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
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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. [] “Regula i y and
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