o [Sun & Liu, 2009] o a model wi h mo e comple e s e ching e ms and o ob ain some s abili y and asymp o ic s abili y p ope ies o his model. 2. Gene al F amewo k. Assume ha we ha e he ollowing si ua ion, a.e. ∈( 0,+∞): E( ), F( )≥0, E0( ) + F( )≤0.(4) Then, E∈Cb[ 0,+∞), is a dec easing unc ion and he e exis s lim →+∞E( ) = E∞≥0. S abili y o Nema ic Liquid C ys als wi h S e ching Te ms 3 On he o he hand, F∈L1( 0,+∞), ha is, Z+∞ 0 F( )d < +∞. In his case, o any δ > 0, he e exis s a la ge enough ime ∗ 1= ∗ 1(δ)≥ 0such ha : Z+∞ ∗ 1 F( )d ≤δ. (5) In pa icula , we can say ha o each δ > 0 he e exis s a la ge enough ime ∗ 1(δ)≥ 0such ha 1 τZ +τ F( )d ≤δ τ,∀τ > 0,∀ ≥ ∗ 1(δ).(6) Lemma 2.1. Le F∈L1( 0,+∞),F≥0in ( 0,+∞), sa is ying (6). Then, ∀δ > 0,∀ ≥ ∗ 1(δ) and ∀τ > 0 he e exis s a ime ¯ ∈[ , +τ]such ha : F(¯ )≤2δ τ.(7) Indeed, he se o poin s ¯ ∈[ , +τ]sa is ying (7) has measu e ≥τ/2. P oo . We ocus on he p oo in he in e al [ ∗ 1, ∗ 1+ τ]. The p oo o ano he in e al o leng h τcon- ained in [ ∗ 1,+∞) is simila . Indeed, we de ine: A={s∈[ ∗ 1, ∗ 1+τ]/ F(s)≥2δ τ}. The e o e, ZA F( )d +ZAc F( )d ≤δ, and hus 2δ τ|A| ≤ δ⇒ |A| ≤ τ 2. Tha is, |Ac| ≥ τ/2. Now, we assume ha he ollowing di e en ial inequali y o F( ) holds: F0( )≤C2(F( )3+ 1).(8) Lemma 2.2. Le F∈L1( 0,+∞)be a unc ion sa is ying he di e en ial inequali y (8). Fo any ε < 1, i F( 0)≤ε/3, hen F( )≤ε∀ ∈ [ 0, 0+T∗(ε)], whe e T∗(ε) = ε 3C2 . P oo . We a gue by con adic ion: Suppose ha he e exis s a ime 1∈[ 0, 0+T∗(ε)] such ha F( )< ε in [ 0, 1) and F( 1) = ε. Then, om eq. (8) we ob ain ha F0<2C2en [ 0, 1]. In eg a ing in [ 0, 1], we ge : F( 1)< F( 0)+2C2( 1− 0) ≤F( 0)+2C2T∗(ε) ≤ε 3+ 2C2 ε 3C2 =ε. This ac con adic s he s a ing hypo hesis. 2.1. Asymp o ic s abili y. Theo em 2.3. Le ε < 1, and F∈L1(0,+∞), F≥0, such ha bo h (8) and inequali y (6) hold o δ=ε2 36C2 , ∗ 1= ∗ 1(δ)and τ=T∗(ε) 2. Then, F( )≤ε, ∀ ≥ ∗ 2= ∗ 1+T∗(ε) 2= ∗ 1(δ)+ ε 6C2 .(9) Rema k 2.4. In pa icula , F∈W1,1( ∗ 2,+∞),→ C[ ∗ 2,+∞). P oo . We a gue by con adic ion: Assume ha he e exis s a ime e > ∗ 2such ha F( )> ε. We conside he in e al [e −T∗(ε)/2,e ]⊂[ ∗ 1,+∞). F om Lemma 2.1 we conclude ha o each in- e al o leng h T∗(ε)/2 con ained in [ ∗ 1,+∞) and ∀ ≥ ∗ 1 he e exis s a ime ¯ 1∈[e −T∗(ε)/2,e ] such ha : F(¯ 1)≤2δ τ=ε2/(18C2) ε/(6C2)=ε 3. Thus, applying Lemma 2.2, one e i ies: F( )≤ε, ∀ ∈[¯ 1,¯ 1+T∗(ε)]. Obse e ha e ∈[¯ 1,¯ 1+T∗(ε)], which gi es us o con adic ion. Co olla y 2.5. Le F∈L1( 0,+∞)be a unc ion sa is ying eq. (8). Then, F( )is a unc ion asymp- o ically s able o 0, ha is, lim →+∞F( )=0. 4B. Climen -Ezque a e al. 2.2. S abili y un il in ini e ime. I we assume ha : (H1) E( 0)≤δ(ε) = ε2 36C2 , hen, om (4) we ge : ∀ 1> 0 Z 1 0 F( )d ≤E( 0)−E( 1)≤δ(ε) In ac , one has (5) o ∗ 1(ε) = 0. Then, applying Theo em 2.3, we ob ain: F( )≤ε∀ ≥ 0+ε 6C2= 0+T∗(ε) 2(10) I , mo eo e , (H2) F( 0)≤ε 3, hen applying Lemma 2.2, we ge : F( )≤ε∀ ∈[ 0, 0+T∗(ε)] (11) In summa y, assuming (H1) and (H2), one has: E( )≤δ(ε), F( )≤ε, ∀ ≥ 0. 