Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 78, pp. 1–21.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.78

STRONG SOLUTIONS TO DENSITY-DEPENDENT INCOMPRESSIBLE

SMECTIC-A LIQUID CRYSTAL EQUATIONS

XUE ZHANG, XIAOPENG ZHAO

Abstract. In this article we study a density-dependent hydrodynamic system that models
smectic-A liquid crystal flow. We establish the existence and uniqueness of local strong solutions

provided that the initial density function has a positive lower bound.

1. Introduction

Smectic liquid crystal is a liquid crystalline phase, which possesses not only some degree of
orientational order like the nematic liquid crystal, but also some degree of positional order (layer
structure) [1, 4, 5, 10]. In [13], the author proposed the following incompressible nonhomogeneous
smectic-A liquid crystals system which is related to the compressibility of fluids/layers and the
thermal effects

ρt +∇ · (ρu) = 0,

ρ(ut + u · ∇u) = ∇ · (−pI + σe + σd),

φt + u · ∇φ = λ(∇ · (ξ∇φ)−K∆2φ), |∇φ| = 1,

∇ · u = 0,

(1.1)

where ρ, u, φ, and p denote the density of the material, flow velocity, layer variable, and pressure,
respectively. The positive constant K which arises in the free energy, the constants µ1 ≥ 0, µ4 > 0,
and µ5 ≥ 0 are dissipative coefficients in the stress tensor, and λ > 0 is elastic relaxation time.
Moreover, D = 1

2 (∇u +∇Tu) represent the symmetric part of the derivative of the velocity, ξ is
the corresponding Lagrange multiplier, n⃗ = ∇φ represents the molecule orientational direction.
The viscous stress tensor σd and the elastic stress tensor (Ericksen tensor) satisfy

σd = µ1(n⃗
TDn⃗)n⃗⊗ n⃗+ µ4D + µ5(Dn⃗⊗ n⃗+ n⊗Dn⃗),

σe = −ξn⃗⊗ n⃗+K∇(∇ · n⃗)⊗ n⃗−K(∇ · n⃗)∇2φ.
(1.2)

It is worth pointing out that the constraint (1.1))4 is the usual incompressibility constraint for
the fluid with the associated pressure p acting as a Lagrange multiplier, the constraint |∇φ| = 1
translates the incompressibility of the layers with the Lagrange multiplier given by ξ [13, 11].
When consider the smectic-A phase, molecules prefer to lie perpendicular to the layers, which
implies that n = ∇φ

|∇φ| = ∇φ [13].

Liu [9] introduced the term f(n) = ∇F (n) = 1
ε2 (|n|

2 − 1)n, with the associated potential

function F (n) = 1
4ε2 (|n|

2 − 1)2 denoting the Ginzburg-Landau potential to relax the constraint

2020 Mathematics Subject Classification. 35Q35, 35B40, 35D35, 76W05.

Key words and phrases. Incompressible Smectic-A liquid crystals; local well-posedness; strong solutions.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted October 3, 2024. Published July 24, 2025.

1



2 X. ZHANG, X. ZHAO EJDE-2025/78

|∇φ| = 1, and studied the density dependent system

ρt +∇ · (ρu) = 0,

ρ(ut + u · ∇u) +∇p = ∇ · σ̃d +∇ · σ̃e,

φt + u · ∇φ = λ
[
∇ ·

( 1

ε2
(|∇φ|2 − 1)∇φ

)
−K∆2φ

]
,

∇ · u = 0,

(1.3)

where

σ̃d = µ1(n⃗
TDn⃗)n⃗⊗ n⃗+ µ4D + µ5(Dn⃗⊗ n⃗+ n⊗Dn⃗),

σ̃e = K∇(∇ · n)⊗ n−K(∇ · n)∇2φ−
( 1

ε2
(|∇φ|2 − 1)n

)
⊗ n.

(1.4)

The author used a no-slip boundary condition for u and time-independent Dirichlet-Neumann
boundary conditions forφ, derived the energy dissipative relation of the system, and proved the
existence of global weak solutions in both two and three dimensions by using a semi-Galerkin
procedure.

If the density ρ is assumed to be a positive constant, then one obtains the incompressible
homogeneous smectic-A liquid crystals equations

ut + u · ∇u+∇p = ∇ · [µ1(n⃗
TDn⃗)n⃗⊗ n⃗+ µ4D + µ5(Dn⃗⊗ n⃗+ n⊗Dn⃗)]

+∇ ·
[
K∇(∇ · n⃗)⊗ n⃗−K(∇ · n⃗)∇2φ−

( 1

ε2
(|∇φ|2 − 1)n⃗

)
⊗ n⃗

]
,

φt + u · ∇φ = λ
[
∇ ·

( 1

ε2
(|∇φ|2 − 1)∇φ

)
−K∆2φ

]
.

(1.5)

Climent-Ezquerra and Guillén-González [2] showed the uniqueness of weak/strong solutions, the
existence of global weak solutions, the existence of weak time-periodic solutions and the existence
of regular solutions for (1.5) together with Dirichlet boundary conditions and the initial condition.
Segatti and Wu [11] considered the long-time behavior of the solutions for the system (1.5) endowed
with periodic boundary conditions within the theory of infinite-dimensional dissipative dynamical
systems. Zhao and Zhou [15] analyzed the local well-posedness, small initial data global well-
posedness and large time behavior of strong solutions for the Cauchy problem of equations (1.5).
On the other hand, Climent-Ezquerra and Guillén-González [3] assumed µ1 = µ5 = 0 in (1.5),
obtained the simple Smectic-A liquid crystal system, proved the existence of global in-time weak
solutions and its convergence to equilibrium of the whole trajectory as time goes to infinity. For
the well-posedness and large time behavior of the Cauchy problem of simple Smectic-A liquid
crystal system, we refer the reader to Zhao and Zhou [14].

In this article, we consider a simple version of (1.3). First, after calculations, σ̃e in (1.4) satisfies
[2]

∇ · σ̃e = − 1

ε2
∇ · [(|∇φ|2 − 1)∇φ]∇φ− 1

4ε2
∇(|∇φ|2 − 1)2 +K∆2φ∇φ−K∇

( |∇φ|2

2

)
.

Moreover, one assumes that µ1 = µ5 = 0, then (1.3)2 can be rewritten as

ρ(ut + u · ∇u)− µ4

2
∆u+∇π =

[
K∆2φ− 1

ε2
∇ ·

(
(|∇φ|2 − 1)∇φ

) ]
∇φ, (1.6)

where π = p + ∇
(K|∇φ|2

2 + 1
4ε2 (|∇φ|2 − 1)2

)
. Combining (1.3), (1.4), and (1.6), we obtain the

system

ρt +∇ · (ρu) = 0,

ρ(ut + u · ∇u)− µ4

2
∆u+∇π =

[
K∆2φ− 1

ε2
∇ · (|∇φ|2 − 1)∇φ

]
∇φ,

φt + u · ∇φ = λ
[ 1

ε2
∇ ·

(
(|∇φ|2 − 1)∇φ

)
−K∆2φ

]
,

∇ · u = 0,

(1.7)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 3

In this article, we consider the existence and uniqueness of local strong solutions for system (1.7)
in T3 = [0, 1]3, thus complement it with the initial condition

(ρ, u, φ)|t=0 = (ρ0, u0, φ0), in T3. (1.8)

For simplicity, we denote Ω := T3 in the following. Next, we give the definition of the strong
solutions for system (1.7)-(1.8) in Ω:

Definition 1.1. Let q ∈ [2,∞) and r ∈ (3, 6]. The time T ∈ (0,∞) is a given positive time. If
the functions

ρ ∈ L∞(0, T ;W 1,q(Ω) ∩W 2,r(Ω) ∩H3(Ω);

u ∈ L∞(0, T ;H3(Ω)) ∩ L2(0, T ;H4(Ω));

ut ∈ L∞(0;T ;H1(Ω)); ∇φt ∈ L∞(0, T ;H2(Ω);

∇φ ∈ L∞(0, T ;H6(Ω)), ∇φtt ∈ L2(0, T ;L2(Ω))

(1.9)

fulfill the initial condition (1.8) and satisfy system (1.7) pointwise, a.e. in Ω × (0, T ), then it is
called a strong solution to system (1.7)-(1.8).

The main results on the existence and uniqueness of local strong solutions are the following.

Theorem 1.2 (Existence). Let q ∈ [2,∞) and r ∈ (3, 6] be fixed constants. Assume that the
initial data (ρ0, u0, φ0) satisfies the regularity conditions

0 < ρ ≤ ρ0 ∈ W 1,q(Ω) ∩W 2,r(Ω) ∩H3(Ω),

∇ · u0 = 0, u0 ∈ H3(Ω), φ0 ∈ Ḣ7(Ω).

Then there exists a positive time T and a strong solution (ρ, u, φ) for system (1.7)-(1.8) in Ω ×
(0, T ).

Theorem 1.3 (Uniqueness). The strong solution established in Theorem 1.2 is unique.

This article is organized as follows. In Section 2 we introduce some preliminary results; In
Section 3, we establish some useful a priori estimates; In Sections 4 and 5, we show the existence
and uniqueness of strong solutions.

2. Preliminaries

We define

Ḣk =
{
w|w ∈ Hk(Ω),

∫
Ω

wdx = 0
}
.

