Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
On Nematic Liquid Crystal Flows
Fanghua Lin
Courant Institute, New York University
The University of Chicago – Carlos Meeting, Sept. 19–21,
2014
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Outline
1
Brief review of the Ericksen-Leslie model
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
2
Simplified EL system in dimensions two
3
Simplified EL system in higher dimensions
4
General Ericksen-Leslie system
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Outline
1
Brief review of the Ericksen-Leslie model
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
2
Simplified EL system in dimensions two
3
Simplified EL system in higher dimensions
4
General Ericksen-Leslie system
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Mean orientation of molecule’s optical axis,
d : Ω → S2 , Ω ⊂ Rn (n = 2, 3),
minimizes Oseen-Frank’s bulk energy functional.
Oseen-Frank density: Assume W (d, ∇d) quadratic in ∇d,
W (−d, −∇d) = W (Qd, Q∇dQ t ) = W (d, ∇d), Q ∈ O(3) ⇒
2W(d,∇d) = k1 (divd)2 + k2 (d · curld + τ )2 + k3 |d × curld|2
+(k2 + k4 )[tr(∇d)2 − (divd)2 ]
k1 , k2 , k3 > 0: splay, twist, and bending constants, k2 ≥ |k4 |,
τ ∈ R. (τ 6= 0: cholesterics; τ = 0: nematics).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
R
Observation (Oseen, Ericksen): [tr(∇d)2 − (divd)2 ] depends
on d|∂Ω (or null-Lagrange).
Equilibrium configuration: Given g : ∂Ω →R S2 , there exists a
d minimizing the energy functional W(d) = Ω W (d, ∇d), i.e.,
n
o
W(d) = min W(e) : e ∈ W 1,2 (Ω, S2 ), e = g on ∂Ω .
The Euler-Lagrange equation (not necessarily elliptic):
δW
d
d + tφ
:= t=0 W(dt ) = 0, dt =
, φ ∈ C0∞ (Ω, R3 ).
δd
dt
|d + tφ|
R
Simple case: k1 = k2 = k3 = 1, k4 = 0 ⇒ W(d) = 12 Ω |∇d|2
δW
= ∆d + |∇d|2 d = 0 (harmonic map to S2 ).
δd
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Regularity of minimizers
1. (Hardt-Lin-Kindelerhrer, 86’s) d ∈ C ∞ (Ω \ Σ, S2 ), with
dimH (Σ) < 1
It covers the point defects observed in experiments and
numerical stimulations:
d(x) =
x
: R3 → S2 (hedgehog)
|x|
2. (Open question) What’s the optimal size estimate of the
singular set Σ? Is Σ a finite set?
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Outline
1
Brief review of the Ericksen-Leslie model
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
2
Simplified EL system in dimensions two
3
Simplified EL system in higher dimensions
4
General Ericksen-Leslie system
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Hydrodynamics (Ericksen and Leslie, 1958-1968): Assume
liquid is homogeneous (density ρ = 1), u : Ω → Rn is the fluid
velocity, based on:
i) Conservation of linear momentum,
ii) Conservation of angular momentum,
iii) Incompressibility of the fluid.
Ericksen-Leslie (EL) system takes the form:
ut + u · ∇u = −∇P + ∇ · σ
∇·u = 0
δW
d×
− γ1 N − γ2 Ad = 0
δd
Fanghua Lin
On Nematic Liquid Crystal Flows
(1)
(2)
(3)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
σ is modeled by the phenomenological relation: σ = σ L + σ E .
∂W
i) σ E – the elastic (Ericksen) stress: σ E = −
· (∇d)t .
∂(∇d)
ii) σ L – the viscous (Leslie) stress (αi ’s - Leslie constants):
(
α1 (d ⊗ d : A)d ⊗ d + α2 d ⊗ N + α3 N ⊗ d
σ L (u, d) =
+α4 A + α5 d ⊗ (A · d) + α6 (A · d) ⊗ d,
A=
iii)
δW
δd
∇u + (∇u)t
∇u − (∇u)t
, N = dt + u · ∇d + ω · d, ω =
.
2
2
– first order variation of W .
γ1 = α3 − α2 , γ2 = α6 − α5
(4)
α2 + α3 = α6 − α5 (Parodi’s condition)
(5)
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
One constant approx. of W:
1
|∇d|2 ⇒
2
δW
∂W
= (∆d + |∇d|2 d),
= ∇d,
δd
∂(∇d)
W (d, ∇d) =
σE = −
∂W
· (∇d)t = −∇d ∇d = −(∇i d · ∇j d)1≤i,j≤n .
∂(∇d)
(EL) system can be written as