3. The Nema ic Liquid C ys al Model 3.1. Weak es ima es I we conside bo h u( ) and w( ) as es unc ions in he u-sys em and d-sys em o (LC) espec i ely, aking in o accoun he equali y: ∇ · ((∇d) ∇d) = −(∇d) w+∇Ee(d), we ob ain: 1 2 d d ku( )k2 L2(Ω) +νk∇u( )k2 L2(Ω) +−λZΩ [(u· ∇)d]·wdx −λZΩβw d+ (1 + β)d w:∇udx= 0 (12) and d d 1 2k∇d( )k2 L2(Ω) +F(d)( )+γkwk2 L2(Ω) +ZΩ [(u· ∇)d]·wdx+ZΩ C(d,∇u)·wdx= 0 (13) o any bounda y condi ions o (u,d) gi en in he In oduc ion ( ha is, Di ichle , Neumann o pe i- odic o d). Then, adding (12) o (13) mul iplied by λ, he las wo e ms o (12) and (13) cancel and he so-called ene gy equali y holds: d d 1 2ku( )k2 L2(Ω) +λEe(d( )) +νk∇u( )k2 L2(Ω) +λγkw( )k2 L2(Ω) = 0 (14) ( ecall ha Ee(d) is gi en in (2)). No e ha , de ining he ime unc ions E( ) and F( ) as: E( ) = 1 2ku( )k2 L2(Ω) +λEe(d( )),(15) F( ) = νk∇u( )k2 L2(Ω) +γλkw( )k2 L2(Ω), (16) he inequali y (4) is deduced om (14). As a con- sequence, we can deduce he exis ence o weak so- lu ions o he p oblem. Mo eo e , by using his weak egula i y and he H2(Ω) and H3(Ω) egu- la i y o he ellip ic p oblem −∆d+ (d) = w wi h app op ia e bounda y condi ions, we can de- duce [Climen -Ezque a e al.,]: kdkH2(Ω) ≤CkwkL2(Ω) + 1, kdkH3(Ω) ≤CkwkH1(Ω) + 1. (17) In he nex sec ion, we will use epea edly hese es ima es. 3.2. S ong es ima es In his sec ion, we only conside he pe iodic bound- a y condi ions case o all a iables (u, p, d). Tak- ing bo h −∆uand −λ∆was es unc ions in he u-sys em and in he d-sys em o (LC) espec i ely, one can ob ain ([Sun & Liu, 2009]): d d k∇uk2 L2(Ω) +νk∆uk2 L2(Ω) ≤ k(u· ∇)uk2 L2(Ω) +Cλ2 νk(∇d) wk2 L2(Ω) +λZΩβwd + (1 + β)dw ∇(∆u)dx (18) S abili y o Nema ic Liquid C ys als wi h S e ching Te ms 5 and λZΩ ∇∂ d:∇wdx+λγk∇wk2 L2(Ω) +λZΩ ∇((u· ∇)d) : ∇wdx +λZΩ ∇C(d,∇u) : ∇wdx= 0 (19) Obse e ha he i s e m o (19) can be ew i en as: ZΩ ∇∂ d:∇wdx=−ZΩ ∆(∂ d)wdx =ZΩ ∂ w w dx−ZΩ 0 (d)(∂ d)wdx =1 2 d d kwk2 L2(Ω) −ZΩ 0 (d)(∂ d)wdx (20) Now, using he d-sys em o (LC), ha is, ∂ d= −(u· ∇)d−C(d,∇u)−γw, one has: −ZΩ 0 (d)(∂ d)wdx =ZΩ 0 (d) ((u· ∇)d+C(d,∇u) + γw)wdx ≤εk∆uk2 L2(Ω) +k∇wk2 L2(Ω) +CεF3( )+1(21) o εsmall enough. The mo e nonlinea e ms o he las e m o (19) a e manipula ed as ollows (he e, he pe iodic bounda y condi ions a e again applied): λZΩβ∇(∇u)d+ (1 + β)∇(∇u) d:∇wdx ≤ −λZΩ ∇(∆u) : βwd + (1 + β)dw dx +CkD2ukL2(Ω)k∇dkL6(Ω)kwkL3(Ω) (22) No e ha he i s e m on he igh -hand side o (22) cancels wi h he las e m in (18). The e- maining pa o he las e m o (19) can be w i en as: λZΩ (β(∇d· ∇)u:∇w +(1 + β)(∇w· ∇)u:∇d)dx ≤C(λ, β)k∇wkL2(Ω)k∇dkL6(Ω)k∇ukL3(Ω) (23) being C(λ, β) a cons an depending on λand β. The e o