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Nematic Liquid Crystals. Existence and Uniqueness of Periodic Solutions to Ericksen-Leslie Equations

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Nematic Liquid Crystals.
Existence and Uniqueness
of Periodic Solutions to
Ericksen-Leslie Equations
G.A. Chechkin,
Department of Differential Equations Faculty of Mechanics and Mathematics
M.V. Lomonosov Moscow State University
T.S. Ratiu,
École polytechnique f édérale de Lausanne, CH-1015 Lausanne, Switzerland & IMISS
M.V. Lomonosov Moscow State University
M.S. Romanov,
Department of Differential Equations Faculty of Mechanics and Mathematics
M.V. Lomonosov Moscow State University
V.N. Samokhin,
Moscow State University of Printing Art 2A, Pryanishnikova ul.
Introduction
Liquid crystals were discovered in 1888 by an Austrian botanist
Friedrich Reinitzer, but the debates about their existence and properties have
continued until the mid XX. Only in 1940, V.N. Tsvetkov formulated modern
concepts of physics of liquid crystals, using viscosity, dielectric and
diamagnetic anisotropy of liquid crystals. Tsvetkov created a general theory
of the mesomorphic state. He created methods allowed to understand the
dynamic properties of the mesophases. In 1963, American James Fergason
used liquid crystals in practical purposes, for detecting thermal fields. After
that, interest in liquid crystals has increased dramatically.
Scientists say thermotropic liquid cristals (formed by heating
the solid) or liotropic liquid cristals (formed in mixtures of solids with solvents).
The thermotropic liquid cristals may be nematics or smectics (see Fig. 1).
In all these types of LC the orientation of dipole molecules in
some direction defined by a unit vector which is called "director".
The study of liquid crystals nematodynamics (hydrodynamics
of nematic liquid crystals) is of our interest.
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Fig. 1. The structure of Nematic (left) and Smectic (right) liquid cristals
Applied questions of the theory of liquid crystals occur in many
fields, including the printing industry. Visualization of products, improving
of image resolution are connected in particular with behavior of liquid crystals.
1. Problem setup
Subject of our research is Ericksen-Leslie system






=

p

(
n) x  F  f , divu = 0
u
u

j

n
x

j
(1)
 
J
q
g
G
2
=
,
|
|=
1
n

n

h

n

Here u is the velocity vector, n is the director;  > 0 is the viscosity,
J > 0 is the moment of inertia; F( x, t ), G( x, t ) are external forces. Terms f and
g corresponds the dissipative part of the stress tensor and dissipative part of
internal body force respectively and depends on
The function
defined as
u , n and their derivatives.
 (n, n ) is called Oseen-Z öcher-Frank free energy and is
 (n, n) = K1n = curln 

1
K11 (divn) 2  K 22 (n  curln) 2  K 33 | n  curln |2  ;

2
h=


(
)x .
n
n x j
j
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The pressure p determine by condition
divu = 0 and the
lagrangian multiplier 2 q by condition | n |= 1.
We are interested in non-dissipative case, i.e.
g = 0, f = 0. Also
K1 = 0, thus the fluid is supposed to be nematic one, and for simplicity we
study one-constant approximation
K11 = K 22 = K 33 = K > 0.
In this case system (1) could be replaced with the system
u  u = p  ( Kn x n ) x  F , divu = 0,
(2)
J  =  K n  n  G ,
(3)
n =   n
(4)
j
j
where  is a new unknown vector function (see [1]).
This system is connected with Ericsen-Leslie one in following way:
if the initial conditions satisfies identities
| n( x, 0) |= 1, n( x, 0)   ( x, 0)
then for any
t >0
| n | 1,  = n  n , 2q = n  h  J | |2
and (2) – (4) turns into (1).
Consider a domain
QT = (0, T )  ,
where  =  2 /[0,1]2 is two-dimensional torus. In this domain
we study system (2) – (4) with initial conditions
(5)
u(0, x ) = u 0 ,  (0, x ) =  0 , n(0, x) = n 0 .
Here u , , n are supposed to be unknown vector-functions,
an unknown scalar function; J, K,  are fixed positive numbers.
p is
2. Notations and definitions
Here and later we use the following notations:
the full time derivative of
f ; bold b stands for a 3- dimensional vector
(b1; b 2 ; b3 ); ai bi means the sum

