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Comptes Rendus
Mécanique
Gregory A. Chechkin, Tudor S. Ratiu and Maxim S. Romanov
On the Eringen model for nematic liquid crystals
Volume 349, issue 1 (2021), p. 21-27.
<https://doi.org/10.5802/crmeca.67>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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Comptes Rendus
Mécanique
2021, 349, n 1, p. 21-27
https://doi.org/10.5802/crmeca.67
Short paper / Note
On the Eringen model for nematic liquid
crystals
Sur le modèle d’Eringen pour les cristaux liquides
nématiques
Gregory A. Chechkin∗, a, b, c, Tudor S. Ratiud , e , f andMaxim S. Romanova
a M. V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991,
Russia
b Institute of Mathematics with Computing Center—Subdivision of the Ufa Federal
Research Center of Russian Academy of Science, Chernyshevskogo St., 112, Ufa,
450008, Russia
c Institute of Mathematics and Mathematical Modeling, Pushkin st. 125, Almaty,
050010, Kazakhstan
d School of Mathematical Sciences and Ministry of Education Laboratory of Scientific
Computing (MOE-LSC), Shanghai Jiao Tong University, 800 Dongchuan Road,
Minhang, Shanghai, 200240, China
e Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64,
1211 Genève 4, Switzerland
f Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015
Lausanne, Switzerland
E-mails: chechkin@mech.math.msu.su (G. A. Chechkin), ratiu@sjtu.edu.cn
(T. S. Ratiu), mcliz@mail.ru (M. S. Romanov)
Abstract. We introduce the three-dimensional Eringen system of equations for the nematodynamics of liquid
crystals, announce the short time existence and uniqueness of strong solutions for the one-dimensional
problem in the periodic case, and show the continuous dependence of the solution on the initial data.
Résumé. Nous présentons le système tridimensionnel d’équations d’Eringen pour la nématodynamique
des cristaux liquides, annonçons l’existence en temps et l’unicité de solutions fortes pour le problème
unidimensionnel dans le cas périodique et montrons la dépendance continue de la solution sur les données
initiales.
Keywords. Liquid crystals, Eringen equations, Nematodynamics, Existence and uniqueness, Conservation
laws, Micromomentum of molecules, Local solution.
∗Corresponding author.
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
22 Gregory A. Chechkin et al.
Mots-clés. Cristaux liquides, Équations d’Eringen, Nematodynamique, Existence et unicité, Lois de conser-
vation, Micromomentum de molécules, Solution locale.
2020 Mathematics Subject Classification. 76A15, 35K61, 35A01, 35A02.
Manuscript received 13th December 2020, revised and accepted 14th December 2020.
1. Introduction
In papers [1–4] and [5], we considered the Ericksen–Leslie system of equations for nematody-
namics and proved the existence and uniqueness theorems. But the Ericksen–Leslie equations
do not take into account micromomentum of molecules. In this paper, we write down the right
Eringen system of equations (see [6] and [7]), which takes into account the micromomentum of
molecules, and study the same question of unique solvability.
In the present paper, we consider the simplest form of Eringen’s system for the nematodynam-
ics of liquid crystals (see Figure 1 for example) and prove the local solvability of the corresponding
initial–boundary value problem.
The existence of global solutions of the general Eringen system is still an open question. One
can highlight two different cases: the existence of strong and weak solutions. For the strong
solutions, it is natural to expect local solvability and for the weak solutions, global solvability.
