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.
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