2
ab;
i =1 i i
Sol () = {v :    3 such as
QT = (0, T )  
141
f  f t  u j f x is
j
v  C  (), div v = 0};
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Sol (QT ) = {v  C  (QT ) : t v(t , )  Sol ()};
Sol2 () is a closure of
Sol ()
in the norm L2 ();
Sol2m () is a closure of
Sol ()
in the norm W2m ();
Definition 1. Quadruple (u, , n, p) is a strong solution of
Q = (0, T )   if u is a vector field of the class
L2 (0, T ; Sol ()), ut  L2 (QT ); n is a vector field from L (0, T ;W21 ()),
 t  L (0, T ; L2 ); n is a vector field from L (0, T ;W22 ()); p  L2 (QT );
u, n, n satisfies initial conditions (5), i.e.
(u, n, )  (u 0 , n 0 , 0 ) weakly in
L2 () as t  0, and the equations (2)–(4) take place almost everywhere.
problem (2)–(5) in domain
2
2
(2)–(5).
Our aim is to prove the existence of the solution for the problem
Theorem 1.
Suppose
Let F, G be equal to zero .
u 0  Sol22 ( ),  0  W21 ( ), n 0  W22 ().
Then for some T > 0 the solution (in terms of Definition 1) of
problem (2)–(4) does exist.
3. Galerkin-type approximations
Select two sequences of subspaces
E 1  E 2   and
2
k
F 1  F 2   such as  E k is dense in Sol2 ( ) and  F is dense in
W22 ( ) (and, consequently, in
W21 () ).
Since embeddings W21 ()  L2 () and Sol21 ()  Sol2 () are
both compact,
 : Sol ()  Sol () and  is a self-adjoint operator in these
spaces, the eigenbases of the operator
 in Sol2 () and L2 () do exist.
Denote E k and F k to be a linear span of first k eigenfunctions in
corresponding spaces. Constructed subspaces consist of smooth functions
and thus are dense in
142
Sol22 ( ) and W22 ( ) respectively.
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Proposition 1.
projection of initial data
Suppose k is fixed and
(u 0 k , 0, k , n 0,k ) is a
(u 0 , 0 , n 0 ) on E k  F k  F k .
Then for some T > 0 a solution (u k , k , n k )  C (0, T ; E k  F k  F k )
of the problem
(u k ,  )t = (ukl u k , x ,  )   ( u k ,  )  K ( n k  n k ,  ), (6)
l
J ( k ,  ) t = ( Ju kl n k , x ,  )  K ( n k  n k ,  ),
(7)
(nk , )t = (ukl n k , x , )  ( k  n k , ),
(8)
(u k , k , n k ) |t =3 D 0 = 3(u0 k , 0 k , n 0 k )
(9)
l
l
does exist, where the identities above take place for all
(u, v) = u  v dx is a scalar multiplication in
  E k ,  ,  F k ,
L2 ().

The system above could be regarded as a Cauchy problem for
ordinary differential equation
X t = 3Df ( X ) in 3k-dimensional space with
continuous function in the right-hand side. Due to Cauchy-Peano theorem
the problem has a solution for T > 0 small enough.
(u k , n k , n k ) in
The next step is to get uniform estimation of
some appropriate norm.
4. Energy conservation
First of all we need the following proposition
For all t > 0 we have
Lemma 1.
t
|u

2
k
2
2
(t ) |  J |  k (t ) |  K | nk (t ) | dx  2   | u |2 dxdt =
0
2
2
2
  | u 0 k |  | 0 k |  | n 0 k | dx.

Proof . In this section we will use u instead of u k for more
comfortable notification.
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In equations (6)–(7) we substitute
( ,  , ) = (u k , k ,  K n k ). Since  : F k  F k ,
this substitution is legal.
Taking the integral over the interval
of all functions we obtain
with respect to periodicity
t
1
2 t
(  | u |2  Knxs nxs u ix ) dxdt .
 2 | u | |0 dx = 
j
i
j
0
(10)
The sum of the integrals of (7) and (8) reads
t
1
0 =  J | |2 |t0 dx  ((h  n)   (  n)  h 
2

0
 Knti nix x  Ku j nix nix x )dxdt =
j
s s
s s
t
1
j
=  ( J |  |2  Knix nix )dx |t0   Ku x nix nix dxdt.
s
s
s
s
j
2
Q
0
(11)
t
Taking the sum of (10) and (11) we obtain
t
1
| u |2  J | |2  K | n |2 dx |t0    | u |2 dxdt = 0.