2. The full system of Eringen equations
To describe the Eringen system for micropolar liquid crystals [6] in a domainΩ⊂R3 we define the
unknowns:ρ : (0, t0)×Ω→R+ is the mass density of the material; u = (u1,u2,u3) : (0, t0)×Ω→R3 is
the velocity;ν= (ν1,ν2,ν3) : (0, t0)×Ω→R3 is the gyration vector, or, equivalently represented as a
matrix ν := ν̂ :=ν×· = [−εklmνm] : (0, t0)×Ω→ so(3), is the gyration tensor; j : (0, t0)×Ω→ gl(3,R)
is the microinertia tensor, a symmetric positive definite 3×3 matrix;γl = (γ1l ,γ2l ,γ3l ) : (0, t0)×Ω→
R3, l = 1,2,3, where γal : (0, t0)×Ω → R, a = 1,2,3, or, equivalently represented as a matrix by
γ̂l :=γl ×· = [−εkr aγal ] : (0, t0)×Ω→ so(3), is the wryness tensor (in our convention, taken from [8]
which is based on geometrical considerations,γl corresponds to −γl in Eringen’s notation [6,7]);
T : (0, t0)×Ω→ R+ is the absolute temperature. The free energy is denoted by Ψ=Ψ(ρ−1, j ,γ,T )
and the dissipation potential by Φ=Φ(ρ−1, j ,γ,T ; a,b,∇T /T, Ṫ ), where
alk := ∂k ul −εlakνa , bal := ∂lνa , Ṫ := ∂t T, Txk := ∂k T, (1)
∇T := (Tx1 ,Tx2 ,Tx3 ), and ∂k := ∂xk := ∂/∂xk . Explicit formulae for Ψ and Φ are given below. De-
note the internal energy density by E =Ψ+Tη and the entropy by η=−(∂Ψ/∂T )−(1/ρ)(∂Ψ/∂Ṫ ).
Thus, in the absence of external forces, the total energy of the liquid crystal is
1
2
∫
Ω
ρ‖u‖2 d3x + 1
2
∫
Ω
ρ( jν) ·νd3x +
∫
Ω
ρE d3x. (2)
With these notations and conventions, Eringen’s equations for micropolar liquid crystals are
given by the following system.
Conservation of mass
∂tρ+div(ρu) = 0. (3)
Balance of momentum for macromotion
ρ∂t u+ρ(u ·∇)u =∇
(
∂Ψ
∂ρ−1
)
−∂l
(
ρ
∂Ψ
∂γal
γa
)
+∂l
∂Φ
∂al
, (4)
where al := (a1l , a2l , a3l ) = ∂l u−
(
ε1alν
a ,ε2alν
a ,ε3alν
a
)
, γa := (γa1 ,γa2 ,γa3 ).
C. R. Mécanique, 2021, 349, n 1, 21-27
Gregory A. Chechkin et al. 23
Figure 1. Schlieren texture of a nematic liquid crystal.
Balance of moment of momentum for micromotion
ρ∂t ( jν)+ρ(u ·∇)( jν) = −div
(
ρ
∂Ψ
∂γ
)
+ργb × ∂Ψ
∂γb
+div
(
∂Φ
∂b
)
+
(
εn1m
∂Φ
∂anm
,εn2m
∂Φ
∂anm
,εn3m
∂Φ
∂anm
)
, (5)
where ba := (ba1 ,ba2 ,ba3 ) = (∂1νa ,∂2νa ,∂3νa).
Conservation of the microinertia tensor
∂t jab +ui∂i jab + (εpca jbp +εpcb jap )νc = 0 ⇐⇒ ∂t j + (u ·∇) j + [ j , ν̂] = 0. (6)
Equations (5) and (6) yield the equation for the gyration vector ν, namely
j∂tν+ j (u ·∇)ν− ( jν)×ν = − 1
ρ
div
(
ρ
∂Ψ
∂γ
)
+γa × ∂Ψ
∂γa
+ 1
ρ
div
(
∂Φ
∂b
)
+ 1
ρ
(
εn1m
∂Φ
∂anm
,εn2m
∂Φ
∂anm
,εn3m
∂Φ
∂anm
)
. (7)
Conversely, this equation and (6) implies (5).
Wryness tensor equation
∂tγ
a
l +ui∂iγal +γai ∂l ui +∂lνa +εabcγbl νc = 0 ⇔ ∂tγ+£uγ+dν̂+ [γ, ν̂] = 0, (8)
where γ := γ̂ is a so(3)-valued one-form on Ω for each t ∈ (0, t0), that is, γ = γ̂l dx l and γ̂l :
(0, t0)×Ω→ so(3) are given by (γ̂l )kr = −εkr aγal , γl = (γ1l ,γ2l ,γ3l ) : (0, t0)×Ω→ R3; £uγ is the Lie
derivative of γ in the direction of the vector field u, and dν̂ is the differential of the so(3)-valued
function ν̂. Thus, this equation is for so(3)-valued one-forms on Ω; γ is a connection one-form
on the trivial bundle Ω×SO(3) and the last two terms are the γ-covariant derivative of ν̂. See [8]
for details.