Note that the total mass of φ(x, t) is conserved, i.e.∫
Ω

φ(x, t)dx =

∫
Ω

φ0 dx = M0, ∀t ≥ 0.

where M0 ≥ 0 is a positive constant. For convenience, one assume that
∫
Ω
φ0(x, t)dx = 0, or else,

we can translate the unknown function

φ̃ = φ−M0.

System (1.7)-(1.8) is reduced to

ρt +∇ · (ρu) = 0, in Ω,

ρ(ut + u · ∇u)− µ4

2
∆u+∇π =

[
K∆2φ̃− 1

ε2
∇ · (|∇φ̃|2 − 1)∇φ̃

]
∇φ̃,

φ̃t + u · ∇φ̃ = λ
[
∇ ·

( 1

ε2
(|∇φ̃|2 − 1)∇φ̃

)
−K∆2φ̃

]
,

∇ · u = 0,

. (2.1)

with the initial condition

(ρ, u, φ̃)|t=0 = (ρ0, u0, φ̃0) = (ρ0, u0, φ0 −M). (2.2)



4 X. ZHANG, X. ZHAO EJDE-2025/78

Instead of problem (1.7)-(1.8), we work on problem (2.1)-(2.2) and still denote φ̃ and φ̃0 by φ, φ0,
etc.

In the proofs of lemmas and theorems, we frequently employ the following Gagliardo-Nirenberg
inequality.

Lemma 2.1 ([6]). Let u ∈ Lq(Ω), ∇mu ∈ Lr(Ω), 1 ≤ q, r ≤ ∞. Then there exists a positive
constant C = C(n,m, j, a, q, r), such that

∥∇ju∥Lp ≤ C∥∇ju∥aWm−j,r∥u∥1−a
Lq ,

where
1

p
=

j

n
+ a(

1

r
− m

n
) + (1− a)

1

q
, 1 ≤ p ≤ ∞, 0 ≤ j ≤ m,

j

m
≤ a ≤ 1.

Next, we introduce the Kato-Ponce inequality which is of great importance in the proof of the
main result.

Lemma 2.2 ([7]). Let 1 < p < ∞, s > 0. There exists a positive constant C such that

∥∇s(fg)− f∇sg∥Lp ≤ C(∥∇f∥Lp1∥∇s−1g∥Lp2 + ∥∇sf∥Lq1 ∥g∥Lq2 ) (2.3)

and

∥∇s(fg)∥Lp ≤ C(∥f∥Lp1∥∇sg∥Lp2 + ∥∇sf∥Lq1 ∥g∥Lq2 ), (2.4)

where p1, q1, p2, q2 ∈ (1,∞) satisfying 1
p = 1

p1
+ 1

p2
= 1

q1
+ 1

q2
.

Also, we give the well-known Gronwall’s Lemma, which will be used later.

Lemma 2.3 (Gronwall Lemma[12]). Let g, h, y, and dy
dt be locally integrable functions on (t0,∞)

such that
dy(t)

dt
≤ g(t)y(t) + h(t), ∀t ≥ t0,

then y(t) satisfies

y(t) ≤ y(t0)exp
(∫ t

t0

g(s)ds
)
+

∫ t

t0

h(s)exp
(∫ t

s

g(τ)dτ
)
ds.

3. A priori estimates

In this section, we establish a priori estimates for strong solutions (ρ, u, φ) to (1.7)-(1.8) in Ω
provided that the initial density function has a positive lower bound ρ0 ≥ ρ > 0. Although these
estimates may have their own interests, we mainly apply them to the approximate solutions to
(1.7)-(1.8) that are constructed by the Galerkin method.

Throughout this paper, we denote by C generic constants that depend on ∥u0∥H3 , ∥φ0∥H7 ,
∥ρ0∥W 1,q∩W 2,r∩H3 and the pressure. We also use the obvious notation

∥ · ∥X1∩···∩Xk
=

k∑
j=1

∥ · ∥Xj

for Banach spaces Xj , 1 ≤ j ≤ k and k ∈ Z+. The notation A ≲ B means that A ≤ CB for a
universal constant C > 0.

Let (ρ, u, φ) be a strong solution of (1.7)-(1.8) in Ω × (0, T ] (or the approximate solutions
(ρm, um, φm) of eqref1-1-(1.8) constructed by the Galerkin method). For 0 < t < T , set

Φ(t) := sup
0≤s≤t

(
∥ρ∥W 2,r∩W 1,q∩H3 + ∥u(s)∥H3 + ∥φ(s)∥H7 + ∥ut∥H1 + ∥φt∥H3 + 1

)
. (3.1)

The main purpose of this section is to bound each term of Φ in terms of some integrals of Φ. In
Section 3 below, we will apply arguments of Gronwall’s type to prove that Φ is locally bounded.
Now, we state the main theorem of this section.



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 5

Theorem 3.1. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-
(1.8), then for any t ∈ (0, T ), it holds that

Φ(t) ≤ exp
[
C

∫ t

0

Φ8(s)ds
]
. (3.2)

The proof of Theorem 3.1 is based on the following lemmas.

Lemma 3.2. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥ρ∥Lr = ∥ρ0∥Lr , for r ∈ [1,∞]. (3.3)

The proof of the above lemma can be found in [8, Theorem 2.1].

Lemma 3.3. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥u∥2L2 +
K

ρ
∥∆φ∥2L2 +

1

2ρε2
∥|∇φ|2 − 1∥2L2

+
2

ρ

∫ t

0

∫
Ω

[
λ
∣∣∣∇( 1

ε2
(|∇φ|2 − 1)∇φ

)
−K∆2φ

∣∣∣2]dx ≤ C.

(3.4)

Proof. By [9, Theorem 2.1], we obtain the basic energy identity

∥√ρu∥2L2 +K∥∆φ∥2L2 +
1

2ε2
∥|∇φ|2 − 1∥2L2

+ 2

∫ t

0

∫
Ω

[
λ
∣∣∣∇( 1

ε2
(|∇φ|2 − 1)∇φ

)
−K∆2φ

∣∣∣2]dx
= ∥√ρ0u0∥2L2 +K∥∆φ0∥2L2 +

1

2ε2
∥|∇φ0|2 − 1∥2L2

(3.5)

Moreover, on the basis of the assumptions of Theorem 1.2 and the result of Lemma 3.2, we easily
obtain

0 < ρ∥u∥2L2 ≤ ∥√ρu∥2L2 ,

∥√ρ0u0∥2L2 ≤ ∥ρ0∥L∞∥u0∥2L2 .

Hence, we obtain (3.4) and complete the proof. □

Lemma 3.4. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥∇ρ∥Lq + ∥∆ρ∥Lr + ∥∇∆ρ∥L2 ≤ exp
(
C

∫ t

0

Φ(s)ds
)
, ∀r ∈ (3, 6], q ∈ [2,∞). (3.6)

Proof. Applying the gradient operator ∇ to (1.7)1, multiplying by q|∇ρ|q−2∇ρ, and integrating
over Ω, we derive that

d

dt

∫
Ω

|∇ρ|qdx ≤ C

∫
Ω

|∇ρ|q|∇u|dx ≤ ∥∇u∥L∞∥∇ρ∥qLq . (3.7)

Then, by Gronwall’s inequality and Sobolev’s inequality, it follows that

∥∇ρ∥qLq ≤ ∥∇ρ0∥qLq exp
(
C

∫ t

0

∥∇u∥L∞ds
)

≤ ∥∇ρ0∥qLq exp
(
C

∫ t

0

∥∇u∥H2ds
)

≤ exp
(
C

∫ t

0

Φ(s)ds
)
.

(3.8)



6 X. ZHANG, X. ZHAO EJDE-2025/78

Applying the Laplacian operator ∆ to (1.7)1, multiplying by r|∆ρ|r−2∆ρ, and integrating the
result over Ω, we obtain

d

dt

∫
Ω

|∆ρ|rdx = −r

∫
Ω

|∆ρ|r−1∆ρ : ∆(u · ∇)ρ dx+

∫
Ω

|∆ρ|r∇ · uds

− 2r

∫
Ω

|∆ρ|r−2∆ρ : ∇(u · ∇)∇ρ dx

≤ C∥∆u∥Lr∥∇ρ∥L∞∥∆ρ∥r−1
Lr + C∥∇u∥L∞∥∆ρ∥rLr

≤ C(∥∇u∥L∞ + ∥∆u∥Lr )∥∆ρ∥rLr .

(3.9)

Using Gronwall’s inequality and Sobolev’s inequality, it follows that

∥∆ρ∥rLr ≤ ∥∆ρ0∥rLr exp
(
C

∫ t

0

(∥∇u∥L∞ + ∥∆u∥Lr )ds
)

≤ C exp
(
C

∫ t

0

∥∇u∥H2ds
)

≤ exp
(
C

∫ t

0

Φ(s)ds
)
.

(3.10)

Applying the operator ∇∆ to (1.7)1, multiplying by ∇∆ρ, and integrating the result over Ω, we
obtain

d

dt
∥∇∆ρ∥2L2 ≤ C(∥∇u∥L∞∥∇∆ρ∥2L2 + ∥∆u∥L3∥∆ρ∥L6∥∇∆ρ∥L2

+ ∥∇∆u∥L2∥∇ρ∥L∞∥∇∆ρ∥L2)

≤ C(∥∇u∥L∞ + ∥∆u∥L3)∥∇∆ρ∥2L2 + C∥∇∆u∥L2∥∆ρ∥1/2L2 ∥∇∆ρ∥
3
2

L2

≤ C(∥∇u∥L∞ + ∥∆u∥L3 + ∥∇∆u∥L2)∥∇∆ρ∥2L2 + C∥∇∆u∥L2∥∆ρ∥2L2 .

