Mathematical problems of the Q-tensor theory of nematic liquid crystals

Mathematical problems of the Q-tensor theory of nematic liquid crystals

Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Mathematical problems of the Q-tensor theory
of nematic liquid crystals
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Arghir Dani Zarnescu
Carnegie Mellon
22 March 2011
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Collaborators:
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
M. Atiyah (Edinburgh), J. Ball (Oxford), E. Feireisl (Prague),
Analogy with
Ginzburg-Landau
R. Ignat(Paris), E. Kirr(Urbana-Champaign), A. Majumdar(Oxford),
The uniform
convergence
(with A.
Majumdar)
L. Nguyen(Princeton), M. Paicu(Bordeaux), J. Robbins (Bristol),
G. Schimperna (Pavia), V. Slastikov (Bristol), M. Wilkinson (Oxford)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Liquid crystals: physics
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
A measure µ such that 0 ≤ µ(A ) ≤ 1 ∀A ⊂ S2
The probability that the molecules are pointing in a direction
contained in the surface A ⊂ S2 is µ(A )
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Liquid crystals: physics
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
A measure µ such that 0 ≤ µ(A ) ≤ 1 ∀A ⊂ S2
The probability that the molecules are pointing in a direction
contained in the surface A ⊂ S2 is µ(A )
Physical requirement µ(A ) = µ(−A ) ∀A ⊂ S2
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Liquid crystals: physics
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
A measure µ such that 0 ≤ µ(A ) ≤ 1 ∀A ⊂ S2
The probability that the molecules are pointing in a direction
contained in the surface A ⊂ S2 is µ(A )
Physical requirement µ(A ) = µ(−A ) ∀A ⊂ S2
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Liquid crystals: physics
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
A measure µ such that 0 ≤ µ(A ) ≤ 1 ∀A ⊂ S2
The probability that the molecules are pointing in a direction
contained in the surface A ⊂ S2 is µ(A )
Physical requirement µ(A ) = µ(−A ) ∀A ⊂ S2
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Landau-de Gennes Q-tensor reduction and earlier
theories
Q-tensor theory
Arghir Dani
Zarnescu
Z
S2
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
p ⊗ p d µ(p ) −
Q=
1
Id
3
Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Landau-de Gennes Q-tensor reduction and earlier
theories
Q-tensor theory
Arghir Dani
Zarnescu
Z
S2
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
p ⊗ p d µ(p ) −
Q=
1
Id
3
Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor
The Q-tensor is:
isotropic is Q = 0
uniaxial if it has two equal eigenvalues
biaxial otherwise
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Landau-de Gennes Q-tensor reduction and earlier
theories
Q-tensor theory
Arghir Dani
Zarnescu
Z
S2
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
p ⊗ p d µ(p ) −
Q=
1
Id
3
Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor
The Q-tensor is:
isotropic is Q = 0
uniaxial if it has two equal eigenvalues
biaxial otherwise
Ericksen’s theory (1991) for uniaxial Q-tensors which can be
written as
!
Q (x ) = s (x ) n(x ) ⊗ n(x ) −
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
1
Id ,
3
s ∈ R, n ∈ S2
Landau-de Gennes Q-tensor reduction and earlier
theories
Q-tensor theory
Arghir Dani
Zarnescu
Z
S2
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
p ⊗ p d µ(p ) −
Q=
1
Id
3
Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor
The Q-tensor is:
isotropic is Q = 0
uniaxial if it has two equal eigenvalues
biaxial otherwise
Ericksen’s theory (1991) for uniaxial Q-tensors which can be
written as
!
Q (x ) = s (x ) n(x ) ⊗ n(x ) −
1
Id ,
3
s ∈ R, n ∈ S2
Oseen-Frank theory (1958) take s in the uniaxial
representation to be a fixed constant s+
Arghir Dani Zarnescu
Q-tensor theory
Landau-de Gennes Q-tensor reduction and earlier
theories
Q-tensor theory
Arghir Dani
Zarnescu
Z
S2
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
p ⊗ p d µ(p ) −
Q=
1
Id
3
Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor
The Q-tensor is:
isotropic is Q = 0
uniaxial if it has two equal eigenvalues
biaxial otherwise
Ericksen’s theory (1991) for uniaxial Q-tensors which can be
written as
!
Q (x ) = s (x ) n(x ) ⊗ n(x ) −
1
Id ,
3
s ∈ R, n ∈ S2
Oseen-Frank theory (1958) take s in the uniaxial
representation to be a fixed constant s+
Arghir Dani Zarnescu
Q-tensor theory
Q-tensors: beyond liquid crystals
Q-tensor theory
Carbon nanotubes:
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
LC states of DNA:
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Active LC: cytoskeletal filaments and motor proteins
Arghir Dani Zarnescu
Q-tensor theory
The big picture
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Full system-weak solutions (work in progress) with G.
Schimperna (Pavia) and E. Fereisl (Prague)
Coupled Navier-Stokes Q-tensor system(weak
solutions and regularity in 2D) (submitted) with M.
Paicu (Bordeaux)
Dynamics for the Q-tensor system only-statistical
dynamics (work in progress) with Ph.D. student M.
Wilkinson (Oxford) and E. Kirr (Urbana-Champaign)
Stationary elliptic system
singular perturbation problem(published) with A.
Majumdar(Oxford) and refinement (submitted) with L.