Balance of energy
ρ∂t E +ρ(u ·∇)E = (divu) ∂Ψ
∂ρ−1
−ρ
((
∂Ψ
∂γa
·∇
)
u
)
·γa −ρν ·
(
γa × ∂Ψ
∂γa
)
−ρ
(
∂Ψ
∂γa
·∇
)
νa +∂k
(
∂Φ
∂(Txk /T )
)
+ank
∂Φ
∂ank
+bak
∂Φ
∂bak
. (9)
C. R. Mécanique, 2021, 349, n 1, 21-27
24 Gregory A. Chechkin et al.
Second law of thermodynamics
ρ∂tη+ρ(u ·∇)η≥ ∂k
(
T −1
∂Φ
∂(Txk /T )
)
. (10)
Based on physical considerations for micropolar liquid crystals, Eringen [6, 7] proposed the
following expressions for the free energy Ψ and the dissipation potential Φ:
Ψ= 1
2ρ
Akmab γ
a
kγ
b
m , (11)
Φ= 1
2
αkml n a
l
k a
n
m +
1
2
βklabb
a
k b
b
l +
1
T
d kla b
a
l ∂k T +
1
2T
κkl∂k T∂l T, (12)
where
d kla = (d1 +d2tr j )ε kla +d3ε kma jmnδln , κkl = (κ1 +κ2tr j )δkl +κ3 jmnδmkδnl ,
the coefficients Akmab , α
km
ln , β
kl
ab are symmetric (i.e., A
km
ab = Amkba , etc.) and are represented as
Aklab = (A1 + A2 jcpδcp )δkaδlb +
A5
2
( jbpδ
plδka + japδpkδlb)
+ 1
2
(A4 + A3 + (A5 + A8) jpqδpq )δkbδla +
1
4
(A6 + A9)( japδplδkb + jbpδpkδla)
+ 1
2
(A4 − A7 + (A5 − A8) jcpδcp )δabδkl +
1
4
(A6 − A9)( jpcδplδckδab + jabδkl )
+ A10
4
( jabδ
kl − jpqδplδqkδab), (13)
with identical expressions for αkmln and β
kl
ab , except that the coefficients Ai are replaced by differ-
ent coefficients αi and βi , respectively. Ai , di , κi , αi , βi are some given functions, depending on
j , T , and ρ.
3. Initial–boundary value problem for the simplest Eringen system
3.1. Assumptions and auxiliary propositions
From now on, we shall assume that the following conditions hold.
Condition 1. Ψ, ∂Φ/∂anm , ∂Φ/∂b
a
m do not depend on temperature T and its gradient ∇T . This
means that the functions Ai , αi , βi do not depend on T and that di ≡ 0.
Under this condition, system (3)–(8) has all coefficients independent of T and is therefore self-
consistent; neither energy, nor temperature appear in this system. Hence, we can exclude (9) and
(10) from consideration and we are left with system (3)–(8). Equations (9) and (10) can be solved
separately.
Condition 2. jab = J (t ,x)δab .
In this case, Equation (6) states that the function J (t ,x) is frozen in the flow. Since the
microinertia tensor [ jab] is a symmetric positive definite 3×3 matrix, the function J is positive.
Due to (6) it is sufficient to suppose that J > 0 at the initial time.
Condition 3. All unknown variables depend only on time t and the first spatial coordinate x.
Even though this is a restrictive hypothesis requiring a one-dimensional spatial domain,
compressibility of the liquid crystal ensures that the solutions of the system (3)–(8) are not
necessarily trivial.
Condition 4. u = (u,0,0), ν= (ν,0,0), γ1 = (γ,0,0), γ2 =γ3 = 0.
C. R. Mécanique, 2021, 349, n 1, 21-27
Gregory A. Chechkin et al. 25
This assumption puts conditions on the form of the solution. If it holds, Equation (8) is
automatically satisfied for a 6= 1 or k 6= 1.
Condition 5. Aklab , β
kl
ab , α
mn
kl are constants.