(3.11)

It then follows from Gronwall’s inequality, Sobolev’s inequality and Taylor’s expansion of ex that

∥∇∆ρ∥2L2 ≤
(
∥∇∆ρ0∥2L2 + sup ∥∆ρ∥2L2

∫ t

0

∥∇∆u∥L2ds
)

× exp
(
C

∫ t

0

(∥∇u∥L∞ + ∥∆u∥L3 + ∥∇∆u∥L2)ds
)

≤ C exp
(
C

∫ t

0

∥∇u∥H2ds
)

≤ exp
(
C

∫ t

0

Φ(s)ds
)
,

(3.12)

which complete the proof. □

Lemma 3.5. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥φ∥2L2 + ∥∇φ∥2L2 +

∫ t

0

(∥∆φ(s)∥2L2 + ∥∇∆φ(s)∥2L2)ds ≤ C

∫ t

0

Φ6(s)ds. (3.13)

Proof. Multiplying (1.7)3 by φ, integrating over Ω, one deduce that

1

2

d

dt
∥φ∥2L2 + λK∥∆φ∥2L2 +

λ

ε2
∥∇φ∥4L4 ≤ λ

ε2
∥∇φ∥2L2 +

∫
Ω

|u||∇φ||φ|dx. (3.14)

We bound the first term on the right-hand side of (3.14) by

λ

ε2
∥∇φ∥2L2 = − λ

ε2

∫
Ω

φ∆φdx ≤ λK

2
∥∆φ∥2L2 + C∥φ∥2L2 . (3.15)

Also, ∫
Ω

|u||∇φ||φ|dx ≤ ∥u∥L3∥∇φ∥L2∥φ∥L6 ≤ C∥u∥H1∥φ∥2H1 ≤ C(∥u∥3H1 + ∥φ∥3H1). (3.16)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 7

Combining (3.14)-(3.16) gives

d

dt
∥φ∥2L2 + λK∥∆φ∥2L2 ≤ C(∥u∥3H1 + ∥φ∥3H1 + 1). (3.17)

Multiplying (1.7)3 by ∆φ, integrating over Ω, and integrating by parts, we arrive at

1

2

d

dt
∥∇φ∥2L2 + λK∥∇∆φ∥2L2 ≤

∫
Ω

|u||∇φ||∆φ|dx+
λ

ε2

∫
Ω

(|∇φ|3 + |∇φ|)|∇∆φ|dx

≤ C∥u∥L6∥∇φ∥L3∥∆φ∥L2 + C∥∇∆φ∥L2(∥∇φ∥3L6 + ∥∇φ∥L2)

≤ C∥u∥H1∥∇φ∥2H1 +
λK

2
∥∇∆φ∥2L2 + C(∥∇φ∥6H1 + ∥∇φ∥2L2)

≤ λK

2
∥∇∆φ∥2L2 + C(∥∇φ∥6H1 + ∥u∥6H1 + 1).

Simple calculations show that

d

dt
∥∇φ∥2L2 + λK∥∇∆φ∥2L2 ≤ C(∥∇φ∥6H1 + ∥u∥6H1 + 1). (3.18)

Adding (3.17) and (3.18), and using Gronwall’s inequality, we obtain (3.13) and complete the
proof. □

Lemma 3.6. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥∇u∥2L2 + ∥∇∆φ∥2L2 + ∥∆2φ∥2L2 +

∫ t

0

(∥√ρut∥2L2 + ∥∇∆2φ∥2L2 + ∥∆3φ∥2L2) ds

≤ C

∫ t

0

Φ6(s)ds.

(3.19)

Proof. Multiplying (1.7)2 by ut, and integrating by parts over Ω, we arrive at

d

dt

∫
Ω

µ4

4
|∇u|2 dx+

∫
Ω

ρ|ut|2dx = −
∫
Ω

(ρu · ∇u)ut dx+

∫
Ω

K∆2φ∇φut dx

− 1

ε2

∫
Ω

[
∇ · [(|∇φ|2 − 1)∇φ]∇φ

]
ut dx−

∫
Ω

∇πut dx

=: I1 + I2 + I3 + I4.

(3.20)

In the following, we estimate the three terms of the right-hand side of (3.20), one by one. The
main tools to bound those terms are Hölder’s inequality and Sobolev’s embedding theorem. Note
that

I1 ≤
∫
Ω

|ρ||u||∇u||ut|dx

≤ C∥√ρ∥L∞∥u∥L6∥∇u∥L3∥√ρut∥L2

≤ C∥√ρ∥L∞(∥u∥L2 + ∥∇u∥L2)∥∇u∥1/2L2 ∥∇u∥1/2H1 ∥
√
ρut∥L2

≤ C∥√ρut∥L2∥u∥2H1 + C∥√ρut∥L2∥u∥
3
2

H1∥∆u∥1/2L2

≤ 1

4
(∥√ρut∥2L2 + ∥∆u∥2L2) + C(∥u∥6H1 + ∥u∥4H1),

(3.21)



8 X. ZHANG, X. ZHAO EJDE-2025/78

and

I2 + I3 ≤ C(∥∆2φ∥L2∥∇φ∥L3∥ut∥L6 + ∥∆φ∥L2∥∇φ∥2L6∥ut∥L6

+ ∥∆φ∥L2∥∇φ∥L3∥ut∥L6)

≤ C∥∆2φ∥L2∥∇φ∥1/2L2 ∥∇φ∥1/2H1 ∥∇ut∥L2 + C∥∆φ∥L2∥∇φ∥2H1∥ut∥H1

+ C∥∇φ∥1/2L2 ∥∇φ∥1/2H1 ∥∆φ∥L2∥ut∥H1

≤ 1

4
∥∇ut∥2L2 + C(∥∆2φ∥2L2∥∇φ∥L2∥∇φ∥H1 + ∥∆φ∥2L2∥∇φ∥4H1

+ ∥∇φ∥L2∥∇φ∥H1∥∆φ∥2L2)

≤ 1

4
∥∇ut∥2L2 + C(∥∆2φ∥6L2 + ∥∆φ∥6L2 + ∥∇φ∥6L2 + 1).

(3.22)

Moreover, (1.7)4 implies that

I4 =

∫
Ω

π∇ · ut dx = 0. (3.23)

Combining (3.20)-(3.23), we derive that

d

dt

∫
Ω

µ4

2
|∇u|2 dx+ ρ

∫
Ω

|ut|2dx

≤ 1

2
(∥∇ut∥2L2 + ∥∆u∥2L2) + C(∥∆2φ∥6L2 + ∥∇φ∥6L2 + ∥∆φ∥6L2 + ∥∇u∥6L2 + 1)

≤ C(∥∇ut∥6L2 + ∥u∥6H2 + ∥φ∥6H4 + 1).

(3.24)

Applying ∇∆ to both side of (1.7)3, multiplying by ∇∆φ, integrating over Ω, we deduce that

1

2

d

dt
∥∇∆φ∥2L2 + λK∥∇∆2φ∥2L2

≤
∫
Ω

|∇(u · ∇φ)||∇∆2φ|dx+ λ

∫
Ω

∣∣∣∇∇ ·
( 1

ε2
(|∇φ|2 − 1)∇φ

)∣∣∣|∇∆2φ| dx

=: I4 + I5.

(3.25)

By using Hölder’s inequality, the Kato-Ponce inequality and Sobolev’e embedding theorem in 3D
bounded domain, the right-hand side of (3.25) can be bounded as

I4 ≤ C∥∇∆2φ∥L2(∥∇u∥L2∥∇φ∥L∞ + ∥u∥L6∥∇2φ∥L3)

≤ C∥∇∆2φ∥L2∥u∥H1∥∇φ∥H2

≤ λK

4
∥∇∆2φ∥2L2 + C∥u∥2H1∥∇φ∥2H2 ,

(3.26)

and

I5 ≤ C∥∇∆2φ∥L2(∥∇2(|∇φ|2∇φ)∥L2 + ∥∇∆φ∥L2)

≤ C∥∇∆2φ∥L2(∥∇φ∥2L6∥∇∆φ∥L6 + ∥∇∆φ∥L2)

≤ C∥∇∆2φ∥L2(∥∇φ∥2H1∥∇∆φ∥H1 + ∥∇∆φ∥L2)

≤ λK

4
∥∇∆2φ∥2L2 + C(1 + ∥∇φ∥4H1)(∥∇∆φ∥2L2 + ∥∆2φ∥2L2).

(3.27)

It then follows from (3.25)-(3.27) that

d

dt
∥∇∆φ∥2L2 + λK∥∇∆2φ∥2L2

≤ C∥u∥2H1∥∇φ∥2H2 + C(1 + ∥∇φ∥4H1)(∥∇∆φ∥2L2 + ∥∆2φ∥2L2)

≤ C(∥u∥6H1 + ∥φ∥6H4 + 1).

(3.28)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 9

Applying ∆2 to both side of (1.7)3, multiplying by ∆2φ, integrating over Ω, we deduce that

1

2

d

dt
∥∆2φ∥2L2 + λK∥∆3φ∥2L2

≤
∫
Ω

|∆(u · ∇φ)||∆3φ| dx+ λ

∫
Ω

∣∣∣∣∆∇ ·
(

1

ε2
(|∇φ|2 − 1)∇φ

)∣∣∣∣ ∣∣∆3φ
∣∣ dx

=: I6 + I7.