L

∂t u + u · ∇u + ∇P = −∇ · (∇d ∇d) + ∇ · (σ (u, d))
∇·u =0


N + γγ12 Ad = |γ11 | (∆d + |∇d|2 d) + γγ21 (d t Ad)d.
(6)
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Z
1
(|u|2 + |∇d|2 ).
Consider Ω =
Set E(t) =
2 Rn
Energy law under Parodi’s condition ( Lin–Liu; W. Wang–P.
Zhang– Z. Zhang):
Z d
2
2 2
2
E(t) +
|∆d + |∇d| d|
α4 |∇u| +
(7)
dt
|γ1 |
Rn
#
Z "
γ22
γ22
2
2
= −2
(α1 − )|A : d ⊗ d| + (α5 + α6 + )|A · d|
γ1
γ1
Rn
Rn .
≤ 0,
provided α1 , · · · , α6 satisfy
γ1 < 0, α1 −
γ22
γ2
≥ 0, α4 > 0, α5 + α6 ≥ − 2 .
γ1
γ1
Fanghua Lin
On Nematic Liquid Crystal Flows
(8)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Basic Questions:
(A) Establish, in dimensions n = 2, 3, the global existence of
Leray-Hopf type weak solutions to the Ericksen-Leslie
system (EL (6)) under general initial-boundary conditions.
(B) (Partial) regularity and uniqueness issues of suitable weak
solutions of (EL (6)).
(C) Global or local well-posedness of the (EL (6)) for rough
initial data belonging to the largest possible function
spaces.
Notation. Leray-Hopf’s weak solutions refer to any weak
solution in the energy space that satisfies a weak form of the
above energy dissipation inequality (7).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Further Simplified EL system (L–, 1989’s):
i) Neglect stretching, rigid rotation, and interacting Leslie:
σ L (u, d) = α4 A N = dt + u · ∇d.
ii)
γ1 = −1, γ2 = 0.
(EL(6)) becomes
ut + u · ∇u − α4 4u + ∇P = −∇ · (∇d ∇d)
∇·u = 0
dt + u · ∇d
(9)
(10)
2
= ∆d + |∇d| d .
|
{z
}
(11)
Remark. (9)-(11) is strongly coupling between Navier-Stokes
equation (NSE) and harmonic heat flow:
i) d constant ⇒ NSE.
ii) u = 0 ⇒ harmonic heat flow to S2 :
dt = 4d + |∇d|2 d, and |∆d · ∇d
{z = 0} (stationarity).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Classical solutions (u, d) to (9)-(11) & (13)-(14) enjoy
Energy Dissipation Inequality:
d
dt
Z
2
2
Z
(|u| + |∇d| ) ≤ −2
Ω
|
{z
}
(µ|∇u|2 + |∆d + |∇d|2 d|2 ) (12)
Ω
Proof. Multiplying (9) by u and (11) by (∆d + |∇d|2 d) and
integrating by parts a couple of times.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Static case: Oseen-Frank model
Dynamic case: Ericksen-Leslie model
Initial-boundary condition:
(u, d)t=0 = (u0 , d0 ) ∈ H × H 1 (Ω, S2 )
(u, d)∂Ω×R+ = (0, d0 ) ∈ C 2,β (∂Ω, S2 )
o
n
H 1 (Ω, S2 ) = d ∈ H 1 (Ω, R3 ) d(x) ∈ S2 a.e. x ∈ Ω .
H = C0∞ (Ω, Rn )
\
{v : ∇ · v = 0} in L2 (Ω, Rn ).
J = C0∞ (Ω, Rn )
\
{v : ∇ · v = 0} in H01 (Ω, Rn ).
Fanghua Lin
On Nematic Liquid Crystal Flows
(13)
(14)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Scaling invariance: For λ > 0, if (u, P, d) solves (9)-(11), so
does
uλ , Pλ , dλ (x, t) = λu, λ2 P, d λx, λ2 t
Dimensions:
u ∼ −1, P ∼ −2, d ∼ 0, x ∼ 1,
t ∼ 2, Pr (= Brn × [−r 2 , 0]) ∼ n + 2
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
"Naive" approach:
i) Approximate initial data (u0 , d0 ) by smooth (u0 , d0 ).
ii) Establish both lower bound of T > 0 and apriori estimates
for short time smooth solutions (u , d ) : Ω × [0, T ] → Rn × S2
to (9)-(11), in terms of the local energy profiles of (u0 , d0 ).
iii) Good news: This approach works for the critical dimension
n = 2!
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
1. ∃ (u0k , d0k ) ∈ C ∞ (Ω, R2 × S2 ), k(u0k − u0 , d0k − d0 )kL2 ×H 1 → 0.
Z
k
k
sup Er0 (u0 , d0 ) = sup max
(|u0k |2 + |∇d0k |2 ) ≤ 20 .
k
k
x∈Ω
Br0 (x)∩Ω
2. Local energy inequality and global energy inequality ⇒
∃ Tk ≥ c0 r02 , (u k , d k ) ∈ C ∞ (Ω × [0, Tk ], R2 × S2 ) satisfy