Let us denote α = α1111 > 0, λ = α1213 −α1312, A = A1111 > 0, β = β1111 > 0. Note that we assume the
positivity of the constants α,β, A. Then, under the five conditions stated above, Equations (3)–(8)
can be written in the following form:
ρt + (ρu)x = 0, (14)
ρut +ρuux = ∂x
(
αux +λν− A
2
γ2
)
, (15)
ρ Jνt +ρ Juνx = ∂x (βνx − Aγ), (16)
(ρ J )t + (uρ J )x = 0, (17)
γt +∂x (uγ+ν) = 0, (18)
where the functions ρ > 0, J > 0.
Let Ω=T, Qt0 =Ω× (0, t0), where T= R/Z, that is, all functions are 1-periodic with respect to
x. In this paper, we study the system (14)–(18).
For simplicity, we use the notation V = (u,ν,ρ,ρ J ,γ). The vector V is prescribed at t = 0,
namely,
V |t=0 =V0. (19)
Notation 6. If µ is a measure on Ω, we denote by Lp (Ω,µ) the Banach space of real-valued
functions onΩwhose p power of the absolute value is integrable relative to µ. If µ is the Lebesgue
measure, we omit µ in the notation above. If E is a Banach space, Lp (0,T ;E) denotes the Banach
space of E-valued Lp -functions on (0,T ) (relative to the Lebesgue measure). W ap (Ω) is the Banach
space of real-valued Lp -functions onΩ that have all derivatives up to and including order a in Lp .
W r ;ap (QT ) is the Banach space of all real-valued Lp -functions on QT that have all derivatives up to
and including order r on (0,T ) and up to and including order a onΩ in Lp . If E is a Hilbert space
and p = 2, all these spaces are Hilbert spaces.
Definition 7. The vector V is a solution to the problem (14)–(18), (19) if
• V ∈ Wt0 , where Wt0 := W2 ×W2 ×W1 ×W1 ×W1, W1 := W 1;12 (Qt0 )∩L∞(0, t0;L2(Ω)), W2 :=
W 2;12 (Qt0 )∩L∞(0, t0;W 12 (Ω));
• Equations (14)–(18) hold almost everywhere in Qt0 ;
• V0 ∈W0 :=W 22 (Ω)×W 22 (Ω)×W 12 (Ω)×W 12 (Ω)×W 12 (Ω);
• ‖V (x, t )−V0(x)‖L2 → 0 if t → 0.
Another useful space is W [t0 := W [2 × W [2 × W [1 × W [1 × W [1 , where W [1 := W
1;0
2 (Qt0 ) ∩
L∞(0, t0;L2(Ω)), W [2 =W 2;02 (Qt0 )∩L∞(0, t0;W 12 (Ω)).
In ensuing analysis we use the following results.
Lemma 8. Let V be a strong solution of the problem (14)–(18), (19). Then
I1(V ; t ) ≤I1(V ;0)exp
{
λ2α−1
∫ t
0
sup
x∈Ω
(ρ(t , x)J (t , x))−1 dt
}
, (20)
where
I1(V ; t ) = sup
(0,t )
(
‖ρ 12 u‖2L2(Ω) +‖(ρ J )
1
2 ν‖2L2(Ω) + A‖γ‖2L2(Ω)
)
+
∫
Qt
(αu2x +βν2x )dx dt .
C. R. Mécanique, 2021, 349, n 1, 21-27
26 Gregory A. Chechkin et al.
Lemma 9. Let V be a smooth strong solution to the problem (14)–(18), (19). Then there exist t1 > 0
and C > 0 depending on the coefficients and the W0-norm of the initial data such that
‖V ‖Wt ≤C for all t ∈ (0, t1).
Let us consider a regularization of the system (14)–(18). Let ε be a small parameter. The
regularized system is
ρt + (ρu)x = ερxx , (21)
ρut +ρuux = ∂x
(
αux +λν− A
2
γ2
)
, (22)
ρ Jνt +ρ Juνx = ∂x (βνx − Aγ), (23)
(ρ J )t + (ρu J )x = ε(ρ J )xx , (24)
γt +∂x (uγ+ν) = εγxx . (25)
Let V ε be a solution of the regularized system satisfying the initial conditions
V ε|t=0 =V ε0 . (26)
Lemma 10. Let V ε0 ∈ C 2+2δ(Ω), δ > 0. Then for any ε > 0 there exists a unique solution V ε ∈
C 2+δ,1+δ(Qt ).
The proof is based on [9, Ch. VII, Theorem 7.1].