(3.29)

The main tools to bound I6 and I7 are also Hölder’s inequality, the Kato-Ponce inequality and
Sobolev’e embedding theorem. Simple calculations show that

I6 ≤ C∥∆3φ∥L2(∥∆u∥L2∥∇φ∥L∞ + ∥u∥L∞∥∇∆φ∥L2)

≤ C∥∆3φ∥L2(∥∆u∥L2∥∇φ∥H2 + ∥u∥H2∥∇∆φ∥L2)

≤ λK

4
∥∆3φ∥2L2 + C(∥∆u∥2L2∥∇φ∥2H2 + ∥∇∆φ∥2L2∥u∥2H2)

≤ λK

4
∥∆3φ∥2L2 + C(∥∇φ∥4H2 + ∥u∥4H2 + 1),

(3.30)

and

I7 ≤ C∥∆3φ∥L2(∥∇3(|∇φ|2∇φ)∥L2 + ∥∆2φ∥L2)

≤ C∥∆3φ∥L2(∥∇φ∥2L∞∥∆2φ∥L2 + ∥∆2φ∥L2)

≤ C∥∆3φ∥L2(∥∇φ∥2H2∥∆2φ∥L2 + ∥∆2φ∥L2)

≤ λK

4
∥∆3φ∥2L2 + C(∥∇φ∥4H2∥∆2φ∥2L2 + ∥∆2φ∥2L2)

≤ λK

4
∥∆3φ∥2L2 + C(1 + ∥∇φ∥6H3).

(3.31)

Summing (3.29)-(3.31), we obtain

d

dt
∥∆2φ∥2L2 + λK∥∆3φ∥2L2 ≤ C(1 + ∥∇φ∥6H3 + ∥u∥6H2). (3.32)

Combining (3.24), (3.28) and (3.32), integrating over (0, t), one obtains (3.19) and completes the
proof. □

Lemma 3.7. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥ut∥2L2 + ∥∇φt∥2L2 + ∥∇∆φt∥2L2 +

∫ t

0

(µ4

2
∥∇ut∥2L2 + ∥∇∆φt∥2L2 + ∥∇φtt∥2L2

)
ds

≤ C

∫ t

0

Φ8(s)ds.

(3.33)

Proof. Differentiating (1.7)2 with respect to t, multiplying the resulting by ut, integrating over Ω,
we arrive at

1

2

d

dt
∥√ρut∥2L2 + ∥∇ut∥2L2 =

∫
Ω

[div(ρu)(ut + u · ∇u)− ρ(ut · ∇u)]ut dx

+

∫
Ω

[(
K∆2φ− 1

ε2
∇ · (|∇φ|2 − 1)∇φ

)
∇φ

]
t
ut dx

=: J1 + J2.

(3.34)



10 X. ZHANG, X. ZHAO EJDE-2025/78

There are two terms on the right-hand side of (3.34). We estimate them by using Höler’s inequality,
Kato-Ponce inequality and Sobolev embedding theorem in the following. For the term J1, we have

J1 ≤ C

∫
Ω

(
ρ|u||∇ut||ut|+ ρ|u||∇u|2|ut|+ ρ|u|2|∆u||ut|

+ ρ|u|2|∇u||∇ut|+ ρ|ut|2|∇u|
)
dx

≤ C(∥√ρut∥L2∥∇ut∥L2∥u∥L∞∥ρ∥1/2L∞ + ∥ρ∥L∞∥u∥L6∥∇u∥2L3∥ut∥L6

+ ∥ρ∥L∞∥u∥2L6∥∆u∥L2∥ut∥L6 + ∥ρ∥L∞∥u∥2L6∥∇u∥L6∥∇ut∥L2

+ ∥∇u∥L3∥√ρut∥L2∥ut∥L6∥ρ∥1/2L∞)

≤ C(∥√ρut∥L2∥∇ut∥L2∥u∥H2 + ∥u∥H1∥∇u∥L2∥∇u∥H1∥ut∥H1

+ ∥u∥2H1∥∆u∥L2∥ut∥H1 + ∥u∥2H1∥∇u∥H1∥∇ut∥L2

+ ∥∇u∥H1∥√ρut∥L2∥ut∥H1)

≤ µ4

28
∥∇ut∥2L2 + C∥√ρut∥2L2∥u∥2H2 + C∥u∥4H1∥∇u∥2H1 + ∥u∥2H1∥∇u∥H1∥ut∥L2

+ C∥√ρut∥L2∥∇u∥H1∥ut∥L2

≤ µ4

28
∥∇ut∥2L2 + C(∥ut∥6L2 + ∥u∥6H2 + 1).

(3.35)

Moreover, J2 can be bounded as

J2 =

∫
Ω

(
K∆2φt∇φ+K∆2φ∇φt +

1

ε2
∆φt −

3

ε2
|∇φ|2∆φt

− 6

ε2
∇φ · ∇φt∆φ

)
ut dx

≤ C

∫
Ω

|∇∆φt||∆φ||ut|dx+ C

∫
Ω

|∇∆φt||∇φ||∇ut|dx+ C

∫
Ω

|∆2φ||∇φt||ut|dx

+ C

∫
Ω

|∇φt||∇ut|dx+ C

∫
Ω

|∇φ|2|∆φt||ut|dx+ C

∫
Ω

|∇φ||∇φt||∆φ||ut|dx

=: J21 + J22 + J23 + J24 + J25 + J26,

(3.36)

where

J21 ≤ ∥∇∆φt∥L2∥∆φ∥L3∥ut∥L6

≤ C∥∇∆φt∥L2∥∆φ∥H1∥ut∥H1

≤ µ4

28
∥∇ut∥2L2 + C∥∇∆φt∥2L2∥∆φ∥2H1 + C∥∇∆φt∥L2∥∆φ∥H1∥ut∥L2

≤ µ4

28
∥∇ut∥2L2 + C(∥∇∆φt∥4L2 + ∥∆φ∥4H1 + ∥ut∥4L2 + 1),

(3.37)

J22 ≤ C∥∇∆φt∥L2∥∇φ∥L∞∥∇ut∥L2

≤ C∥∇∆φt∥L2∥∇φ∥H2∥∇ut∥L2

≤ µ4

28
∥∇ut∥2L2 + C∥∇∆φt∥2L2∥∇φ∥2H2

≤ µ4

28
∥∇ut∥2L2 + C(∥∇∆φt∥4L2 + ∥∇φ∥4H2),

(3.38)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 11

J23 ≤ C∥∆2φ∥L2∥∇φt∥L3∥ut∥L6

≤ C∥∆2φ∥L2∥∇φt∥1/4H2 ∥∇φt∥3/4L2 ∥ut∥H1

≤ C∥∆2φ∥L2(∥∇∆φt∥1/4L2 + ∥∇φt∥1/4L2 )∥∇φt∥3/4L2 (∥ut∥L2 + ∥∇ut∥L2)

≤ µ4

28
∥∇ut∥2L2 + C∥∆2φ∥2L2∥∇∆φt∥1/2L2 ∥∇φt∥

3
2

L2 + C∥∆2φ∥2L2∥∇φt∥2L2

+ C∥∆2φ∥L2(∥∇∆φt∥1/4L2 + ∥∇φt∥1/4L2 )∥∇φt∥3/4L2 ∥ut∥L2

≤ µ4

28
∥∇ut∥2L2 + C(∥∇∆φt∥4L2 + ∥∇φt∥4L2 + ∥∆2φ∥4L2 + ∥ut∥4L2 + 1),

(3.39)

J24 ≤ C∥∇ut∥L2∥∇φt∥L2 ≤ µ4

28
∥∇ut∥2L2 + C∥∇φt∥2L2 , (3.40)

J25 ≤ C∥∆φt∥L2∥∇φ∥2L6∥ut∥L6

≤ C∥∇∆φt∥1/2L2 ∥∇φt∥1/2L2 ∥∇φ∥2H1∥ut∥H1

≤ µ4

28
∥∇ut∥2L2 + C∥∇∆φt∥L2∥∇φt∥L2∥∇φ∥4H1

+ C∥∇∆φt∥1/2L2 ∥∇φt∥1/2L2 ∥∇φ∥2H1∥ut∥L2

≤ µ4

28
∥∇ut∥2L2 + C(∥ut∥6L2 + ∥∇∆φt∥6L2 + ∥∇φt∥6L2 + ∥∇φ∥6H1 + 1),

(3.41)

J26 ≤ C∥∇φt∥L2∥∇φ∥L6∥∆φ∥L6∥ut∥L6 ≤ C∥∇φt∥L2∥∇φ∥2H2∥ut∥H1

≤ µ4

28
∥∇ut∥2L2 + C∥∇φt∥2L2∥∇φ∥4H2 + C∥∇φt∥L2∥∇φ∥2H2∥ut∥L2

≤ µ4

28
∥∇ut∥2L2 + C(∥∇∆φ∥6L2 + ∥∇φ∥6L2 + ∥∇φt∥6L2 + ∥ut∥6L2 + 1).