k
k
2
sup
0≤t≤c0 r02 Er0 (u (t), d (t)) ≤ 20 ,
2
R
R
 c0 r0 (|∇u k |2 + |∆d k + |∇d k |2 d k |2 ) . 1.
0
Ω
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
3. Ladyzhenskaya “≤" & Calderon-Zgymund L2 -est. on P k ⇒
Z
(|u k |4 +|∇u k |2 +|∇d k |4 +|∇2 d k |2 +|P k |2 ) . 20 .
max
x∈Ω
Br0 (x)×[0,c0 r02 ]
Ladyzhenskaya inequality (n = 2):
R
Ω |f |
4
. supx∈Ω
R
2
Ω∩Br (x) |f |
R
Ω |∇f |
2
+ r −2 supx∈Ω
Equation of P:
∆P = −div2 u ⊗ u + ∇d ∇d
Fanghua Lin
On Nematic Liquid Crystal Flows
R
Ω∩Br (x) |f |
2 2
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Lemma 1 (0 -regularity lemma: n = 2). There exists 0 > 0
2
2 1
2
2 2
2
such that if u ∈ L∞
t Lx ∩ Lt Hx (P1 , R ), d ∈ Lt Hx (P1 , S ), and
2
P ∈ L (P1 ) solves (9)-(11), satisfying
Φ(u, d, P, r ) := kukL4 (Pr ) + k∇ukL2 (Pr ) + k∇dkL4 (Pr )
+k∇2 dkL2 (Pr ) + kPkL2 (Pr ) ≤ 0 ,
(15)
for some r > 0, then (u, d) ∈ C ∞ (P r ), and
2
k(u, ∇d)kC k (P r ) ≤ C(k )0 r −k , ∀ k ≥ 0.
2
Fanghua Lin
On Nematic Liquid Crystal Flows
(16)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Theorem 2 (regularity, with J. Lin, C.Y.Wang).
2
2 1
2
2
2
2
If u ∈ L∞
t Lx ∩ Lt Hx (ΩT , R ) and d ∈ L ([0, T ], H (Ω, S )) solves
(9)-(11), then (u, d) ∈ C ∞ (Ω × (0, T ]).
If (u, d)|∂Ω = (0, d0 ) for d0 ∈ H 1 (Ω, S2 ) ∩ C 2,β (∂Ω, S2 ) with
β ∈ (0, 1), then (u, d) ∈ Cβ2,1 (Ω × (0, T ]).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Theorem 3 (existence, with J. Y. Lin, C.Y.Wang).
For u0 ∈ H, d0 ∈ H 1 (Ω, S2 ) ∩ C 2,β (∂Ω, S2 ), there exists
u ∈ L∞ ([0, ∞), H) ∩ L2 ([0, ∞), J), d ∈ L∞ ([0, ∞), H 1 (Ω, S2 ))
solve (9)-(14) such that
(i) ∃L ∈ N depending on (u0 , d0 ) and 0 < T1 < · · · < TL s.t.
\
(u, d) ∈ C ∞ (Ω × (R+ \ {Ti }Li=1 )) Cβ2,1 (Ω × (R+ \ {Ti }Li=1 )).
(ii) Each Ti can be characterized by
Z
lim max
(|u|2 + |∇d|2 )(y , t)dy ≥ 8π, ∀r > 0. (17)
t↑Ti x∈Ω
Ω∩Br (x)
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
(iii) ∃ xki → x0i ∈ Ω, tki ↑ Ti , rki ↓ 0, constant 6= ωi ∈ C ∞ (R2 , S2 )
harmonic map of finite energy such that
2
(uki , dki ) → (0, ωi ) in Cloc
(R2 × [−∞, 0]),
(
uki (x, t) = rki u(xki + rki x, tki + (rki )2 t),
dki (x, t) = d(xki + rki x, tki + (rki )2 t).
(iv) ∃ tk ↑ +∞ and harmonic d∞ ∈ C ∞ (Ω, S2 ) ∩ Cd2,β
(Ω, S2 ) s.t.
0
u(·, tk ) → 0 in H 1 (Ω), d(·, tk ) * d∞ in H 1 (Ω),
2
2
|∇d(·, tk )| dx * |∇d∞ | dx +
l
X
8πmi δxi
i=1
for some l ∈ N, {xi }li=1 ⊂ Ω, and {mi }li=1 ⊂ N.
Fanghua Lin
On Nematic Liquid Crystal Flows
(18)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Z
(v) If either
Ω
(|u0 |2 + |∇d0 |2 ) ≤ 8π or d03 ≥ 0 , then
(u, d) ∈ C ∞ (Ω × (0, +∞)) ∩ Cβ2,1 (Ω × (0, +∞)),
∃ tk ↑ +∞ and a harmonic d∞ ∈ C ∞ (Ω, S2 ) ∩ Cd2,β
(Ω, S2 )
0
s.t.
(u(·, tk ), d(·, tk )) → (0, d∞ ) in C 2 (Ω).
Theorem 4 (uniqueness, with C.Y.Wang).
For n = 2 and 0 < T < +∞, suppose that for i = 1, 2,
2
2 1
2
2
∞ 1
2 2
2
2
(ui , di ) ∈ L∞
t Lx ∩ Lt Hx (RT , R ) × (Lt Hx ∩ Lt Hx )(RT , S )
solves (9)-(11), under the same initial-boundary condition
(13)-(14). Then (u1 , d1 ) ≡ (u2 , d2 ) .
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
“Tangible" approach (Ginzburg-Landau approximation)
For > 0, consider the Ginzburg-Landau functional:
Z
Z
1
1
|∇d|2 + 2 (1 − |d|2 )2 , d : Ω → R3 .
E (d) =
e (d) :=
4
Ω
Ω 2
The system (9)-(11) becomes