e, adding (18) o (19) and aking in o accoun es ima es (20)-(23), we ob ain: d d k∇uk2 L2(Ω) +λkwk2 L2(Ω) +νk∆uk2 L2(Ω) +λγk∇wk2 L2(Ω) ≤ k(u· ∇)uk2 L2(Ω) +Cλ2 νk(∇d) wk2 L2(Ω) +λZΩ ∇((u· ∇)d) : ∇wdx +CF3( )+1 +CkD2ukL2(Ω)k∇dkL6(Ω)kwkL3(Ω) +C(λ, β)k∇wkL2(Ω)k∇dkL6(Ω)k∇ukL3(Ω) =P6 i=1 Ii (24) Using Sobole ’s inequali ies, he es ima es o he Ii- e ms can be summa ized as ollows: I1≤ kuk2 L6(Ω)k∇uk2 L3(Ω) ≤εk∆uk2 L2(Ω) +Cεk∇uk6 L2(Ω) I2≤ k∇dk2 L6(Ω)kwk2 L3(Ω) ≤εk∇wk2 L2(Ω) +Cεkwk6 L2(Ω) I3≤ k∇ukL3(Ω)k∇dkL6(Ω)k∇wkL2(Ω) +kukL6(Ω)k∆dkL3(Ω)k∇wkL2(Ω) =I31 +I32 whe e e m I31 as he same ype o es ima es as e m I6. I32 ≤ kukL6(Ω)k∆dkL3(Ω)k∇wkL2(Ω) ≤εk∇wkL2(Ω) +Cεk∇uk4 L2(Ω)kwk2 L2(Ω) I5≤εk∆uk2 L2(Ω) +k∇wk2 L2(Ω)+Cεkwk6 L2(Ω) I6≤εk∆uk2 L2(Ω) +k∇wk2 L2(Ω) +Cεkwk4 L2(Ω)k∇uk2 L2(Ω) Now, we in oduce unc ion G( ) de ined as: G( ) = νk∆uk2 L2(Ω) +λγk∇wk2 L2(Ω) (25) Obse e ha , inally, (24) can be w i en in he ollowing o m: F0( ) + G( )≤C1 + F3( ), 6B. Climen -Ezque a e al. being F( ) and G( ) he unc ions de ined in (16) and (25) espec i ely. The e o e, unc ion F( ) sa - is ies (8). 4. Applica ions o he Gene al F amewo k o Nema ic Liquid C ys al. 4.1. Asymp o ic s abili y. Le (u0,d0)∈H1(Ω) ×H2(Ω) be wo gi en unc- ions and (u( ),d( )) a weak solu ion o sys em (LC) wi h pe iodic bounda y condi ions and ini ial da a (u0,d0). Since (4) and (8) hold, applying he esul s om Sec ion 2, one has: (E( )↓E∞(≥0) in R, ↑+∞ F( )→0 in R, ↑+∞ hence:          E0( )→0 in R ↑+∞ u( )→0 in H1 0(Ω) ↑+∞ w( )→0 in L2(Ω) ↑+∞ Mo eo e , o each subsequence j↑+∞, he e ex- is s a subsequence ( jk)⊂( j) such ha : d( jk)*¯ din H2(Ω)-weak o k↑+∞. being ¯ da c i ical poin o he elas ic ene gy Ee(d), ha is, a solu ion o he s a iona y p oblem: −∆¯ d+ ε(¯ d) = 0in Ω, wi h pe iodic bounda y condi ions on ∂Ω. No e ha , E∞=λ 2|∇¯ d( )|2 2+ 2 ZΩ F(¯ d( ))=λEe(¯ d) ha is, e e y possible limi o he di ec o ield ¯ d when ↑+∞is a c i ical poin o he elas ic ene gy and all hese possible limi s ha e he same elas ic ene gy E∞. 4.2. S abili y o cons an di ec o ields. I (u0,d0) a e such ha : (H1) 1 2ku0k2 L2(Ω) +λ 2k∇d0k2 L2(Ω) +λZΩ F(d0)≤δ(ε) (in pa icula , kdc e −d0k2 H1(Ω) ≤C ε2 o a con- s an ec o ¯ dc e wi h |¯ dc e|= 1) and (H2) νk∇u0k2 L2(Ω) +γλkw0k2 L2(Ω) ≤ε 3, whe e w0=−∆d0+ (d0) hen o each ≥ 0, applying he esul s om Sec ion 2 one has: 1 2ku( )k2 L2(Ω)+λ 2k∇d( )k2 L2(Ω)+λZΩ F(d( )) ≤δ(ε) and νk∇u( )k2 L2(Ω) +γλkw( )k2 L2(Ω) ≤ε 3. Acknowledgemen s The au ho s ha e been pa ially suppo ed by P ojec s MTM2006-07932 and P06-FQM-02373. Re e ences Climen -Ezque a, B., Guill´en-Gonz´alez, F. & Rojas-Meda , M.A. [2006] “Rep oduc i i y o a nema ic liquid c ys al model,” Z. Angew. Ma h. Phys. 576, No. 6, 984-998. Climen -Ezque a, B., Guill´en-Gonz´alez, F. & Mo eno-I abe e, M. J. 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