(3.42)

Summing (3.34)-(3.42), it yields that

d

dt
∥√ρut∥2L2 +

µ4

2
∥∇ut∥2L2 ≤ C(∥ut∥6L2 + ∥∇u∥6L2 + ∥∆u∥6L2 + ∥∇∆u∥6L2 + ∥∇φt∥6L2

+ ∥∇∆φt∥6L2 + ∥∆2φ∥6L2 + ∥∇∆φ∥6L2 + 1).
(3.43)

Differentiating (1.7)3 with respect to t, multiplying the resulting by ∆φt, integrating over Ω, one
arrive that

1

2

d

dt
∥∇φt∥2L2 + λK∥∇∆φt∥2L2 = −

∫
Ω

ut · ∇φ ·∆φt dx−
∫
Ω

u · ∇φt ·∆φt dx

+
λ

ε2

∫
Ω

∇ ·
(
(3|∇φ|2 − 1)∇φt

)
∆φt dx

=: J3 + J4 + J5.

(3.44)

There are three terms on the right-hand side of (3.44). The main tools to bound them are Höler’s
inequality, Kato-Ponce inequality and Sobolev embedding theorem. Note that

J3 ≤ C∥ut∥L6∥∇φ∥L3∥∆φt∥L2

≤ C∥ut∥H1∥∇φ∥1/2L2 ∥∇φ∥1/2H1 ∥∇φt∥1/2L2 ∥∇∆φt∥1/2L2

≤ 1

6
(∥∇ut∥2L2 + ∥∇∆φt∥2L2) + C∥∇φ∥2L2∥∇φ∥2H1∥∇φt∥2L2

+ C∥ut∥L2∥∇φ∥H1∥∇φt∥1/2L2 ∥∇∆φt∥1/2L2

≤ 1

6
(∥∇ut∥2L2 + ∥∇∆φt∥2L2) + C(∥∇φ∥6H1 + ∥ut∥6L2 + ∥∇φt∥6L2 + 1),

(3.45)



12 X. ZHANG, X. ZHAO EJDE-2025/78

J4 ≤ C∥u∥L6∥∇φt∥L3∥∆φt∥L2

≤ C∥u∥H1∥∇φt∥5/4L2 ∥∇φt∥3/4H2

≤ C∥u∥H1∥∇φt∥5/4L2 (∥∇φt∥3/4L2 + ∥∇∆φt∥3/4L2 )

≤ 1

6
∥∇∆φt∥2L2 + C∥u∥8/5H1 ∥∇φt∥2L2 + C∥u∥H1∥∇φt∥2L2

≤ 1

6
∥∇∆φt∥2L2 + C(∥u∥6H1 + ∥∇φt∥6L2 + 1),

(3.46)

J5 ≤ C∥∇∆φt∥L2(∥∇φt∥L2 + ∥∇φ∥2L∞∥∇φt∥L2)

≤ C∥∇∆φt∥L2(∥∇φt∥L2 + ∥∇φ∥H2∥∇φt∥L2)

≤ 1

6
∥∇∆φt∥2L2 + C∥∇φt∥2L2(1 + ∥∇φ∥2H2)

≤ 1

6
∥∇∆φt∥2L2 + C(∥∇∆φ∥6L2 + ∥∇φ∥6L2 + ∥∇φt∥6L2 + 1).

(3.47)

Summing (3.44)-(3.47), we find that

d

dt
∥∇φt∥2L2 + λK∥∇∆φt∥2L2 ≤ 1

2
∥∇ut∥2L2 + C(∥u∥6H1 + ∥∇φ∥6L2 + ∥∇φt∥6L2 + ∥∇∆φ∥6L2 + 1)

≤ C(∥∇ut∥6L2 + ∥u∥6H1 + ∥∇φ∥6H2 + ∥∇φt∥6L2 + 1).

(3.48)
Differentiating (1.7)3 with respect to t, multiplying the resulting by ∆φtt, integrating over Ω, one
arrive at

λK

2

d

dt
∥∇∆φt∥2L2 + ∥∇φtt∥2L2 = −

∫
Ω

ut · ∇φ ·∆φttdx−
∫
Ω

u · ∇φt ·∆φttdx

+
λ

ε2

∫
Ω

(
∇ · (3|∇φ|2 − 1)∇φt

)
∆φttdx

= J6 + J7 + J8.

(3.49)

On the basis of the Kato-Ponce inequality, Höler’s inequality and Sobolev embedding theorem,
one obtains

J6 ≤ C∥∇φtt∥L2∥∇(ut · ∇φ)∥L2

≤ C∥∇φtt∥L2(∥∇ut∥L2∥∇φ∥L∞ + ∥ut∥L6∥∆φ∥L3)

≤ C∥∇φtt∥L2(∥∇ut∥L2∥∇φ∥H2 + ∥ut∥H1∥∆φ∥H1)

≤ 1

6
∥∇φtt∥2L2 + C∥ut∥2H1∥∇φ∥2H2

≤ 1

6
∥∇φtt∥2L2 + C(∥∇ut∥4L2 + ∥ut∥4L2 + ∥∇φ∥4H2),

(3.50)

J7 ≤ C∥∇φtt∥L2∥∇(u · ∇φt)∥L2

≤ ∥∇φtt∥L2(∥∇u∥L3∥∇φt∥L6 + ∥u∥L∞∥∆φt∥L2)

≤ C∥∇φtt∥L2(∥∇u∥H1∥∇φt∥H1 + ∥∇u∥H1∥∆φt∥L2)

≤ 1

6
∥∇φtt∥2L2 + C∥∇u∥2H1∥∇φt∥2H1

≤ 1

6
∥∇φtt∥2L2 + C(∥∇u∥4H1 + ∥∇φt∥4L2 + ∥∇∆φt∥4L2),

(3.51)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 13

J8 ≤ C∥∇φtt∥L2

(
∥∇2 · (|∇φ|2∇φt)∥L2 + ∥∇∆φt∥L2

)
≤ C∥∇φtt∥L2

(
∥∇φ∥2L∞∥∇∆φt∥L2 + ∥∇φ∥L6∥∇∆φ∥L6∥∇φt∥L6 + ∥∇∆φt∥L2

)
≤ C∥∇φtt∥L2

(
∥∇φ∥2H2∥∇∆φt∥L2 + ∥∇φ∥2H3∥∇φt∥H1 + ∥∇∆φt∥L2

)
≤ 1

6
∥∇φtt∥2L2 + C∥∇φ∥4H3(∥∇φt∥2L2 + ∥∇∆φt∥2L2) + C∥∇∆φt∥2L2

≤ 1

6
∥∇φtt∥2L2 + C(∥∇φ∥6H3 + ∥∇∆φt∥6L2 + ∥∇φt∥6L2 + 1).

(3.52)

Summing (3.49)-(3.52), we deduce that

λK
d

dt
∥∇∆φt∥2L2+∥∇φtt∥2L2 ≤ C(∥∇u∥6H1+∥ut∥6H1+∥∇φt∥6L2+∥∇∆φt∥6L2+∥∇φ∥6H3+1). (3.53)

Combining (3.43), (3.48) and (3.53) gives

d

dt
(∥√ρut∥2L2 + ∥∇φt∥2L2 + ∥∇∆φt∥2L2) + ∥∇ut∥2L2 + ∥∇∆φt∥2L2 + ∥∇φtt∥2L2

≤ C(∥u∥6H1 + ∥ut∥6H1 + ∥φ∥6H4 + ∥∇φt∥6L2 + ∥∇∆φt∥6L2 + 1)

≤ CΦ6(t).

(3.54)

Integrating over (0, t), note that 0 < ρ ≤ ρ, we obtain (3.33) and complete the proof. □

Lemma 3.8. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥∆u∥2L2 ≤ CΦ8(t). (3.55)

Proof. Multiplying equation (1.7)2 by ∆u and integrating over Ω yields that

µ4

2
∥∆u∥2L2 =

∫
Ω

(ρut + ρu · ∇u) ·∆u dx−
∫
Ω

K∆2φ∇φ∆u dx

+
1

ε2

∫
Ω

∇ · [(|∇φ|2 − 1)∇φ]∇φ∆u dx

=: L1 + L2 + L3.

(3.56)

Next, we estimate L1, L2 and L3. The mail tools are also Hölder’s inequality, Kato-Ponce inequal-
ity and Sobolev’e embedding theorem. We have

L1 ≤
∫
Ω

|√ρ||√ρut||∆u|dx+

∫
Ω

|ρ||u||∇u||∆u|dx

≤ C∥√ρut∥L2∥∆u∥L2 + C∥u∥L6∥∇u∥L3∥∆u∥L2

≤ C∥√ρut∥L2∥∆u∥L2 + C∥u∥
3
2

H1(∥∇u∥L2 + ∥∆u∥L2)1/2∥∆u∥L2

≤ µ4

8
∥∆u∥2L2 + C(∥u∥6H1 + ∥√ρut∥2L2 + 1),

(3.57)

and

L2 + L3 ≤ ∥∆u∥L2∥[K∆2φ− 1

ε2
∇ · [(|∇φ|2 − 1)∇φ]∇φ∥L2

≤ C∥∆u∥L2∥∇φ∥L∞(∥∆2φ∥L2 + ∥∇ · [(|∇φ|2 − 1)∇φ]∥L2)

≤ C∥∆u∥L2∥∇φ∥L∞(∥∆2φ∥L2 + ∥∆φ∥L2 + ∥∇ · (|∇φ|2)∇φ∥L2)

≤ C∥∆u∥L2∥∇φ∥L∞(∥∆2φ∥L2 + ∥∆φ∥L2 + ∥∇φ∥2L∞∥∆φ∥L2)

≤ C∥∆u∥L2∥∇φ∥H2 [∥∆2φ∥L2 + ∥∆φ∥L2 + ∥∇φ∥2H2∥∆φ∥L2 ]

≤ µ4

8
∥∆u∥2L2 + C∥∇φ∥2H2 [∥∆2φ∥L2 + ∥∆φ∥L2 + ∥∇φ∥2H2∥∆φ∥L2 ]2

≤ µ4

8
∥∆u∥2L2 + C(∥∇φ∥8H3 + 1).