ut + u · ∇u − µ∆u + ∇P = −∇ · (∇d ∇d)
∇·u =0


dt + u · ∇d = ∆d + 12 (1 − |d|2 )d.
(19)
Remark 1. L–& C. Liu (95-01’s) showed, under (13)-(14),
(i) Global well-posedness of (19) for n = 2, for n = 3 if µ 1;
(ii) Partial regularity for suitable weak solutions of (19) for n = 3,
similar to Caffarelli-Kohn-Nirenberg on NSE.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Creation of defect measures as ↓ 0:
Let (u , d ) be a sequence of solutions to (19) such that
u * u in L2 ; d * d in H 1 .
⇓
∇ · u = 0 & dt + u · ∇d = ∆d + |∇d|2 d.
But u may not solve (9), since
(∇d ∇d ) dxdt * (∇d ∇d) dxdt + ν as ↓ 0,
for some n × n matrix ν of signed Radon measures in
Ω × (0, +∞) (defect measure).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Question 3: How large is the size of support of ν? Under what
conditions is supp(ν) empty or dviν is given by a gradient?
Theorem 5 (with C.Y.Wang, preprint). For n = 2, 3, there exists
δ0 such that if
|d | ≤ 1, d3 ≥ −1 + δ, ∀ > 0
then ν ≡ 0 and (u , d ) → (u, d) strongly in L2 × H 1 .
Theorem 6 (with C.Y.Wang, preprint).
For n = 3, if (u0 , d0 ) ∈ H × H 1 (Ω, S2 ), with d03 ≥ 0, then there is
a Leray-Hopf type weak solution (u, d) : Ω × [0, +∞) → R3 × S2
of the initial and boundary value problem of the simplifed
Ericksen-Leslie system (9)-(11).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
For δ ∈ (0, 2) and L1 , L2 > 0, define