Nguyen (Princeton)
existence and energetic stability(work in progress)
index 12 -defects (with V. Slastikov (Bristol) and J.
Robbins (Bristol)
existence and energetic stability(work in progress)
radial hegehog (with R. Ignat (Paris), L. Nguyen and
V. Slastikov)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Q-harmonic maps: topological issues (submitted and
in progress) with M. Atiyah, J. Ball (Oxford)
Arghir Dani Zarnescu
Q-tensor theory
Topological aspects: line fields (constrained
Q-tensors)
Q-tensor theory
Arghir Dani
Zarnescu
Use functions Q : Ω → {s+ n ⊗ n − 13 Id } with s+ = s+ (α, T , b , c )
and n ∈ S2 .
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Theorem
(JM Ball-AZ) In a simply connected domain a line field in W 1,p is
orientable if p ≥ 2. For p < 2 there exist line fields that are not
orientable.
Arghir Dani Zarnescu
Q-tensor theory
A complex topology
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Theorem (JM Ball, AZ) Let G be a domain with holes in the plane. A
line field in W 1,p for p ≥ 2 is orientable if and only if its restriction to
the boundary is orientable.
Arghir Dani Zarnescu
Q-tensor theory
A domain with holes and partial boundary
conditions
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Comparison between line fields and vector fields
global energy minimizers
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Beyond constrained Q-tensors
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Energy functionals in the three theories:
Landau-de Gennes:
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
with Q (x ) : Ω → {M ∈ R3 , M = M t , trM = 0} a Q-tensor
Ericksen’s theory:
Z
FE [s , n] =
s (x )2 |∇n(x )|2 + k |∇s (x )|2 + W0 (s (x )) dx
f B (Q ) =
Ω
with (s , n) ∈ R × S2
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Beyond constrained Q-tensors
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Energy functionals in the three theories:
Landau-de Gennes:
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
with Q (x ) : Ω → {M ∈ R3 , M = M t , trM = 0} a Q-tensor
Ericksen’s theory:
Z
FE [s , n] =
s (x )2 |∇n(x )|2 + k |∇s (x )|2 + W0 (s (x )) dx
f B (Q ) =
Ω
with (s , n) ∈ R × S2
Oseen-Frank:
Z
FOF [n] =
ni ,k (x )ni ,k (x ) dx ,
Ω
Arghir Dani Zarnescu
Q-tensor theory
n ∈ S2
Beyond constrained Q-tensors
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Energy functionals in the three theories:
Landau-de Gennes:
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
with Q (x ) : Ω → {M ∈ R3 , M = M t , trM = 0} a Q-tensor
Ericksen’s theory:
Z
FE [s , n] =
s (x )2 |∇n(x )|2 + k |∇s (x )|2 + W0 (s (x )) dx
f B (Q ) =
Ω
with (s , n) ∈ R × S2
Oseen-Frank:
Z
FOF [n] =
ni ,k (x )ni ,k (x ) dx ,
Ω
Arghir Dani Zarnescu
Q-tensor theory
n ∈ S2
Analogy with Ginzburg Landau
Q-tensor theory
Arghir Dani
Zarnescu
We can denote f̃B (Q ) = fB (Q ) − min fB (Q ) and we have
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Z
F̃LG [Q ] =
Ω
f̃B (Q )
|∇Q |2
+
dx
2
L
f̃B (Q ) ≥ 0 and f̃B (Q ) = 0 ⇔ Q ∈ {s+ n ⊗ n − 31 Id } with
s+ = s+ (α, T , b , c ) and n ∈ S2 .
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Analogy with Ginzburg Landau
Q-tensor theory
Arghir Dani
Zarnescu
We can denote f̃B (Q ) = fB (Q ) − min fB (Q ) and we have
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Z
F̃LG [Q ] =
Ω
f̃B (Q )
|∇Q |2
+
dx
2
L
f̃B (Q ) ≥ 0 and f̃B (Q ) = 0 ⇔ Q ∈ {s+ n ⊗ n − 31 Id } with
s+ = s+ (α, T , b , c ) and n ∈ S2 .
Experimentally L << 1
Ginzburg-Landau: u : Rn → Rn and energy functional
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Analogy with Ginzburg Landau
Q-tensor theory
Arghir Dani
Zarnescu
We can denote f̃B (Q ) = fB (Q ) − min fB (Q ) and we have
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Z
F̃LG [Q ] =
Ω
f̃B (Q )
|∇Q |2
+
dx
2
L
f̃B (Q ) ≥ 0 and f̃B (Q ) = 0 ⇔ Q ∈ {s+ n ⊗ n − 31 Id } with
s+ = s+ (α, T , b , c ) and n ∈ S2 .
Experimentally L << 1
Ginzburg-Landau: u : Rn → Rn and energy functional
Z
FGL [u] =
Ω
|∇u(x )|2
1
+ 2 (1 − |u|2 )2 dx
2
ε
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Analogy with Ginzburg Landau
Q-tensor theory
Arghir Dani
Zarnescu
We can denote f̃B (Q ) = fB (Q ) − min fB (Q ) and we have
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Z
F̃LG [Q ] =
Ω
f̃B (Q )
|∇Q |2
+
dx
2
L
f̃B (Q ) ≥ 0 and f̃B (Q ) = 0 ⇔ Q ∈ {s+ n ⊗ n − 31 Id } with
s+ = s+ (α, T , b , c ) and n ∈ S2 .
Experimentally L << 1
Ginzburg-Landau: u : Rn → Rn and energy functional
Z
FGL [u] =
Ω
|∇u(x )|2
1
+ 2 (1 − |u|2 )2 dx
2
ε
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Analogy with Ginzburg Landau
Q-tensor theory
Arghir Dani
Zarnescu
We can denote f̃B (Q ) = fB (Q ) − min fB (Q ) and we have
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Z
F̃LG [Q ] =
Ω
f̃B (Q )
|∇Q |2
+
dx
2
L
f̃B (Q ) ≥ 0 and f̃B (Q ) = 0 ⇔ Q ∈ {s+ n ⊗ n − 31 Id } with
s+ = s+ (α, T , b , c ) and n ∈ S2 .
Experimentally L << 1
Ginzburg-Landau: u : Rn → Rn and energy functional
Z
FGL [u] =
Ω
|∇u(x )|2
1
+ 2 (1 − |u|2 )2 dx
2
ε
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Boundary conditions and the W 1,2 convergence
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
We denote
!
Qmin = {s+
1
n(x ) ⊗ n(x ) − Id , n ∈ S2 }
3
so that f̃B (Q ) = 0 ⇔ Q ∈ Qmin .
Boundary conditions:
Qb (x ) = s+ nb (x ) ⊗ nb (x ) − 13 Id , nb (x ) ∈ C ∞ (∂Ω, S2 )
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Boundary conditions and the W 1,2 convergence
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
We denote
!
Qmin = {s+
1
n(x ) ⊗ n(x ) − Id , n ∈ S2 }
3
so that f̃B (Q ) = 0 ⇔ Q ∈ Qmin .
Boundary conditions:
Qb (x ) = s+ nb (x ) ⊗ nb (x ) − 13 Id , nb (x ) ∈ C ∞ (∂Ω, S2 )
Recall: F̃LG =
R
Ω
|∇Q |2
2
+
f̃B (Q )
L
dx
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Boundary conditions and the W 1,2 convergence
Q-tensor theory
Arghir Dani
Zarnescu
We denote
!
Qmin = {s+
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
1
n(x ) ⊗ n(x ) − Id , n ∈ S2 }
3
so that f̃B (Q ) = 0 ⇔ Q ∈ Qmin .
Boundary conditions:
Qb (x ) = s+ nb (x ) ⊗ nb (x ) − 13 Id , nb (x ) ∈ C ∞ (∂Ω, S2 )
Recall: F̃LG =
Q
(L )
→Q
(0)
R
|∇Q |2
Ω
in W
2
1,2
+
f̃B (Q )
L
dx
on a subsequence, as L → 0.
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Boundary conditions and the W 1,2 convergence
Q-tensor theory
Arghir Dani
Zarnescu
We denote
!
Qmin = {s+
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
1
n(x ) ⊗ n(x ) − Id , n ∈ S2 }
3
so that f̃B (Q ) = 0 ⇔ Q ∈ Qmin .
Boundary conditions:
Qb (x ) = s+ nb (x ) ⊗ nb (x ) − 13 Id , nb (x ) ∈ C ∞ (∂Ω, S2 )
Recall: F̃LG =
Q
(L )
→Q
(0)
R
|∇Q |2
Ω
in W
2
1,2
+
f̃B (Q )
L
dx
on a subsequence, as L → 0.
(0)
Q a global energy minimizer of
W 1,2 (Ω, Qmin )
|∇Q |2
R
Ω
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
2
in the space
Boundary conditions and the W 1,2 convergence
Q-tensor theory
Arghir Dani
Zarnescu
We denote
!
Qmin = {s+
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
1
n(x ) ⊗ n(x ) − Id , n ∈ S2 }
3
so that f̃B (Q ) = 0 ⇔ Q ∈ Qmin .
Boundary conditions:
Qb (x ) = s+ nb (x ) ⊗ nb (x ) − 13 Id , nb (x ) ∈ C ∞ (∂Ω, S2 )
Recall: F̃LG =
Q
(L )
→Q
(0)
R
|∇Q |2
Ω
in W
2
1,2
+
f̃B (Q )
L
dx
on a subsequence, as L → 0.
(0)
Q a global energy minimizer of
W 1,2 (Ω, Qmin )
Q (0) = s+ n(0) ⊗ n(0) − 13 Id and
|∇Q |2
R
Ω
R
Ω
2
|∇Q (0) |2
2
with n(0) a global minimizer of FOF [n] =
W 1,2 (Ω, S2 )
Arghir Dani Zarnescu
Q-tensor theory
in the space
R
2
dx = 2s+
R
|∇n|2 dx in
Ω
Ω
|∇n(0) |2
2
dx
Boundary conditions and the W 1,2 convergence
Q-tensor theory
Arghir Dani
Zarnescu
We denote
!
Qmin = {s+
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
1
n(x ) ⊗ n(x ) − Id , n ∈ S2 }
3
so that f̃B (Q ) = 0 ⇔ Q ∈ Qmin .
Boundary conditions:
Qb (x ) = s+ nb (x ) ⊗ nb (x ) − 13 Id , nb (x ) ∈ C ∞ (∂Ω, S2 )
Recall: F̃LG =
Q
(L )
→Q
(0)
R
|∇Q |2
Ω
in W
2
1,2
+
f̃B (Q )
L
dx
on a subsequence, as L → 0.
(0)
Q a global energy minimizer of
W 1,2 (Ω, Qmin )
Q (0) = s+ n(0) ⊗ n(0) − 13 Id and
|∇Q |2
R
Ω
R
Ω
2
|∇Q (0) |2
2
with n(0) a global minimizer of FOF [n] =
W 1,2 (Ω, S2 )
Arghir Dani Zarnescu
Q-tensor theory
in the space
R
2
dx = 2s+
R
|∇n|2 dx in
Ω
Ω
|∇n(0) |2
2
dx
The uniform convergence: obtaining uniform W 1,∞
bounds-the general mechanism
Q-tensor theory
Recall the energy functional:
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
The energy inequality:
Z
Z
f̃B (Q )
f̃B (Q )
1
|∇Q |2
1
|∇Q |2
+
dx ≤
+
dx
r Br 2
L
R BR 2
L
f B (Q ) =
for r < R
Bochner-type inequality:
−∆eL ≤ eL2
where eL =
R
Br
|∇Q |2
2
+
f̃B (Q )
L
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
The uniform convergence: obtaining uniform W 1,∞
bounds-the general mechanism
Q-tensor theory
Recall the energy functional:
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
The energy inequality:
Z
Z
f̃B (Q )
f̃B (Q )
1
|∇Q |2
1
|∇Q |2
+
dx ≤
+
dx
r Br 2
L
R BR 2
L
f B (Q ) =
for r < R
Bochner-type inequality:
−∆eL ≤ eL2
R
2
f̃ (Q )
|
where eL = B |∇Q
+ BL
2
r
Combine the two into a standard rescaling and blow-up
argument. Troubles near the boundary.
Arghir Dani Zarnescu
Q-tensor theory
The uniform convergence: obtaining uniform W 1,∞
bounds-the general mechanism
Q-tensor theory
Recall the energy functional:
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
The energy inequality:
Z
Z
f̃B (Q )
f̃B (Q )
1
|∇Q |2
1
|∇Q |2
+
dx ≤
+
dx
r Br 2
L
R BR 2
L
f B (Q ) =
for r < R
Bochner-type inequality:
−∆eL ≤ eL2
R
2
f̃ (Q )
|
where eL = B |∇Q
+ BL
2
r
Combine the two into a standard rescaling and blow-up
argument. Troubles near the boundary.
Remark that uniform W 1,∞ bounds cannot hold in the whole Ω
Arghir Dani Zarnescu
Q-tensor theory
The uniform convergence: obtaining uniform W 1,∞
bounds-the general mechanism
Q-tensor theory
Recall the energy functional:
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Z
FLG [Q ] =
Ω
L
Qij ,k (x )Qij ,k (x ) + fB (Q (x )) dx
2
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
The energy inequality:
Z
Z
f̃B (Q )
f̃B (Q )
1
|∇Q |2
1
|∇Q |2
+
dx ≤
+
dx
r Br 2
L
R BR 2
L
f B (Q ) =
for r < R
Bochner-type inequality:
−∆eL ≤ eL2
R
2
f̃ (Q )
|
where eL = B |∇Q
+ BL
2
r
Combine the two into a standard rescaling and blow-up
argument. Troubles near the boundary.
Remark that uniform W 1,∞ bounds cannot hold in the whole Ω
Arghir Dani Zarnescu
Q-tensor theory
Heuristical bookkeeping: work with spectral
quantitities
Q-tensor theory
Arghir Dani
Zarnescu
Examples
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
f B (Q ) =
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 ) ×
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 )