(3.58)



14 X. ZHANG, X. ZHAO EJDE-2025/78

Combining (3.56), (3.57) and (3.58), we find that

µ4

2
∥∆u∥2L2 ≤ C(∥∇u∥8L2 + ∥∇φ∥8H3 + ρ∥ut∥2L2 + 1) ≤ CΦ8(t), (3.59)

this complete the proof. □

Lemma 3.9. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-(1.8),
then for any t ∈ (0, T ), it holds that

∥∇∆u∥2L2 + ∥∇ut∥2L2 +

∫ t

0

(∥∆2u∥2L2 + ∥∆ut∥2L2)ds ≤ C

∫ t

0

Φ8(s)ds. (3.60)

Proof. We remark that (1.7)2 is equivalent to

ρ(ut + u · ∇u)− µ4

2
∆u+∇π = − 1

λ
(φt + u · ∇φ)∇φ. (3.61)

Applying ∇∆ to (3.61), multiplying the resulting equation by ∇∆u, integrating the resultant over
Ω, one derives that

1

2

d

dt
∥√ρ∇∆u∥2L2 + ∥∆2u∥2L2

≤
∫
Ω

|∆ρ||∇ut||∇∆u|dx+

∫
Ω

|∇ρ||∆ut||∇∆u|dx+

∫
Ω

|∆(ρu · ∇u)||∆2u|dx

+
1

λ

∫
Ω

|∆(φt∇φ)||∆2u|dx+
1

λ

∫
Ω

|∆[(u · ∇φ)∇φ]||∆2u|dx+

∫
Ω

|ut||∆ρ|∆2u|dx

=: K1 +K2 +K3 +K4 +K5 +K6,

(3.62)

where we used that∫
Ω

ut∇∆ρ · ∇∆u dx ≤
∫
Ω

|∆ρ|(|∇ut||∇∆u|dx+ |ut||∆2u|)dx.

In the following, using Hölder’s inequality, Kato-Ponce inequality and Sobolev’e embedding theo-
rem, we estimate the sixth terms of the right-hand side of (3.62) term by term. Note that

K1 ≤ ∥∇ut∥L2∥∆ρ∥L3∥∇∆u∥L6 ≤ ∥∇ut∥L2∥∆ρ∥L3∥∇∆u∥H1

≤ 1

12
∥∆2u∥2L2 + C∥∇ut∥2L2∥∆ρ∥2L3 + C∥∇ut∥L2∥∆ρ∥L3∥∇∆u∥L2

≤ 1

12
∥∆2u∥2L2 + CΦ4(t) + CΦ3(t),

(3.63)

K2 ≤ C∥∆ut∥L2∥∇ρ∥L∞∥∇∆u∥L2 ≤ η∥∆ut∥2L2 + C∥∇ρ∥2L∞∥∇∆u∥2L2

≤ η∥∆ut∥2L2 + CΦ4(t),
(3.64)

K3 ≤ C∥∆2u∥L2∥∆(ρu · ∇u)∥L2

≤ C∥∆2u∥L2(∥ρ∥L∞∥∆u∥L6∥∇u∥L3 + ∥ρ∥L∞∥u∥L∞∥∇∆u∥L2

+ ∥∆ρ∥L6∥u∥L3∥∇u∥L2)

≤ 1

12
∥∆2u∥2L2 + C∥ρ∥2H3∥u∥4H3

≤ 1

12
∥∆2u∥2L2 + CΦ6(t),

(3.65)

K4 ≤ C∥∆2u∥L2∥∆(φt∇φ)∥L2

≤ C∥∆2u∥L2(∥∆φt∥L6∥∇φ|L3 + ∥φt∥6∥∇∆φ∥L3)

≤ 1

12
∥∆2u∥2L2 + C∥∇φ∥2H1∥∆φt∥2H1 + C∥∇φt∥2L2∥∇∆φ∥2H1

≤ 1

12
∥∆2u∥2L2 + CΦ4(t),

(3.66)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 15

K5 ≤ C∥∆2u∥L2∥∆[(u · ∇φ)∇φ]∥L2

≤ C∥∆2u∥L2(∥∆u∥L2∥∇φ|2L∞ + ∥u∥6∥∇∆φ∥L6∥∇φ∥L6)

≤ 1

12
∥∆2u∥2L2 + C∥∇φ∥4H2∥∆u∥2L2 + C∥u∥2H1∥∇φ∥4H3

≤ 1

12
∥∆2u∥2L2 + CΦ6(t),

(3.67)

K6 ≤ C∥∆2u∥L2∥ut∥L3∥∆ρ∥L6

≤ C∥∆2u∥L2∥ut∥1/2L2 (∥ut∥L2 + ∥∇ut∥L2)1/2∥∆ρ∥L6

≤ 1

12
∥∆2u∥2L2 + ∥ut∥L2(∥ut∥L2 + ∥∇ut∥L2)∥∆ρ∥2L6

≤ 1

12
∥∆2u∥2L2 + CΦ4(t).

(3.68)

Adding (3.62)-(3.68) together gives

d

dt
∥√ρ∇∆u∥2L2 + ∥∆2u∥2L2 ≤ CΦ6(t). (3.69)

Differentiating (1.7)2 with respect to t, multiplying the resulting by −∆ut, integrating over Ω, one
arrive that

1

2

d

dt
∥√ρ∇ut∥2L2 + ∥∆ut∥2L2

≤
∫
Ω

|∇ρ||u||∇ut|dx+

∫
Ω

|u||∇ρ||ut||∆ut|dx+

∫
Ω

|u||∇ρ||u||∇u||∆ut|dx

+

∫
Ω

|ρ||ut||∇u||∆ut|dx+

∫
Ω

|ρ||u||∇ut||∆ut|dx+

∫
Ω

|φt||∇φt||∆ut|dx

+

∫
Ω

|u||∇φ||∇φt||∆ut|dx+

∫
Ω

|∇φtt||∇φ||∇ut|dx+

∫
Ω

|φtt||∆φ||∇ut|dx

+

∫
Ω

|ut||∇φ|2|∆ut|dx+

∫
Ω

|u||∇φt||∇φ||∆ut|dx

=: K7 +K8 +K9 +K10 +K11 +K12 +K13 +K14 +K15 +K16 +K17,

(3.70)

where we used that ∫
Ω

φtt∆ut∇φdx ≤
∫
Ω

|∇ut|(|∇φtt||∇φ|+ |φtt||∆φ|)dx.

In the following, we estimate the eleven terms on the right-hand side of (3.70) one by one. The main
tools we use are Hólder’s inequality, Kato-Ponce inequality and Sobolev’s embedding theorem.
Note that

K7 ≤ ∥∇ut∥L2∥∇ρ∥L3∥ut∥L6 ≤ C∥∇ut∥2L2∥∇ρ∥L3 ≤ CΦ3(t), (3.71)