Lemma 11. Let limε→0 V ε0 =V0 ∈W0, the limit taken in the W0-norm. Then there exists t0 > 0 such
that V ε are uniformly bounded in Wt0 -norm.
3.2. Existence theorem
Theorem 12. Let V0 ∈W0. Then for some t0 > 0 there exists a strong solution V ∈Wt0 .
Proof. Let us consider a family of regularized problems (21)–(25), (26), V ε0 ∈ C 2+2δ(Ω), W0-
limε→0 V ε0 =V0. By Lemma 11, there exists t0 > 0 such that this family of solutions V ε is uniformly
bounded in the Wt0 -norm. Hence, we can choose a sequence V
εn such that V εn →V *-weakly in
Wt0 and, in particular,
• (uεn ,νεn )* (u,ν) weakly in W 2,12 (Qt0 );
• (ρεn , (ρ J )εn ,γεn )* (ρ,ρ J ,γ) weakly in W 1,02 (Qt0 );
• (uεn ,νεn ) â (u,ν), that is, converges uniformly on Q t0 .

So, for any Ξ := (ψ,η,ϕ,ζ,θ) ∈C∞0 (Qt0 ), we have
(A (V εn ),Ξ) → (A (V ),Ξ),
where
(A (V ),Ξ) ≡
∫
QT
ρ(ϕt +uϕx )dx dt
+
∫
QT
[
ρu(ψt +uψx )−αuxψx −λνψx + A
2
γ2ψx
]
dx dt
+
∫
QT
[ρ Jν(ηt +uηx )−βνxηx + Aγηx ]dx dt
+
∫
QT
ρ J (ζt +uζx )dx dt
+
∫
QT
γθt + (uγ+ν)θx dx dt = 0, (27)
C. R. Mécanique, 2021, 349, n 1, 21-27
Gregory A. Chechkin et al. 27
where Ξ := (ψ,η,ϕ,ζ,θ) ∈ C∞0 (Qt0 ) (smooth functions on Qt0 with compact support) is a test
vector. By the definition, we have
(A (V ε),Ξ) =−ε
∫
Qt
(ρεxϕx + (ρ J )εxζx +γεxθx )dx dt . (28)
The right-hand side of (28) tends to be zero if ε→ 0, since the norm of (ρεx , (ρ J )εx ,γεx ) is uniformly
bounded in L2(Qt0 ) and Ξ is fixed smooth vector. Hence
(A (V ),Ξ) = 0,
which proves that V is a weak solution of the system (14)–(18) (in the sense of integral identity).
Since V ∈ Wt0 it is easy to prove, that it is also a strong solution. Let us check that the initial
conditions hold. By Lemma 11 and the compact embedding Wt0 ,→ C (0,T ;L2(Ω)), the solution
V (t ) is a continuous map from [0, t0] to L2(Ω) and
max
t∈[0,t0]
‖V ε(t )−V (t )‖L2(Ω) → 0.
In particular, V (0) = (L2)-limε→0 V ε. On the other hand, V ε(0) →V0. The theorem is proved.
3.3. Continuous dependence on initial data and uniqueness
Define V0 :=W 12 (Ω)×W 12 (Ω)×L2(Ω)×L2(Ω)×L2(Ω), Vt0 :=W [1 ×W [1 ×L∞,2 ×L∞,2 ×L∞,2, where
L∞,2 := L∞(0, t0;L2(Ω)). The following theorem holds.
Theorem 13. Let V1, V2 be two strong solutions to the system (14)–(18) in the domain Qt0 . Then,
for any T > 0, there exists C > 0 depending on ‖Vi‖Wt0 , infρ1 > 0, inf(Jρ)1 > 0, α,β, A, and λ such
that
‖V1 −V2‖VT ≤C‖V1(0)−V2(0)‖V0 .
In particular, problem (14)–(18), (19) has unique strong solution.
Acknowledgements
Work of GAC and MSR was partially supported by the Russian Science Foundation (project 20-11-
20272). TSR was partially supported by the National Natural Science Foundation of China grant
number 11871334 and the NCCR SwissMAP grant of the Swiss National Science Foundation. We
also express our thanks to the Committee of Science of the Ministry of Education and Science of
the Republic of Kazakhstan for the support (AP08855579).
References
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