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Heuristical bookkeeping: work with spectral
quantitities
Q-tensor theory
Arghir Dani
Zarnescu
Examples
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
f B (Q ) =
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 ) ×
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 )
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 ) Qij
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Heuristical bookkeeping: work with spectral
quantitities
Q-tensor theory
Arghir Dani
Zarnescu
Examples
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
f B (Q ) =
α(T − T ∗ ) 2 b 3 c 2 2
tr Q − tr Q +
trQ
2
3
4
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 ) ×
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 )
h
i
α(T − T ∗ )Qij − b Qil Qlj − δij tr(Q )2 + cQij tr(Q 2 ) Qij
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Apriori L ∞ bounds
Q-tensor theory
Arghir Dani
Zarnescu
!
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
2
L ∆Qij = −a Qij −b
2
1
Qip Qpj − trQ 2 δij +c 2 trQ 2 Qij , i , j = 1, 2, 3
3
Multiply by Qij sum over repeated indices and obtain:
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
L ∆(Qij Qij ) − 2LQij ,l Qij ,l = L ∆Qij Qij ≥ Lg (|Q |)
b2
def
g (|Q |) = −a 2 |Q |2 − √ |Q |3 + c 2 |Q |4
6
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Apriori L ∞ bounds
Q-tensor theory
Arghir Dani
Zarnescu
!
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
2
L ∆Qij = −a Qij −b
2
1
Qip Qpj − trQ 2 δij +c 2 trQ 2 Qij , i , j = 1, 2, 3
3
Multiply by Qij sum over repeated indices and obtain:
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
L ∆(Qij Qij ) − 2LQij ,l Qij ,l = L ∆Qij Qij ≥ Lg (|Q |)
b2
def
g (|Q |) = −a 2 |Q |2 − √ |Q |3 + c 2 |Q |4
6
On the other hand g (|Q |) > 0
for |Q | >
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
q
2
s
3 +
Apriori L ∞ bounds
Q-tensor theory
Arghir Dani
Zarnescu
!
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
2
L ∆Qij = −a Qij −b
2
1
Qip Qpj − trQ 2 δij +c 2 trQ 2 Qij , i , j = 1, 2, 3
3
Multiply by Qij sum over repeated indices and obtain:
Analogy with
Ginzburg-Landau
L ∆(Qij Qij ) − 2LQij ,l Qij ,l = L ∆Qij Qij ≥ Lg (|Q |)
The uniform
convergence
(with A.
Majumdar)
b2
def
g (|Q |) = −a 2 |Q |2 − √ |Q |3 + c 2 |Q |4
6
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
On the other hand g (|Q |) > 0
for |Q | >
q
2
s
3 +
Hence L ∆ |Q |2 (x ) > 0 for all interior points x ∈ Ω, where
|Q (x )| >
q
2
s .
3 +
Arghir Dani Zarnescu
Q-tensor theory
Apriori L ∞ bounds
Q-tensor theory
Arghir Dani
Zarnescu
!
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
2
L ∆Qij = −a Qij −b
2
1
Qip Qpj − trQ 2 δij +c 2 trQ 2 Qij , i , j = 1, 2, 3
3
Multiply by Qij sum over repeated indices and obtain:
Analogy with
Ginzburg-Landau
L ∆(Qij Qij ) − 2LQij ,l Qij ,l = L ∆Qij Qij ≥ Lg (|Q |)
The uniform
convergence
(with A.
Majumdar)
b2
def
g (|Q |) = −a 2 |Q |2 − √ |Q |3 + c 2 |Q |4
6
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
On the other hand g (|Q |) > 0
for |Q | >
q
2
s
3 +
Hence L ∆ |Q |2 (x ) > 0 for all interior points x ∈ Ω, where
|Q (x )| >
q
2
s .
3 +
Arghir Dani Zarnescu
Q-tensor theory
The uniform convergence result
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Proposition
(A. Majumdar, AZ) Let Ω ⊂ R3 be a simply-connected bounded open
set with smooth boundary. Let Q (L ) denote a global minimizer of the
energy
Z
L
F̃LG [Q ] =
Qij ,k (x )Qij ,k (x ) + f̃B (Q (x )) dx
Ω 2
with Q ∈ W 1,2 subject to boundary conditions Qb ∈ C ∞ (∂Ω), with
Qb (x ) = s+ n ⊗ n − 13 Id , n ∈ S2 . Let Lk → 0 be a sequence such
that Q (Lk ) → Q (0) in W 1,2 (Ω).
Let K ⊂ Ω be a compact set which contains no singularity of Q (0) .
Then
lim Q (Lk ) (x ) = Q (0) (x ), uniformly for x ∈ K
(1)
k →∞
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Beyond the small L limit
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
Heuristically: Q (L ) ∼ Q (0) + LR (L ) + h .o .t
Beyond the first order
term:
biaxial Q = s n ⊗ n − 31 Id + r m ⊗ m − 13 Id
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Beyond the small L limit
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
Heuristically: Q (L ) ∼ Q (0) + LR (L ) + h .o .t
Beyond the first order
term:
biaxial Q = s n ⊗ n − 31 Id + r m ⊗ m − 13 Id
The constrained
de Gennes’
theory (with JM
Ball)
β(Q ) = 1 −
Analogy with
Ginzburg-Landau
S. Kralj, E.G. Virga, J. Phys. A (2001)
2
6(tr(Q 3 ))
(tr(Q 2 ))
3
-biaxiality parameter
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Beyond the small L limit
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
Heuristically: Q (L ) ∼ Q (0) + LR (L ) + h .o .t
Beyond the first order
term:
biaxial Q = s n ⊗ n − 31 Id + r m ⊗ m − 13 Id
The constrained
de Gennes’
theory (with JM
Ball)
β(Q ) = 1 −
Analogy with
Ginzburg-Landau
S. Kralj, E.G. Virga, J. Phys. A (2001)
2
6(tr(Q 3 ))
(tr(Q 2 ))
3
-biaxiality parameter
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Defects:A physical experiment
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
A possible interpretation of defects in Landau-de
Gennes theory
Q-tensor theory
Arghir Dani
Zarnescu
A matrix depending smoothly on a parameter can have discontinuous eigenvectors
Example: A real analytic matrix