K8 ≤ ∥∆ut∥L2∥ut∥L6∥∇ρ∥L6∥u∥L6 ≤ C∥∆ut∥L2∥ut∥H1∥∇ρ∥L6∥u∥H1

≤ µ4

28
∥∆ut∥2L2 + C∥ut∥2H1∥∇ρ∥2L6∥u∥2H1 ≤ δ∥∆ut∥2L2 + CΦ6(t),

K9 ≤ ∥∆ut∥L2∥u∥L6∥u∥L∞∥∇ρ∥L6∥∇u∥L6

≤ ∥∆ut∥L2∥u∥3H2∥∇ρ∥L6

≤ µ4

28
∥∆ut∥2L2 + C∥u∥6H2∥∇ρ∥2L6

≤ µ4

28
∥∆ut∥2L2 + CΦ8(t),

K10 ≤ ∥∆ut∥L2∥ut∥L6∥∇u∥L3∥ρ∥L∞ ≤ ∥∆ut∥L2∥ut∥H1∥∇u∥H1∥ρ∥H2

≤ µ4

28
∥∆ut∥2L2 + C∥ut∥2H1∥∇u∥2H1∥ρ∥2H2

≤ µ4

28
∥∆ut∥2L2 + CΦ6(t),



16 X. ZHANG, X. ZHAO EJDE-2025/78

K11 ≤ ∥∆ut∥L2∥∇ut∥L2∥ρ∥L∞∥u∥L∞

≤ ∥∆ut∥L2∥∇ut∥L2∥u∥H2∥ρ∥H2

≤ µ4

28
∥∆ut∥2L2 + C∥∇ut∥2L2∥u∥2H2∥ρ∥2H2

≤ µ4

28
∥∆ut∥2L2 + CΦ6(t),

K12 ≤ ∥∆ut∥L2∥φt∥L6∥∇φt∥L3

≤ ∥∆ut∥L2∥∇φt∥L2∥∇φt∥3/4L2 ∥∇∆φt∥1/4L2

≤ µ4

28
∥∆ut∥2L2 + C|∇φt∥

7
2

L2∥∇∆φt∥1/2L2

≤ µ4

28
∥∆ut∥L2 + CΦ4(t),

K13 ≤ ∥∆ut∥L2∥u∥L6∥∇φ∥L6∥∇φt∥L6

≤ ∥∆ut∥L2∥u∥H1∥∇φ∥H1∥∇φt∥H1

≤ µ4

28
∥∆ut∥2L2 + C∥u∥2H1∥∇φ∥2H1∥∇φt∥2H1

≤ µ4

28
∥∆ut∥2L2 + CΦ6(t),

K14 ≤ ∥∇φtt∥L2∥∇φ∥L∞∥∇ut∥L2

≤ C∥∇φtt∥L2∥∇φ∥H2∥∇ut∥L2

≤ 1

3
∥∇φtt∥2L2 + C∥∇φ∥2H2∥∇ut∥2L2

≤ 1

3
∥∇φtt∥2L2 + CΦ4(t),

K15 ≤ ∥φtt∥L6∥∆φ∥L3∥∇ut∥L2

≤ C∥∇φtt∥L2∥∆φ∥H1∥∇ut∥L2

≤ 1

3
∥∇φtt∥2L2 + C∥∆φ∥2H1∥∇ut∥2L2

≤ 1

3
∥∇φtt∥2L2 + CΦ4(t),

K16 ≤ ∥ut∥L6∥∇φ∥L6∥∆ut∥L2

≤ C∥ut∥H1∥∇φ∥2H1∥∆ut∥L2

≤ 1

3
∥∆ut∥2L2 + C∥ut∥2H1∥∇φ∥4H1

≤ 1

3
∥∆ut∥2L2 + CΦ6(t),

K17 ≤ ∥u∥L6∥∇φt∥L6∥∇φ∥L6∥∆ut∥L2

≤ C∥u∥H1∥∇φ∥H1∥∇φt∥H1∥∆ut∥L2

≤ µ4

28
∥∆ut∥2L2 + C∥u∥2H1∥∇φ∥2H1∥∇φt∥2H1

≤ µ4

28
∥∇φtt∥2L2 + CΦ6(t).

Summing, we obtain

d

dt
(∥√ρ∇∆u∥2L2 + ∥√ρ∇ut∥2L2) + ∥∆2u∥2L2 + ∥∆ut∥2L2 ≤ C(Φ8(t) + ∥∇φtt∥2L2), (3.72)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 17

Integrating (3.72) over (0, t), we obtain

∥√ρ∇∆u∥2L2 + ∥√ρ∇ut∥2L2 +

∫ t

0

(∥∆2u∥2L2 + ∥∆ut∥2L2)ds

≤ C

∫ t

0

(
Φ8(s) + ∥∇φtt∥2L2

)
ds ≤ C

∫ t

0

Φ8(s)ds.

(3.73)

Note that ρ ≥ ρ, then by using (3.33), we obtain (3.73) and complete the proof. □

Lemma 3.10. If (ρ, u, φ) is the unique strong solution stated in Definition 1.1 to system (1.7)-
(1.8), then for any t ∈ (0, T ), it holds that

∥∇∆3φ∥L2 ≤ C

∫ t

0

Φ8(s)ds. (3.74)

Proof. Applying ∇∆ to (1.7)3, we easily obtain

∥∇∆3φ∥L2 ≤ ∥∇∆φt∥L2 + ∥∇∆(u · ∇φ)∥L2 + ∥∆∇ · (|∇φ|2∇φ−∇φ)∥L2

≤ ∥∇∆φt∥L2 + ∥u∥L∞∥∆2φ∥L2 + ∥∇∆u∥L2∥∇φ∥L∞ + ∥∆2φ∥L2 + ∥∇φ∥2L∞∥∆2φ∥L2

≤ ∥∇∆φt∥L2 + ∥u∥H2∥∆2φ∥L2 + ∥∇∆u∥L2∥∇φ∥H2 + ∥∆2φ∥L2 + ∥∇φ∥2H2∥∆2φ∥L2

≤ C

∫ t

0

Φ8(s)ds,

and complete the proof. □

Proof of Theorem 3.1. It follows from (3.4), (3.6), (3.13), (3.19), (3.33), (3.70) and (3.74) that

Φ(t) ≤ C

∫ t

0

Φ8(s)ds+ exp

{
C

∫ t

0

Φ(s)ds

}
. (3.75)

Moreover, from the Taylor expansion

ex = 1 + x+
x2

2!
+

x3

3!
+ . . . ,

we obtain that

C

∫ t

0

Φ8(s)ds ≤ exp
{
C

∫ t

0

Φ8(s)ds
}
. (3.76)

It is readily seen that the conclusion (3.2) follows from (3.75) and (3.76). □

4. Proof of Theorem 1.2

In this section, we employ Galerkin’s method to obtain a sequence of approximate solutions to
system (1.7)-(1.8), which will converge to a strong solution to (1.7)-(1.8).

To implement Galerkin’s method, we take the function space to be X := H2(Ω) ∩H1
0 (Ω) and

its finite dimensional subspaces as

Xm := span{ξ1, ξ2, · · · , ξm}, m ≥ 1,

where {ξm} ⊂ X is an orthonormal base of H1(Ω). Now, we outline Galerkin’s scheme into several
steps:

Step 1: mth approximate solutions. Fix 2 ≤ q < ∞ and 3 < r ≤ 6. For m ≥ 1 and some
0 < T = T (m) < +∞ to be determined below, we let

um
0 =

m∑
k=1

(u0, ξk)ξk



18 X. ZHANG, X. ZHAO EJDE-2025/78

and look for the triple

ρm ∈ C([0, T ];W 1,q ∩W 2,r ∩H3),

um(x, t) =

m∑
k=1

um
k (t)ξk(x) ∈ C([0, T ];H3),

φm ∈ C([0, T ];H7)

(4.1)

as solution of the problem

ρmt +∇ · (ρmum) = 0,

(ρm(um
t + um · ∇um), ξk) +

µ4

2
(∇um,∇ξk) + (∇πm, ξk)

=
([

K∆2φm − 1

ε2
∇ · (|∇φm|2 − 1)∇φm

]
∇φm, ξk

)
,

φm
t + um · ∇φm = λ

[
∇ ·

( 1

ε2
(|∇φm|2 − 1)∇φm

)
−K∆2φm

]
,

∇ · um = 0,

(ρm, um, φm)|t=0 = (ρ0, u
m
0 , φ0).

(4.2)

The existence of a solution (ρm, um, φm) to (4.2) over Ω × [0, T (m)] for some T (m) > 0 can be
obtained by the fixed point theorem. Since the process is standard, we only sketch the argument.
First, observe that for any given 0 < T < ∞ and um ∈ C([0, T ];H3), it is standard to show that
there exist

(1) a solution ρm ∈ C([0, T ];W 1,q ∩W 2,r ∩H3) of (4.2)1 along with ρm|t=0 = ρ0.
(2) 0 < tm ≤ T , depending on um and ∥φ0∥H6 , and φm ∈ C([0, tm], H7(Ω)).

It is well known that

ρm(x, t) ≥ ρ > 0.

The coefficients um
k (t) can be determined by the following system of m first order ODEs: 1 ≤ k ≤

m,
m∑
i=1

(ρmξi, ξk)u̇
m
i = Gk

(
um
l (t),

∫ t

0

um
l (s)ds, t

)
; um

k (0) = (u0, ξk), (4.3)

where Gk denotes the right-hand side of (4.2)2. Note that ρm is strictly positive, the determinant
of the m×m matrix (ρmξi, ξk)1≤i,k≤m is positive. Therefore, (4.3) can be reduced to

u̇m
k = Fk(u

m
l , bml , t), ḃmk = um

k ; um
k (0) = (u0, ξk), bmk (0) = 0, (4.4)

where Fk is a regular function of um
l and bml . Hence, on the basis of the standard existence theory

of ODEs, we conclude that there is a time T, ∈ (0, tm] and a solution um
k (t) to (4.4), which in turn

implies the existence of solution ρm, φm of (4.2)1 and (4.2))3 on the same time interval.

Step 2: A priori estimates. We also need to show that there exist 0 < T0 < +∞ and C > 0,
depending only on the initial data ρ0, u0 and φ0, but independent of the parameter m and the
size of the domain Ω, such that for any m ≥ 1, (ρm, um, φm) satisfies

Φm(t) ≤ exp
[
C

∫ t

0

(Φm(s))
8
ds
]
, 0 < t ≤ T0, (4.5)

where Φm(t) is defined by (3.1)) with (ρ, u, φ) replaced by (ρm, um, φm). Since the argument to
obtain (4.5)) is similar to the proof of Theorem 3.1, we omit it here.

Step 3: Convergence. By (4.5), we obtain that for any m ≥ 1,

sup
0≤t≤T0

(∥um
t ∥2H1 + ∥ρm∥2W 1,q∩W 2,r + ∥um∥2H3 + ∥φm∥2H7 + ∥φm

t ∥2H3)

+

∫ T0

0

(∥∇um∥2H3 + ∥∇∆φm∥2H3 + ∥um
t ∥2H2 + ∥∇∆φm

t ∥2L2 + ∥∇φm
tt ∥2L2)ds ≤ C.

(4.6)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 19

On the basis of the estimate (4.6)), we can deduce that after taking subsequences, there exists
(ρ, u, φ) such that

ρm → ρ weak* in L∞(0, T0;W
1,q ∩W 2,r),

um → u weak* in L∞(0, T0;H
3),

um → u weak in L2(0, T0;H
4),

um
t → ut weak* in L∞(0, T0;H

1),

um
t → ut weak in L2(0, T ;H2),

∇φm → ∇φ weak* in L∞(0, T ;H6),

φm
t → φt weak* in L∞(0, T ;H3),

∇∆φm → ∇∆φ weak in L2(0, T ;H3),

∇∆φm
t → ∇∆φt weak in L2(0, T ;L2),

∇φm
tt → ∇φtt weak in L2(0, T ;L2).

