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-
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[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]
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