2
0
The proof is complete.
Corollary 1.
Problem (6)–(9) has a solution for every T
> 0.
5. Estimates of higher derivatives
Regretful, results of Lemma 1 are not sufficient to prove the
convergence of
estimates.
(u k , k , n k ) to the solution of (2)–(5). We need more precise
Theorem 2.
initial data, J, K,  such as
|| u k ||
L2
|| u k ,t ||
There exists T > 0 and C > 0 depending only on
(0,T ;W22 )
L2
(QT )
,||  k ||
,|| k ,t ||
L2
L
(QT )
(0,T ;L2 )
,|| n k ||
,|| n k ,t ||
L2
L
(QT )
(0,T ;W22 )
C
C
Proof . Set in the equation (7)  =  and integrate over the
domain (0, T) with respect to Green's identities. We have
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J
J j
|  |2 dx |T0 =  ( u x  xi  xi  K (n  n x )  x 

l
k
2
2 l j l
Q
T
(12)
 K (n x  n)  x )dxdt.
l
k
The integral of (8) with
  K  2n could be written as
K i
n x x nix x dx |T0 =
2 T k j k j
j
= K  (u j nix x nix  u x nix nix 
k
k
j
k
k j
QT
 (  n x   x  n)  n x )dxdt =
k
k
k
j
= K  (2u x n x x n  u j n x n 
k
j k
j
QT
( x  n x )  n  (n  x )  n x dxdt.
k
k
k
k
(13)
To estimate the second derivatives of u we should set w = u
and take an integral over (0, T):
1
| u |2 dx |T0    | u |2 dxdt =
QT
2 
j
  (u x u x u x  Kn x  nu i )dxdt.
k
QT
j
k
i
The sum of (12)–(14) reads
2
1
2
(
J
|


|

K
| n x x |2  | u |2 )dx |T0 


j k
2
j ,k =3 D1
   | u |2 dxdt =
QT
=
 ( Ju
j
xk
 x j xk  2 Ku xjk n x x n  ( xk  n xk ) 
j k
QT
j
n  u x u x u x )dxdt.
k
145
j
k
(14)
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We can estimate the integral in the right-hand side as
2
1
2
(
J
|


|

K
| n x x |2  | u |2 )dx |T0    | u |2 dxdt 

j k
2 
j , k =1
Q
T
T
2
 C1 (esssup | u(t ) | (||  (t ) ||22   || n x x (t ) ||= 22  || u ||22 ) 
i
i , j =1
0
j
(15)
 esssup | n(t ) | (||  (t ) ||2 || n(t ) ||2 ))dt .

Due to the fact that
ij
|| u x x ||22 =|| u ||2 (see [2, III, Section
i j
u in L2 (0, T ;W ) -norm.
2
2
8]), we have estimated
We need to estimate
esssup | n (t ) |
via highest derivative.
Regretful, W22 - norm is not enough, so we repeat similar procedure for
( ,  , )  ( 2u,  2 ,  K 3n). We have
2
1
2
2
J