 1

y

Q (L ) (x , y , z ) = 
Liquid crystal
modeling
|
0
y
1
0
{z
0
0
−2
 
 
 
 + 
}
def (0)
=Q
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
0
Lx
0
0
−Lx
0
|
{z
def
= LR (1)
0
0
0




}
A possible interpretation of defects in Landau-de
Gennes theory
Q-tensor theory
A matrix depending smoothly on a parameter can have discontinuous eigenvectors
Example: A real analytic matrix
Arghir Dani
Zarnescu

 1

y

Q (L ) (x , y , z ) = 
Liquid crystal
modeling
|
 
 
 
 + 
0
0
−2
{z
}
On y = 0 and L , 0 we have eigenvectors 
Analogy with
Ginzburg-Landau




 
  0
 
,  1
0
 
  1
1
 

−1 ,  1
 




 


1
0
0
 
  0
 
,  0
1
 
  0
,  0
 
0
Lx
0
0
−Lx
0
|
{z
0
0
0




}
def
def (0)
=Q
The constrained
de Gennes’
theory (with JM
Ball)
The uniform
convergence
(with A.
Majumdar)
y
1
0
0
= LR (1)






. On x = 0 and L , 0 we have eigenvectors 

1
−1
0


1   0 



 
1 ,  0 . If L = 0 we have that 
 are eigenvectors everywhere.


 
1
0
0
1
0
Consistent with the interpretation of PG De Gennes (Comptes Rendus Hebdomadaires des Seances de l’Academie des
sciences, Serie B, 275(9) 1972)
A. Sonnet, A. Killian and S. Hess Phys. Rev. E 52 (1995)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory



,
A possible interpretation of defects in Landau-de
Gennes theory
Q-tensor theory
A matrix depending smoothly on a parameter can have discontinuous eigenvectors
Example: A real analytic matrix
Arghir Dani
Zarnescu

 1

y

Q (L ) (x , y , z ) = 
Liquid crystal
modeling
|
 
 
 
 + 
0
0
−2
{z
}
On y = 0 and L , 0 we have eigenvectors 
Analogy with
Ginzburg-Landau




 
  0
 
,  1
0
 
  1
1
 

−1 ,  1
 




 


1
0
0
 
  0
 
,  0
1
 
  0
,  0
 
0
Lx
0
0
−Lx
0
|
{z
0
0
0




}
def
def (0)
=Q
The constrained
de Gennes’
theory (with JM
Ball)
The uniform
convergence
(with A.
Majumdar)
y
1
0
0
= LR (1)






. On x = 0 and L , 0 we have eigenvectors 

1
−1
0


1   0 



 
1 ,  0 . If L = 0 we have that 
 are eigenvectors everywhere.


 
1
0
0
1
0
Consistent with the interpretation of PG De Gennes (Comptes Rendus Hebdomadaires des Seances de l’Academie des
sciences, Serie B, 275(9) 1972)
A. Sonnet, A. Killian and S. Hess Phys. Rev. E 52 (1995)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory



,
The regularity of eigenvectors in our case
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Proposition
(i) Let Q (L ) be a global minimizer of F̃LG [Q ]. Then there exists a set
of measure zero, possibly empty, Ω0 in Ω such that the eigenvectors
of Q (L ) are smooth at all points x ∈ Ω \ Ω0 . The uniaxial-biaxial ,
isotropic-uniaxial or isotropic-biaxial interfaces are contained in Ω0 .
(ii) Let K ⊂ Ω be a compact subset of Ω that does not contain
singularities of the limiting map Q (0) . Let n(L ) denote the leading
eigenvector of Q (L ) Then, for L small enough
(depending
on K ), the
leading eigendirection n(L ) ⊗ n(L ) ∈ C ∞ K ; M 3×3 .
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
The geometry of the Q-tensor (I)
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
Recall ∆Q (L ) =
The limit Q
−a 2 Q − b 2 Q 2 −
tr(Q 2 )
3
Id + c 2 Q tr(Q 2 )
∼ Q + LR + h .o .t
= s+ n ⊗ n − 13 Id and Q (0) ∆Q (0) = ∆Q (0) Q (0)
Heuristically: Q
(0)
1
L
(L )
(0)
(1)
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
The geometry of the Q-tensor (I)
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Recall ∆Q (L ) =
The limit Q
−a 2 Q − b 2 Q 2 −
tr(Q 2 )
3
Id + c 2 Q tr(Q 2 )
∼ Q + LR + h .o .t
= s+ n ⊗ n − 13 Id and Q (0) ∆Q (0) = ∆Q (0) Q (0)
Heuristically: Q
(0)
1
L
(L )
(0)
(1)
After some geometry...
P
∆Q (0) = − s42 (Q (0) − 16 Id ) 3α=1 (∇α Q (0) )2
+
Equivalently the initial equation is
P
∆QLk 0 = − s42 (QLk 0 − 16 Id ) 3α=1 (∇α QLk 0 )2 + Lk 0 RLk 0
+
def
We want to determine R (1) = limL →0 L1 (Q (L ) − Q (0) )
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
The geometry of the Q-tensor (I)
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Recall ∆Q (L ) =
The limit Q
−a 2 Q − b 2 Q 2 −
tr(Q 2 )
3
Id + c 2 Q tr(Q 2 )
∼ Q + LR + h .o .t
= s+ n ⊗ n − 13 Id and Q (0) ∆Q (0) = ∆Q (0) Q (0)
Heuristically: Q
(0)
1
L
(L )
(0)
(1)
After some geometry...
P
∆Q (0) = − s42 (Q (0) − 16 Id ) 3α=1 (∇α Q (0) )2
+
Equivalently the initial equation is
P
∆QLk 0 = − s42 (QLk 0 − 16 Id ) 3α=1 (∇α QLk 0 )2 + Lk 0 RLk 0
+
def
We want to determine R (1) = limL →0 L1 (Q (L ) − Q (0) )
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
The geometry of the Q-tensor (II)
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Lemma
For a point Q ∈ S∗ , the tangent and normal spaces to S∗ at Q are
sym
TQ S∗ = {Q̇ ∈ M3×3 :
Analogy with
Ginzburg-Landau
(TQ S∗ )⊥ = {Q ⊥ ∈ M3sym
×3
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
1
s+ Q̇ = Q̇ Q + Q Q̇ },
3
: Q ⊥ Q = Q Q ⊥ }.
Lemma
Let Q be a point in S∗ . For X , Y in TQ S∗ and Z, W in (TQ S∗ )⊥ , we
have XY + YX , ZW + WZ ∈ (TQ S∗ )⊥ and XZ + ZX ∈ TQ S∗ .
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
(2)
(3)
The equation for the next term in the asymptotic
expansion
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Proposition
(L. Nguyen, AZ) Assume that QLk ∈ C 2 (Ω, S0 ) is a critical point of ILk , and that as Lk
→ 0, QLk converges on compact subsets of Ω in C 2 -norm to Q∗ ∈ C 2 (Ω, S∗ ) which is
a critical point of I∗ and L1 (QLk − Q∗ ) converges in C 2 -norm to some Q• ∈ C 2 (Ω, S0 ).
(i)
Q•⊥
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
k
is given by
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
k
k
If we write Q• = Q• + Q•⊥ with Q• ∈ TQ∗ S∗ and Q•⊥ ∈ (TQ∗ S∗ )⊥ , then
Q•⊥ = −
2
3
b 2s2
+
6a 2
X
1
6
1
|∇Q∗ |2 c 2 Q∗ + b 2 Id Q∗ − s+ Id −
(∇α Q∗ )2 ,
2
3
6
+ b s+
α=1
(4)
k
(ii) and Q• satisfies in Ω the equations
6c 2
|∇Q∗ |2 Q•k
+ b 2 s+
k i
k
k
1
4 h
− 2 ∇Q•k ∇Q∗ + ∇Q∗ ∇Q•k Q∗ − s+ Id − ∆Q•⊥ ,
6
s+
∆Q•k = − b 2 (Q•k Q•⊥ + Q•⊥ Q•k ) −
k k
where ∇Q• and ∆Q•⊥
respectively.
k
6a 2
k
are the tangential components of ∇Q• and ∆Q•⊥ ,
Arghir Dani Zarnescu
Q-tensor theory
(5)
Dynamical issues: statistical aspects of
coarsening
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
The quenching from the isotropic into the nematic occurs
through the creation of nematically ordered islands into the
ambient isotropic fluid.
A scaling phenomenon: the pattern of domains at a later time
looks statistically similar to that at an earlier time, up to a
time-dependent change of scale.
hTr [Q (x + r , t )Q (x , t )]i
(6)
hTr [Q (x , t )Q (x , t )]i
where the brackets h, i denote an average over x ∈ Rd and over
the initial conditions.
The statistical scaling hypothesis states that for late enough
times the correlation function C (r , t ) will assume a scaling form:
C (r , t ) =
C (r , t ) ∼ f (
r
)
L (t )
where L (t ) is the time-dependent length scale of the nematic
domains.
Arghir Dani Zarnescu
Q-tensor theory
(7)
Statistical solutions and the correlation function
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
The
flow of the energy
R Lsimplest gradient
a
b
2
2
|∇
Q
|
+
tr(
Q
)
−
tr(Q 3 ) + c4 tr2 (Q 2 ) dx
2
3
Rd 2
∂t Qij = ∆Qij + a Qij + b
2
2
Qil Qlj −
δij
3
!
tr(Q ) − c 2 Qij tr(Q 2 ), i , j = 1, 2, 3
2
Consider an averaging measure µ0 on the infinite-dimensional
functional space of initial datas, let us call it H. If
Probability that Q0 ∈ A = µ0 (A ), for a Borel set A ⊂ H then one can
rigorously define the time-dependent family of measures
def
µt (A ) = µ0 ({S (t , Q0 ) ∈ A }) = µ0 (S (t )−1 A )(where S (t , Q0 ) is the
solution with initial data Q0 at time t) and study the evolution of these
measures.
We define then C (r , t ):
R R
def
C (r , t ) =
Tr [Q (x + r )Q (x )] dx d µt (Q )
R R
Tr [Q (x )Q (x )] dx d µt (Q )
H
R3
H
Arghir Dani Zarnescu
R3
Q-tensor theory
The evolutionary equation: just a bistable gradient
system
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
High dimensional (in the domain and target space) version of:
∂t u = uxx − F 0 (u), u(t , x ) : R+ × R → R
Choosing a suitable initial data, our system reduces to the scalar
equation above.
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
The equation is: ut = uxx − au + bu2 − c 2 u3 with a > 0 in the
shallow quenching and a < 0 in the deep quenching
We continue referring just to the shallow quenching regime!
Arghir Dani Zarnescu
Q-tensor theory
The evolutionary equation: just a bistable gradient
system
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
High dimensional (in the domain and target space) version of:
∂t u = uxx − F 0 (u), u(t , x ) : R+ × R → R
Choosing a suitable initial data, our system reduces to the scalar
equation above.
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
The equation is: ut = uxx − au + bu2 − c 2 u3 with a > 0 in the
shallow quenching and a < 0 in the deep quenching
We continue referring just to the shallow quenching regime!
Arghir Dani Zarnescu
Q-tensor theory
Statistical solutions as averages of individual
solutions and the individual behaviour I
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
(A. Zlatos, JAMS, 19 (2006)) For initial data u(0, x ) = χ[−L ,L ] we have
that there exists L0 > 0 so that
If L < L0 u(t , x ) → 0 uniformly on compacts
If L = L0 u(t , x ) → U uniformly on compacts, with U a stationary
solution
If L > L0 u(t , x ) → 1 uniformly on compacts
R R
J (n ) (n )
Σj =1 θj H R3 Tr [Q (x + r )Q (x )] dx d δQ (n)
j
Cn ( r , t ) =
→ C (r , t ) (8)
R R
[
Tr
Q
(
x
)
Q
(
x
)]
dx
d
σ
d
δ
(n)
3
H
R
Q
j
J (n ) (n )
where Σj =1 θj
= 1.
Understanding the behaviour of individual solution helps to understand
the statistical solutions. But it might not be necessary...
Arghir Dani Zarnescu
Q-tensor theory
Statistical solutions as averages of individual
solutions and the individual behaviour II
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
For small enough initial data we have a representation
2
Qij (t , x ) = Aij (Q )
e
− 4(|tx+| 1)
3/2
e a t (4π(t + 1))
2
+ wij (t , x )
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
where wij decays faster than the first term.
def
Let Cδ (r , t ) =
R
( RR3 Tr[Q (x +r )Q (x )] dx ) d δQ (t ) (Q )
. Then
Tr[Q (x )Q (x )] dx ) d δQ (t ) (Q )
H ( R3
R
H
R
|r |2
kCδ (r , t ) − e − 8(t +1) kL ∞ (dr ) = o (1) as t → ∞.
1
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Thus for small initial data, L (t ) ∼ t 2 , but this scaling only
captures the underlying brownian motion.
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Statistical solutions as averages of individual
solutions and the individual behaviour II
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
For small enough initial data we have a representation
2
Qij (t , x ) = Aij (Q )
e
− 4(|tx+| 1)
3/2
e a t (4π(t + 1))
2
+ wij (t , x )
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
where wij decays faster than the first term.
def
Let Cδ (r , t ) =
R
( RR3 Tr[Q (x +r )Q (x )] dx ) d δQ (t ) (Q )
. Then
Tr[Q (x )Q (x )] dx ) d δQ (t ) (Q )
H ( R3
R
H
R
|r |2
kCδ (r , t ) − e − 8(t +1) kL ∞ (dr ) = o (1) as t → ∞.
1
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Thus for small initial data, L (t ) ∼ t 2 , but this scaling only
captures the underlying brownian motion.
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Statistical solutions as averages of individual
solutions and the individual behaviour III
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
If u0 (x ) is spherically symmetric u0 (x ) = χB (0,R ) for R large enough we
have u(t , x ) ∼ χB (0,c̄t ) and Cδ (r , t ) ∼ P ( rt ) as t → ∞ (where P is a third
order polynomial).
Thus for some large enough initial data L (t ) ∼ t as t → ∞
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Statistical solutions as averages of individual
solutions and the individual behaviour III
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
If u0 (x ) is spherically symmetric u0 (x ) = χB (0,R ) for R large enough we
have u(t , x ) ∼ χB (0,c̄t ) and Cδ (r , t ) ∼ P ( rt ) as t → ∞ (where P is a third
order polynomial).
The constrained
de Gennes’
theory (with JM
Ball)
Thus for some large enough initial data L (t ) ∼ t as t → ∞
Analogy with
Ginzburg-Landau
Averaging over Zlatos’ initial data
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Statistical solutions as averages of individual
solutions and the individual behaviour III
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
If u0 (x ) is spherically symmetric u0 (x ) = χB (0,R ) for R large enough we
have u(t , x ) ∼ χB (0,c̄t ) and Cδ (r , t ) ∼ P ( rt ) as t → ∞ (where P is a third
order polynomial).
The constrained
de Gennes’
theory (with JM
Ball)
Thus for some large enough initial data L (t ) ∼ t as t → ∞
Analogy with
Ginzburg-Landau
Averaging over Zlatos’ initial data
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
The Q-tensors+NSE system of equations
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The flow equations:
∂t u + u∇u = ν∆u + ∇p + ∇ · τ + ∇ · σ
∇·u =0
where we have the symmetric part of the additional stress tensor:
(
The constrained
de Gennes’
theory (with JM
Ball)
!
τ = −ξ Q +
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
+2ξ(Q +
1
1
Id H − ξ H Q + Id
3
3
!
tr(Q 2 )
1
Id )QH − L ∇Q ∇Q +
Id
3
3
!
and an antisymmetric part σ = QH − HQ where
tr(Q 2 )
Id ] − cQ tr(Q 2 )
3
The equation for the liquid crystal molecules, represented by functions with
values in the space of Q-tensors (i.e. symmetric and traceless d × d matrices):
H = L ∆Q − aQ + b [Q 2 −
(∂t + u · ∇)Q − S (∇u, Q ) = ΓH
with
def
S (∇u, Q ) = (ξ D + Ω)(Q +
Arghir Dani Zarnescu
1
1
1
Id ) + (Q + Id )(ξ D − Ω) − 2ξ(Q + Id )tr(Q ∇u)
3
3
3
Q-tensor theory
Energy dissipation and weak solutions-apriori
bounds
Q-tensor theory
Arghir Dani
Zarnescu
def
Z
E (t ) =
Rd
Liquid crystal
modeling
L
a
b
c
|∇Q |2 + tr(Q 2 ) − tr(Q 3 ) + tr2 (Q 2 ) dx
2
2
3
4
|
{z
}
free energy of the liquid crystal molecules
The constrained
de Gennes’
theory (with JM
Ball)
+
1
2
|
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Z
Rd
|u|2 (t , x ) dx
{z
}
kinetic energy of the flow
is decreasing
d
dt
E (t ) ≤ 0.
Note that this does not readily provide L p norm estimates.
Proposition
For d = 2, 3 there exists a weak solution (Q , u) of the coupled system, with restrictions
c > 0 and |ξ| < ξ0 s, subject to initial conditions
Q (0, x ) = Q̄ (x ) ∈ H 1 (Rd ), u(0, x ) = ū(x ) ∈ L 2 (Rd ), ∇ · ū = 0 in D0 (Rd )
∞ (R ; H 1 ) ∩ L 2 (R ; H 2 ) and
The solution (Q , u) is such that Q ∈ Lloc
+
+
loc
∞ (R ; L 2 ) ∩ L 2 (R ; H 1 ).
u ∈ Lloc
+
+
loc
Arghir Dani Zarnescu
Q-tensor theory
(9)
Regularity difficulties: the maximal derivatives
and “the co-rotational parameter”
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Recall the system:









