By the lower semicontinuity, (4.6)) implies that for 0 ≤ t ≤ T0, (ρ, u, φ) satisfies

sup
0≤t≤T0

(∥ut∥2H1 + ∥ρ∥2W 1,q∩W 2,r + ∥u∥2H3 + ∥∇φ∥2H6 + ∥φt∥2H3)

+

∫ T0

0

(∥∇u∥2H3 + ∥∇∆φ∥2H3 + ∥ut∥2H2 + ∥∇∆φt∥2L2 + ∥∇φtt∥2L2)ds ≤ C,

(4.7)

this completes the proof of existence of strong solutions.

5. Proof of Theorem 1.3

This section is devoted to show the uniqueness of the local strong solutions obtained in the
above section.

Let (ρi, ui, φi) (i = 1, 2) be two strong solutions on Ω × (0, T ] of system (1.7))-(1.8)). Set
ρ̄ = ρ1 − ρ2, ū = u1 − u2, π̄ = π1 − π2 and φ̄ = φ1 − φ2. Then we have

ρ̄t + u1 · ∇ρ̄+ ū · ∇ρ2 = 0,

ρ1ūt + ρ1u1 · ∇ū+ ρ1ū · ∇u2 + ρ̄u2 · ∇u2 + ρ̄u2t +∇π̄ +
µ4

2
∆ū

= K∆2φ̄∇φ1 +K∆2φ2∇φ̄− 1

ε2
∇ ·

(
|∇φ1|2∇φ1 −∇φ1

)
∇φ̄

− 1

ε2
∇ ·

(
|∇φ1|2∇φ̄+ (∇φ1 +∇φ2) · ∇φ̄∇φ2 −∇φ̄

)
∇φ2,

φ̄t + u1 · ∇φ̄+ ū · ∇φ2 = −λK∆2φ̄+
λ

ε2
∇ ·

(
|∇φ1|2∇φ̄+ (∇φ1 +∇φ2) · ∇φ̄∇φ2 +∇φ̄

)
,

∇ · ū = 0,

(ρ̄, ū, φ̄)|t=0 = 0.

(5.1)
Multiplying (5.1))1 by 2ρ̄, integrating over Ω and using integration by parts, we obtain

d

dt
∥ρ̄∥2L2 ≤

∫
Ω

|ρ̄ū · ∇ρ2|dx

≤ C∥ρ̄∥L2∥∇ρ2∥L3∥ū∥L6

≤ C∥ρ̄∥L2∥∇ū∥L2

≤ µ4

8
∥∇ū∥2L2 + C∥ρ̄∥2L2 .

(5.2)



20 X. ZHANG, X. ZHAO EJDE-2025/78

Multiplying (5.1))2 by ū, integrating over Ω and using integration by parts, we deduce that

1

2

d

dt
∥√ρ1ū∥2L2 +

µ4

2
∥∇ū∥2L2

≤
∫
Ω

|ρ1||u1||∇ū||ū|dx+

∫
Ω

|ρ1||ū||∇u2||ū|dx+

∫
Ω

|ρ̄||u2||∇u2||ū|dx

+

∫
Ω

|ρ̄||u2t||ū|dx+K

∫
Ω

|∆2φ̄||∇φ1ū|dx+K

∫
Ω

|∆2φ2||∇φ̄||ū|dx

+
1

ε2

∫
Ω

|∇ · (|∇φ1|2∇φ1 −∇φ1)||∇φ̄||ū|dx

+
1

ε2

∫
Ω

∣∣∇ ·
(
|∇φ1|2∇φ̄+ (∇φ1 +∇φ2) · ∇φ̄∇φ2 −∇φ̄

)∣∣ |∇φ2||ū|dx

≤ ∥√ρ1ū∥L2∥√ρ1∥L∞∥u1∥L∞∥∇ū∥L2 + ∥√ρ1ū∥2L2∥∇u2∥L∞

+ ∥ū∥L6∥ρ̄∥L2∥u2∥L6∥∇u2∥L6 + ∥ρ̄∥L2∥u2t∥L3∥ū∥L6

+
1
√
ρ
∥∆2φ̄∥L2∥∇φ1∥L∞∥√ρ1ū∥L2 + ∥ū∥L6∥∇φ̄∥L2∥∆2φ2∥L3

+ ∥ū∥L6∥∇φ̄∥L2(∥∆φ1∥L3 + ∥∇φ1∥2L∞∥∆φ1∥L3)

+ ∥ū∥L6∥∇φ2∥L∞(∥∇φ1∥2L6∥∆φ̄∥L2

+ ∥∇φ̄∥L2∥∇φ1∥L6∥∆φ1∥L6 + ∥∇φ1 +∇φ2∥L6∥∇φ̄∥L2∥∆φ2∥L6

quad+ ∥∆(φ1 + φ2)∥L6∥∇φ̄∥L2∥∇φ2∥L6 + ∥∇(φ1 + φ2)∥L∞∥∆φ̄∥L6∥∇φ2∥L2)

≤ µ4

8
∥∇ū∥2L2 +

λK

4
(∥∇∆φ̄∥2L2 + ∥∆2φ̄∥2L2)

+ C(∥ρ̄∥2L2 + ∥√ρ1ū∥2L2 + ∥∇φ̄∥2L2 + ∥∆φ̄∥2L2).

(5.3)

Multiplying (5.1))3 by ∆φ̄, integrating over Ω and using integration by parts, we deduce that

1

2

d

dt
∥∇φ̄∥2L2 + λK∥∇∆φ̄∥2L2 ≤ ∥u1∥L3∥∇φ̄∥L2∥∆φ̄∥L6 +

1
√
ρ
∥√ρ1ū∥L2∥∇φ2∥L3∥∆φ̄∥L6

+ ∥∇φ1∥2L∞∥∇φ̄∥L2∥∇∆φ̄∥L2 + ∥∇φ̄∥L2∥∇∆φ̄∥L2

+ ∥∇φ1 +∇φ2∥L∞∥∇φ̄∥L2∥∇φ2∥L∞∥∇∆φ̄∥L2

≤ λK

4
∥∇∆φ̄∥2L2 + C(∥∇φ̄∥2L2 + ∥√ρ1ū∥2L2).

(5.4)

Applying ∆ to (5.1))3, multiplying by ∆φ̄, integrating over Ω and using integration by parts, we
deduce that

1

2

d

dt
∥∆φ̄∥2L2 + λK∥∆2φ̄∥2L2

≤ ∥u1∥L∞∥∇φ̄∥L2∥∆2φ̄∥L2 +
1
√
ρ
∥√ρ1ū∥L2∥∇φ2∥L∞∥∆2φ̄∥L2

+ ∥∆2φ̄∥L2(∥∇φ1∥2L∞∥∆φ̄∥L2 + ∥∇φ1∥L6∥∆φ1∥L6∥∇φ̄∥L6

+ ∥∆(φ1 + φ2)∥L6∥∇φ̄∥L6∥∆φ2∥L6 + ∥∇(φ1 + φ2)∥L∞∥∇φ2∥L∞∥∆φ̄∥L2

+ ∥∇(φ1 + φ2)∥L6∥∇φ̄∥L6∥∇φ2∥L6 + ∥∆φ̄∥L2)

≤ λK

4
∥∆2φ̄∥2L2 + C(∥∇φ̄∥2L2 + ∥∆φ̄∥2L2 + ∥√ρ1ū∥2L2).

(5.5)

Summing, one find that

d

dt
(∥ρ̄∥2L2 + ∥√ρ1ū∥2L2 + ∥∇φ̄∥2L2 + ∥∆φ̄∥2L2) + ∥∇ū∥2L2 + ∥∇∆φ̄∥2L2 + ∥∆2φ̄∥2L2

≤ C(∥ρ̄∥2L2 + ∥√ρ1ū∥2L2 + ∥∇φ̄∥2L2 + ∥∆φ̄∥2L2).
(5.6)



EJDE-2025/78 DENSITY-DEPENDENT INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTALS 21

By (5.6)), Gronwall’s inequality and (ρ̄0, ū0, φ̄0) = 0, we arrive at

∥ρ̄∥2L2 + ∥√ρ1ū∥2L2 + ∥∇φ̄∥2L2 + ∥∆φ̄∥2L2

+

∫ t

0

eC(t−s)(∥∇ū∥2L2 + ∥∇∆φ̄∥2L2 + ∥∆2φ̄∥2L2)ds = 0,
(5.7)

which implies that
(ρ̄, ū,∇φ̄,∆φ̄) = 0. (5.8)

To see that φ̄ = 0, we observe that after substituting (5.8)) from (5.1))3, we have

φ̄t = 0, φ̄|t=0 = 0,

this implies φ̄ = 0. This completes the proof of uniqueness.

Acknowledgements. The authors would like to thank the anonymous referee for his/her helpful
suggestions.

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Xue Zhang

College of Sciences, Northeastern University, Shenyang 110004, China

Email address: 2200151@stu.neu.edu.cn

Xiaopeng Zhao
College of Sciences, Northeastern University, Shenyang 110004, China

Email address: zhaoxiaopeng@mail.neu.edu.cn


	1. Introduction
	2. Preliminaries
	3. A priori estimates
	4. Proof of Theorem 1.2
	5. Proof of Theorem 1.3
	Acknowledgements

	References