u

K
(
|
|
|
|
| (n |2 )dx |t0    | (u) |2 dxdt 


QT
2
j , k =3 D1
j
   u x (2 J x xk   2u x xk u  K n x n xk  2n xi x j x n xi ) 
k
j
j
j
k
Q
T
u j ( J x   u x u  Kn x j x n xk )  Ku xjk xi n x j x n xi 
j
j
k
k
 K u xjk (n x j n xk  n xk n x j  n x j xk n) 
 K ( 2n  n  2(n xk  n xk ))  K (  n) 2n 
 K (2 x j xk  n x j  2 x j  n x j xk  xk  n   n xk )n xk dxdt 
T
 C  (esssup | u(t ) | (|| (t ) ||W2 2  || u(t ) ||W2 2  || n(t ) ||W2 3 ) 
2
2
2
0
(||  (t ) ||W2 1  || u(t ) ||W2 2 )(||  (t ) ||W 1  || u(t ) ||W 1  || n(t ) ||W 2 )
4
4
4
4
4
(||  (t ) ||W 2  || u (t ) ||W 2  || n (t ) ||W 3 ) || u (t ) ||W 3 || n (t ) ||W2 2 
2
2
2
2
4
 esssup | u(t ) | (|| (t ) ||W 2 || n(t ) ||W 3 ) 
2
2
W23
 esssup |  (t ) |  || n (t ) || ) dt.
146
2
(16)
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Denote
 (t ) =|| u(t ) ||W2 2  J || (t ) ||W2 2  K || nx x (t ) ||W2 3 .
2
i j
2
2
Since
|| u(t ) ||L  C (|| u(t ) ||L  || u(t ) ||L ), as
p
2
p<
2
d
,
d 2
and
max | u | C (|| u (t ) ||L  || u (t ) ||L ), as
p
p  d,
p
we have from (14), (15) and (16) is followed by the inequality
t
 (t )   (0)  C4   2 (t )dt , as t < T0 ,
(17)
0
where T0 depends only on
j, K, .
Here we need some simple lemma.
Lemma 2.
Suppose that for almost all t we have
t
0  Y (t )  Y (0)  k  Y 2 dt.
0
Then
Y (t ) 
Y (0)
.
1  ktY (0)
Proof . Consider the function
X( t ) of the form
t
X (t ) = Y (0)    k  X 2 dt <=>
0
i.e. X = 3D
The difference
X  = kX 2 , X (0) = Y (0)   ,
Y (0)  
.
1  kt (Y (0)   )
W = Y  X satisfies the inequality
t
W (t )    k  ( X  Y )Wdt.
0
Since Y and X are measurable functions, the function
t
f (t ) =    ( X  Y )Wdt is a continuous one. Suppose
0
of f . Since X  Y > 0, the condition
on positive measure subset of (0,
t < t0 , consequently
147
t 0 is the least zero
f (t0 ) = 0 means that W is non-negative
t 0 ). But it contradicts with
f (t ) < 0 for all t > 0.
f (t ) < 0 as
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Tending  to zero, we prove the lemma.
Thus for any
t < min{T0 ,1/(C4  (0))}
 (t )  C5 ,
(16)
where C5 depends only on k,, J,  (0).
Estimate time-derivative

Qt
u t . Set  = u t and rewrite (6) as
1
| u t |2    || u || dx |t0 =  (u j u x u t  nn x utj )dxdt .
Qt
j
j
2
Since
u j u x and nn x j are uniformly bounded in
j
L2 (Qt ) (it
follows from (16) and the the embedding theorems), we derive

Qt
| ut |2    || u || dx |t0 || u j u x ut ||22  || nn ||2 .
j
The same inequalities could be obtained for
t
and n t .
The theorem is proved
6. Convergence of the approximations
Theorem 2 provides the existence of measurable functions
u,  , n such that
u k  u weakly in L2 (0, T ; Sol22 ()),
u k ,t  ut
*
k

 n
*
nk
weakly in Sol2 (QT ),
*  weakly in L (0, T ;W21 ()),
*  weakly in L (0, T ;W22 ()).
u k ,t  u t ,  k ,t   t ,  = n k ,t  n t weakly in
Moreover, due to the embedding theorems
u k  u strongly in Sol2 (QT ),
 k   strongly in L2 (QT ),
and
n k  n strongly in L2 (QT ).
148
L2 (QT ).
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  C1 (0, T ; E k ),
 (t), (t)  C 1 (0, T ; F k )
integrate (6)–(8) over (0,T). Passing to the limits as
k   we have
Fix