(∂t + u · ∇)Q − ξD (u) + Ω(u) (Q + 31 Id ) + (Q + 31 Id ) ξD (u) − Ω(u)
1
−2ξ(Q + 3 Id )tr(Q ∇u) = ΓH
∂t u + u∇u = ν∆uα + ∇p + ∇ · QH − HQ
−∇ · ξ Q + 13 Id H + ξH Q + 13 Id
+2ξ∇ · (Q + 13 )QH − L ∇ · ∇Q ∇Q + 31 tr(Q 2 )
∇·u =0
Refined
description of
minimizers (with
Luc Nguyen)
with H = L ∆Q − aQ + b [Q 2 −
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Where’s the difficulty?
tr(Q 2 )
3
Id ] − cQ tr(Q 2 ).
Worse than Navier-Stokes
If ξ = 0 maximal derivatives only
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
Regularity difficulties: the maximal derivatives
and “the co-rotational parameter”
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Recall the system:









































(∂t + u · ∇)Q − ξD (u) + Ω(u) (Q + 31 Id ) + (Q + 31 Id ) ξD (u) − Ω(u)
1
−2ξ(Q + 3 Id )tr(Q ∇u) = ΓH
∂t u + u∇u = ν∆uα + ∇p + ∇ · QH − HQ
−∇ · ξ Q + 13 Id H + ξH Q + 13 Id
+2ξ∇ · (Q + 13 )QH − L ∇ · ∇Q ∇Q + 31 tr(Q 2 )
∇·u =0
Refined
description of
minimizers (with
Luc Nguyen)
with H = L ∆Q − aQ + b [Q 2 −
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Where’s the difficulty?
Q-tensors+NSE
(with M. Paicu)
tr(Q 2 )
3
Id ] − cQ tr(Q 2 ).
Worse than Navier-Stokes
If ξ = 0 maximal derivatives only
If ξ , 0 maximal derivatives+high power of Q
Arghir Dani Zarnescu
Q-tensor theory
Regularity difficulties: the maximal derivatives
and “the co-rotational parameter”
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Recall the system:









































(∂t + u · ∇)Q − ξD (u) + Ω(u) (Q + 31 Id ) + (Q + 31 Id ) ξD (u) − Ω(u)
1
−2ξ(Q + 3 Id )tr(Q ∇u) = ΓH
∂t u + u∇u = ν∆uα + ∇p + ∇ · QH − HQ
−∇ · ξ Q + 13 Id H + ξH Q + 13 Id
+2ξ∇ · (Q + 13 )QH − L ∇ · ∇Q ∇Q + 31 tr(Q 2 )
∇·u =0
Refined
description of
minimizers (with
Luc Nguyen)
with H = L ∆Q − aQ + b [Q 2 −
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Where’s the difficulty?
Q-tensors+NSE
(with M. Paicu)
tr(Q 2 )
3
Id ] − cQ tr(Q 2 ).
Worse than Navier-Stokes
If ξ = 0 maximal derivatives only
If ξ , 0 maximal derivatives+high power of Q
Arghir Dani Zarnescu
Q-tensor theory
Regularity difficulties: the maximal derivatives
and “the co-rotational parameter”
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Recall the system:









































(∂t + u · ∇)Q − ξD (u) + Ω(u) (Q + 31 Id ) + (Q + 31 Id ) ξD (u) − Ω(u)
1
−2ξ(Q + 3 Id )tr(Q ∇u) = ΓH
∂t u + u∇u = ν∆uα + ∇p + ∇ · QH − HQ
−∇ · ξ Q + 13 Id H + ξH Q + 13 Id
+2ξ∇ · (Q + 13 )QH − L ∇ · ∇Q ∇Q + 31 tr(Q 2 )
∇·u =0
Refined
description of
minimizers (with
Luc Nguyen)
with H = L ∆Q − aQ + b [Q 2 −
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Where’s the difficulty?
Q-tensors+NSE
(with M. Paicu)
tr(Q 2 )
3
Id ] − cQ tr(Q 2 ).
Worse than Navier-Stokes
If ξ = 0 maximal derivatives only
If ξ , 0 maximal derivatives+high power of Q
Arghir Dani Zarnescu
Q-tensor theory
The regularity result, in 2D
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Theorem
Let s > 0 and (Q̄ , ū) ∈ H s +1 (R2 ) × H s (R2 ). There exists a global a
solution (Q (t , x ), u(t , x )) of the coupled system, with restrictions
c > 0 and |ξ| < ξ0 , subject to initial conditions
Q (0, x ) = Q̄ (x ), u(0, x ) = ū(x )
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
2
∞
and Q ∈ Lloc
(R+ ; H s +2 (R2 )) ∩ Lloc
(R+ ; H s +1 (R2 )),
2
s +1
2
∞
u ∈ Lloc (R+ ; H (R ) ∩ Lloc (R+ ; H s ). Moreover, we have:
L k∇Q (t , ·)k2H s (R2 )
+
ku(t , ·)k2Hs (R2 )
e eeCt
≤ C e + kQ̄ kHs+1 (R2 ) + kūkHs (R2 )
(10)
where the constant C depends only on Q̄ , ū, a , b , c, Γ and L . If ξ = 0
the increase in time of the norms above can be made to be only
doubly exponential.
Arghir Dani Zarnescu
Q-tensor theory
The full system (in progress with E. Feireisl and
G. Schimperna)
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
Q-LC molecules, u-fluid velocity, θ-temperature
h
i Q
h
i
1
Q
1
(∂t + u · ∇)Q − (ξ + 1)∇u + (ξ − 1)∇ut (
+
Id ) − (
+
Id ) (ξ + 1)∇u + (ξ − 1)∇ut
2
2d
2
2d
∂ψ(Q )
1
1
∂ψ
Id )tr(Q ∇u) = Γ L ∆Q −
+ tr
Id
d
∂Q
d
∂Q
!
+2ξ(Q +
The constrained
de Gennes’
theory (with JM
Ball)
|
{z
}
def
=H
Analogy with
Ginzburg-Landau
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
∂t u + ∇ · (u ⊗ u) − ∇ · λ(θ)[QH − HQ ]−pId −
|
µ(θ)
2
!
(∇u + ∇ut )
{z
}
def
= T1
"
#!
1
1
1
= ∇ · λ(θ) −ξ(Q + Id )H − ξH (Q + Id ) + 2ξ(Q + )QH − L ∇Q ∇Q +f
d
|
d
d
{z
def
= T2
∇·u =0
∂t θ + ∇ · (θu) + ∇ · (J (Q , θ)∇θ)) = T1 + T2 : ∇u
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory
!
}
Q-tensor theory
Arghir Dani
Zarnescu
Liquid crystal
modeling
The constrained
de Gennes’
theory (with JM
Ball)
Analogy with
Ginzburg-Landau
THANK YOU!
The uniform
convergence
(with A.
Majumdar)
Refined
description of
minimizers (with
Luc Nguyen)
Coarsening and
statistical
dynamics (with
E. Kirr and M.
Wilkinson)
Q-tensors+NSE
(with M. Paicu)
Arghir Dani Zarnescu
Q-tensor theory