2
| )

∆d + (1−|d
d = f

2



3
|d | ≤ 1, d ≥ −1 + δ
d ∈ H 1 (Ω, R3 ) ∈ X(δ, L1 , L2 ) ⇔

kf kL2 (Ω) ≤ L1



E (d ) = R e (d ) ≤ L .
2
Ω 1 (Ω, R3 ).
Lemma 2. X(δ, L1 , L2 ) is precompact in Hloc
Sketch of proof: Assume d * d in H 1 , and
e (d ) dx * µ :=
1
|∇d|2 dx + ν
2
for some Radon measure ν ≥ 0.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
ν≡0
Step 1 (almost monotonicity). d ∈ X(δ, L1 , L2 ) satisfies
Z
Φ (R) ≥ Φ (r ) +
|x|−1 |
BR \Br
∂d 2
| , ∀ 0 < r ≤ R,
∂|x|
where
1
Φ (r ) :=
r
Z
Br
1
e (d ) − hx · ∇d , f i +
2
Fanghua Lin
Z
|x||f |2 .
Br
On Nematic Liquid Crystal Flows
(20)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Step 2 (δ0 -compactness). ∃ δ0 > 0, α0 ∈ (0, 1), C0 > 0 such
that for d ∈ X(δ, L1 , L2 ), it holds
Φ (r0 ) ≤ δ0 ⇒ d → d in H 1 (B r0 ) and ν = 0 in B r0 .
2
Proof. ω =
d
|d |
2
solves
div(|d |2 ∇ω × ω ) = div(∇d × d ) = τ × d .
⇓
 
 ω C α0 (B r0 ) ≤ C0
h
i
2

∇ω
.
∇ω
+
τ

Lp (B r )
L2 (Br )
L2 (Br ) ,
0
2
0
0
for 2 < p < 3.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Step 3 (almost monotonicity for µ)
Θ1 (µ, r ) :=
1
µ(Br ) ≤ Θ1 (µ, R) + C0 (R − r ), ∀ 0 < r ≤ R. (21)
r
⇒ Θ1 (µ) = lim Θ1 (µ, r ) exists and is USC.
r ↓0
Step 4 (concentration set)
n
o
Σ = x ∈ Ω : Θ1 (µ) ≥ δ0
is 1-rectifiable, closed subset, with H 1 (Σ) < +∞.
supp(ν) ⊂ Σ; and δ0 ≤ Θ1 (ν, x) ≤ C0 , H 1 a.e. x ∈ Σ.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Step 5 (stratification and blow-up). Pick generic x0 ∈ Σ:
lim ri−1
r ↓0
i
Z
|∇d|2 = 0; Θ1 (ν, ·) is H 1 approx. continu. at x0 ,
Bri (x0 )
and Σ has a tangent line at x0 .
Set di (x) = di (x0 + ri x), νi (A) = ri−1 ν(x0 + ri A). Then
νi * ν0 := θ0 H 1 LY , ei (di ) dx * ν0 = θ0 H 1 LY ,
for some θ0 > 0 and line Y = {(0, 0, x3 ) : x3 ∈ R}.
Z
∂di 2
(20) ⇒
|
| →0
∂x
3
B2
⇒ ∃ ω(x) = ω(x1 , x2 ) : R3 → S2−1+δ nontrivial, smooth harmonic
map. This is impossible.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Sketch of Proof of Theorem 6:
Lemma 1. Assume u ∈ L2loc (R+ , L2 ) and d ∈ L2loc (R+ , H 1 )
solve the transported GL:

1
2


 ∂t d + u · ∇d = ∆d + 2 (1 − |d | )d in Ω × R+ ,
(22)
∇ · u = 0
in Ω × R+ ,



d = g
on ∂p (Ω × R+ ).
If g ∈ H 1 (Ω, R3 ) satisfies
|g (x)| ≤ 1 and g3 (x) ≥ 0, a.e. x ∈ Ω.
Then
|d (x, t)| ≤ 1 and d3 (x, t) ≥ 0, a.e. (x, t) ∈ Ω × R+ .
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Global energy inequality:
Z h
i
1
sup sup
|u |2 + |∇d |2 + 2 (1 − |d |2 )2
2
>0 0<t<+∞ Ω
Z ∞Z h
2 i
1
+2
|∇u |2 + ∆d + 2 (1 − |d |2 )d Ω
Z 0
≤
|u0 |2 + |∇d0 |2 := C0
(23)
Ω
⇒ ∃ p > 3 such that for 0 < T < +∞,
h
sup k∂t u k 5
5
2
>0
+k∂t d k
L 4 ([0,T ],L 4 )+L ([0,T ],H −1 )+L2 ([0,T ],W −1,p )
i
5
5
L2 ([0,T ],L2 )+L 4 ([0,T ],L 4 )
Fanghua Lin
< +∞.
On Nematic Liquid Crystal Flows
(24)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
2
2 1
3
Aubin-Lions’ Lemma ⇒ ∃ u ∈ L∞
t Lx ∩ Lt Hx (Ω × R+ , R ) and
∞
1
2
d ∈ Lt Hx (Ω × R+ , S ) such that
(
(u , d ) → (u, d) in L2loc (Ω × R+ ),
(∇u , ∇d ) * (∇u, ∇d) in L2t L2x (Ω × R+ ).
Fatou’s lemma ⇒
Z ∞
Z 2
1
lim inf ∆d + 2 (1 − |d |2 )d ≤ C0 .
→0
0
Ω
(25)
For Λ >> 1, define
Z 2
n
o
1
T
2
GΛ := t ∈ [0, T ] : lim inf ∆d + 2 (1 − |d | )d (t) ≤ Λ ,
→0
Ω
C
0
BΛT = [0, T ] \ GΛT ⇒ BΛT ≤
.
Λ
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Claim 1. For any t ∈ GΛT , it holds
1
d (t) → d(t) in Hloc
(Ω), e (d (t)) dx *
1
|∇d(t)|2 dx.
2
(26)
Proof. For t ∈ GΛT , since
{d (t)} ⊂ X(C0 , Λ, 1; Ω),
by the precompactness theorem, (26) holds.
d
|d |
Claim 2. |d (x, t)| ≥ 12 , and ω :=
1 (Ω)).
→ d in L2 (GΛT , Hloc
2, 5
Claim 3. For t ∈ GΛT , ω (t) ∈ Wloc4 (Ω, S2 ) and
≤ C δ0 , K , ∀K ⊂⊂ Ω.
ω (t) 2, 5
W
4 (K )
Fanghua Lin
On Nematic Liquid Crystal Flows
(27)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Claim 4. It holds
∇ω − ∇d L2 (K ×GΛT )
→ 0, ∀K ⊂⊂ Ω.
(28)
2, 5
1 (Ω) compact, H 1 (Ω) ⊂ H −2 (Ω).
Proof. Wloc4 (Ω) ⊂ Hloc
loc
loc
Applying Aubin-Lions’ Lemma yields Claim 4.
Claim 5. It holds
Z
∇|d |2 + 1 (1 − |d |2 )2 → 0, ∀K ⊂⊂ Ω.
2
K ×GΛT
Combining (28) and (29) yields
Z
2
+
∇d − ∇d 2
T
L
K ×GΛ
Fanghua Lin
K ×GΛT
1
(1 − |d |2 )2 → 0.
2
On Nematic Liquid Crystal Flows
(29)
(30)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Claim 6. It holds
2
∇d − ∇d 2
L
Z
≤ 2 sup
t>0
Ω
Z
K ×BΛT
+
K ×BΛT
1
(1 − |d |2 )2
2
e (d )(t) BΛT ≤ CΛ−1 .
Putting all these estimates together yields we have
Z
i
1
lim
|∇d − ∇d|2 + 2 (1 − |d |2 )2 ≤ CΛ−1 .
→0 K ×[0,T ]
Fanghua Lin
On Nematic Liquid Crystal Flows
(31)
(32)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Well-posedness for rough initial data:
Definition 1. f ∈ BMO(Rn ) ⇔ f ∈ L1 (Rn ) satisfies
loc
h i
n1 Z
o
f
:= sup inf
|f − c| < +∞.
n
BMO(Rn )
x∈Rn ,r >0 c∈R r
Br (x)
f ∈ BMO−1 (Rn ) ⇔ f ∈ L1 (Rn ) and ∃ (f1 , · · · , fn ) ∈ BMO(Rn )
loc
such that
n
X
∂fi
,
f =
∂xi
i=1
h i
f
BMO
−1
(Rn )
= inf
n
n
nX
X
∂fi o
[fi ]BMO(Rn ) : f =
.
∂xi
i=1
Fanghua Lin
i=1
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
Theorem 7(C.Y.Wang et al) ∃ 0 > 0 s.t. if
(u0 , d0 ) ∈ BMO−1 (Rn ) × BMO(Rn ) has
h
u0
i
h
i
+
d
≤ 0 ,
0
−1
BMO (Rn )
BMO(Rn )
(33)
∃ ! (u, d) ∈ C ∞ (Rn × (0, +∞), Rn × S2 ) of (9)-(11), with
(u, d)t=0 = (u0 , d0 ).
Remark 2. (i) For n ≥ 2, (33) holds, provided
Z
2−n
sup
r
(|u0 |2 + |∇d0 |2 ) ≤ 20 .
x∈Rn ,r >0
Br (x)
(ii) Reduces to Koch-Tataru’s classical theorem on NSE.
(iii) Answer an open question by Sverak on harmonic heat flow.
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
General Ericksen-Leslie system in 2D:
Consider the Ericksen-Leslie system in R2 :