(u  u  nn) dxdt = 0,

( J/K    (n  n)   )dxdt = 0,

= n   (  n)  dxdt = 0.
QT
QT
QT
Since C 1 (0, T ; E k ) and
and
(17)
 C 1 (0, T ; F k ) are dense in the
corresponding spaces, these equations give us (2)–(4).
Denote p to be a projection of
u  u  nn  L2 (QT )
L2 (0, T ; Sol2 ()). It's obvious that
on the orthogonal complement of the
p  L2 (QT ).
Verify initial conditions (5). Fix
of functions
f k (t ) = (uk (t ),  )
L2
( )
  Sol2 ()
and consider a family
. Since f k = (uk ,t (t ),  ) L ()  (ut ,  ) L ()
2
2
weakly in L2 (0, T ), f k tends to (u,  )
L2
() in C(0,T). Thus u (t , ) tends to
lim u0 k = u0 weakly in L2 ().
k 
Other initial conditions are cheking in the same way.
Q.e.d.
7. Uniqueness
Theorem 3.
Suppose (u1 , 1 , n1 , p1 ) and (u 2 , 2 , n 2 , p2 ) are
solutions of the problem (2)–(5) in the domain
Then for some
QT .
T0 > 0
(u 2 , 2 , n 2 , p2 ) = (u1 , 1 , n1 , p1 )
almost everywhere in
Qt .
0
Proof . First of all, every solution of the problem satisfies identities
(17) for all
  W21 (QT ),   W21 (Qt ),   L2 (0, T ;W22 ).
149
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Denote
 = u1  u 2 ,
f =  1  2 ,
g = n1  n 2
and
set
( ,  , ) = ( w , f ,  K g ). The identities (17) becomes the following:
 (w
t
 w  u2i w x  w  w i u1, x  w  ww 
i
i
Qt
 K n1g x w i  K gn 2,= x w j ) dxdt = 0,
j
j
i
2 xi
 ( J (f f  u f
t
f  w i 1, x f )  K (g  n1  n 2  g )  f )dxdt = 0,
i
Qt
 K  (g t g  u1i g x g  w i n 2, x g 
i
i
Qt
(18)
( 1  g  f  n 2 )g )dxdt = 0
and
1
 2 | w (t ) |

2
dx    | w |2 dxdt =
Qt
  ( w i u1, x  w  K (n1g x w i  gn 2,= x w j )) dxdt ,
i
j
j
Qt
J
| f (t ) |2 dx =   ( Jw i 1, x f  K (g  n 2  n1  g )  f )dxdt ,

i
2
Q
t
K
| g (t ) |2 dx = K  (w i n 2, x g  u1,i x g x g x 

i
k
i
k
2 
Q
t
(19)
 ( 1, x  g )  g x  (f  n 2 )  g )dxdt.
k
k
The sum of the last three identities reads
1
|| w (t ) ||2  J/K || f (t ) ||2  K || g (t ) ||2 dx    || w ||2 dxdt =

2
=   w j u1,i x w i  K n1g x w i  Jw i 1, x f  K (n1  g )  f 
j
j
i
Qt
 Ku1,i x g x g x  K ( 1, x  g)  g x dxdt.
k
150
i
k
k
k
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Due to embedding theorems and H
ölder inequalities
1
|| w (t ) ||2  J/K || f (t ) ||2  K || g (t ) ||2 dx    ||  w ||2 dxdt 

2
1
 Ct 2 esssup (|| u1 (t ) ||
L2
()
 || n1 ||
L2
()
(esssup (|| w (t ) ||22  || f ||22  || g ||22 ) || w ||2
L2
 ||  1 ||
(Qt )
L2
()
 || w (t ) ||2
)
L2
(Qt )
).
Thus if t is sufficiently small, then
(w, f ,  K g) = (0, 0, 0),
consequently
(u1 , 1 , n1 ) = (u 2 , 2 , n 2 ) and, since p i is a
  u  nn, we have
projection of u
p1  p2 .
8. The presence of external forces
The same results take place for
Theorem 4.
F , G  0.
Suppose u 0  Sol22 (),  0  W21 (), n 0  W22 ()
and F  L2 (QT ), G  L1 (0, T ; L2 (QT )).
Then for some 0 < T0 < T the solution (in terms of Definition 1)
of problem (2)–(4) exists and is unique in
QT .
0
The proof is similar to the proof of Theorems 1 and 3.
Acknowledgments
The work was supported by the Government grant of the Russian
Federation under the Resolution No. 220 ``On measures designed to attract
leading scientists to Russian institutions of higher education'' according to
the Agreement No. 11.G34.31.0054, signed by the Ministry of education and
science of the Russian Federation, the leading scientist, and Lomonosov
Moscow State University (on the basis of which the present research is
organized). The work of the first author was also supported in part by RFBR
grant (12-01-00445).
References
[1] François Gay-Balmaz, Tudor S. Ratiu. The geometric structure
of complex fluids. Advances in Applied Mathematics 42 (2009) 176–275.
[2] O. Ladyzhenskaya, N. Uraltseva, Linear and Quasilinear Elliptic
Equations, Academic Press, New York (1968).
151
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