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
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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.
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[5] F. Guill´en-Gonz´alez, M.A. Rojas-Meda , M.A. Rod ´ıguez-Bellido Su icien condi ions o
egula i y and uniqueness o a 3D nema ic liquid c ys al model. Submi ed.
[6] F.H. Lin Nonlinea heo y o de ec s in nema ic liquid c ys als: phase ansi ion and low
phenomena, Comm. Pu e Appl. Ma h., 42 (1989), 789-814.
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Comm. Pu e Appl. Ma h., 4 (1995), 501-537.
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16