L

∂t u + u · ∇u + ∇P = −∇ · (∇d ∇d) + ∇ · (σ (u, d))
∇·u =0


N + γγ21 Ad = |γ11 | (∆d + |∇d|2 d) + γγ12 (d t Ad)d.
(34)
Assume
γ1 = α3 − α2 , γ2 = α6 − α5
(35)
α2 + α3 = α6 − α5 (Parodi’s condition)
(36)
and
γ1 < 0, α1 −
γ22
γ2
≥ 0, α4 > 0, α5 + α6 ≥ − 2 .
γ1
γ1
Fanghua Lin
On Nematic Liquid Crystal Flows
(37)
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
The 0 -regularity lemma also hold for (34):
Theorem 8 (existence of Ericksen-Leslie, with J. Huang, C. Y.
Wang).
For n = 2, assume (35), (36), and (37). For any u0 ∈ L2 (R2 )
with ∇ · u0 = 0 and d0 ∈ He10 (R2 , S2 ), there exists a global weak
solution (u, d) to EL (34) along with the initial condition (u0 , d0 ),
such that
(i) ∃ T0 = 0 < T1 < · · · < TL < TL+1 = +∞ such that
2
2
(u, d) ∈ C ∞ (R2 × ((0 + ∞) \ {Ti }L+1
i=0 ), R × S ).
Fanghua Lin
On Nematic Liquid Crystal Flows
Brief review of the Ericksen-Leslie model
Simplified EL system in dimensions two
Simplified EL system in higher dimensions
General Ericksen-Leslie system
2
2 1
2
∞ 1
2
(ii) u ∈ L∞
t L ∩ Lt H (R × (0, +∞)), d ∈ Lt He0 (R ), and
d∈
\
L2t ([T1 , Ti+1 − δ], He20 (R2 )), ∀i = 1, · · · , L + 1.
δ>0
(iii) Each Ti can be characterized by
Z
(|u|2 + |∇d|2 )(t) ≥ 8π, ∀r > 0.
lim max
t↑Ti x∈R2
Br (x)
(iv) E(0) ≤ 8π or d03 ≥ 0 ⇒ (u, d) ∈ C ∞ (R2 × (0, +∞)).
Fanghua Lin
On Nematic Liquid Crystal Flows