Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 101, pp. 1–17.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.101

GLOBAL WELL-POSEDNESS OF 3D INHOMOGENEOUS INCOMPRESSIBLE

NEMATIC LIQUID CRYSTAL SYSTEMS IN CRITICAL BESOV SPACES

WITH INITIAL DENSITY PERTURBED AROUND THE EQUILIBRIUM

XINRUI WU, XINGYU LIANG

Abstract. In this article, we establish the existence of unique global solutions for three-

dimensional inhomogeneous incompressible nematic liquid crystal systems. Our analysis does
not assume that the initial density ρ0 is endowed with any regularity, requiring only that the

fluid density satisfy ∥ρ0 − 1∥L∞(R3) ≤ c for a sufficiently small constant c. While the initial

velocity u0 and the gradient of the initial molecular orientation ∇d0 belong to the critical space

Ḃ
−1+ 3

p

p,1 (R3). This work extends the result by Danchin and Wang [15] for the Navier-Stokes

equations to the three-dimensional inhomogeneous nematic liquid crystal system.

1. Introduction

This study is devoted to the mathematical analysis of the three-dimensional inhomogeneous
incompressible nematic liquid crystal system

∂tρ+ u · ∇ρ = 0,

ρ(∂tu+ u · ∇u)− µ∆u+∇π = −λ div(∇d⊙∇d),

∂td + u · ∇d = θ(∆d + |∇d|2d),
div u = 0,

|d| = 1,

(ρ, u,d)|t=0 = (ρ0, u0,d0).

(1.1)

In this system, ρ and u denote the fluid density and velocity field respectively, and π represents
the pressure field. The molecular orientation of the system (1.1) is described by the director field
d ∈ S2, where S2 = {d ∈ R3 : |d| = 1} is the unit sphere in R3. The stress tensor ∇d ⊙ ∇d is
a 3 × 3 matrix with components (∇d ⊙∇d)ij = ∂id · ∂jd for 1 ≤ i, j ≤ 3. System (1.1) contains
three positive viscosity parameters: µ (shear viscosity), λ (elastic relaxation), and θ (orientational
diffusivity). The initial data satisfy the constraints that div u0 = 0 (divergence-free condition),
and |d0| ≡ 1 (unit length constraint).

For the special case of homogeneous molecular alignment where d(x, t) ≡ d0 (constant), imply-
ing ∇d ≡ 0, the coupled system (1.1) degenerates to the inhomogeneous Navier-Stokes equations
(INS) describing variable-density fluid motion:

∂tρ+ div(ρu) = 0,

ρ(∂tu+ u · ∇u)−∆u+∇P = 0,

div u = 0,

(ρ, u)|t=0 = (ρ0, u0).

(1.2)

2020 Mathematics Subject Classification. 35Q35, 76D03.
Key words and phrases. Inhomogeneous incompressible nematic liquid crystal; critical Besov spaces;

maximal regularity estimates.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted July 18, 2025. Published October 27, 2025.

1



2 X. WU, X. LIANG EJDE-2025/101

The notion of critical spaces with scaling invariance matching the natural scaling of scaling,

uλ(t, x) := λu(λ2t, λx), λ > 0, (1.3)

plays a pivotal role in modern analysis. Because of the scaling invariance property of the system
(1.2), there has been growing interest in studying their behavior in critical spaces. When the initial
density ρ0 possesses some regularity, Abidi [1] and Danchin [11] obtained a global solution to system
(1.2) provided that ρ0 is a small perturbation of a positive constant in the critical Besov space

Ḃ
d
p

p,1(Rd) with the initial velocity ∥u0∥
Ḃ

d
p
−1

p,1 (Rd)
≤ ε0 for p ∈ (1, 2d). This result was later extended

by Danchin and Mucha [13], who showed through Lagrangian coordinate techniques that global
existence and uniqueness hold for the system (1.2) when the integral index p belongs to [1, 2d)
under the condition that the initial density ρ0 remains close to a constant in the multiplier space

M(Ḃ
−1+ d

p

p,1 (Rd)) and the initial velocity u0 keeps in the critical space Ḃ
−1+ d

p

p,1 (Rd). These existing
results fundamentally assume either density continuity or small perturbations around a positive
constant, thus precluding the analysis of discontinuous densities with bounded cases. Zhang [28]
pioneered the study of system (1.2) with merely bounded initial density ρ0 (having a positive
lower bound), establishing global existence when the initial datum u0 has sufficiently small norm

in the critical Besov space Ḃ
1/2
2,1 (Rd). The uniqueness of Zhang’s solutions [28] was subsequently

established by Danchin and Wang [15] who showed ∇u ∈ L(0, T ;L∞(R3)) through careful analysis
in Lorentz spaces. Concurrently, the authors [15] proved global existence and uniqueness for the
system (1.2) under the specific initial data requirements that ∥ρ0 − 1∥L∞(R3) ≤ c for a sufficiently

small constant c and u0 belongs to Ḃ
−1+ 3

p

p,1 (R3) if 1 < p ≤ 2 or u0 belongs to Ḃ
−1+ 3

p

p,1 (R3) if

2 < p < 3. The Lipschitz property of the fluid flow (i.e. ∇u ∈ L(0, T ;L∞(R3)) plays a pivotal
role in establishing the uniqueness. For technical details, we can also refer to the recent work
of Danchin [12]. More related developments on the Navier-Stokes equations in critical function
spaces are documented in [2, 3, 4, 10, 19, 21, 27].

For the special case of non-vanishing density ρ ≡ 0, the coupled system (1.1) reduces to the
inhomogeneous incompressible nematic liquid crystal flow equations. There exists an extensive
literature concerning the well-posedness of system (1.1) in critical function spaces. In 2015, De
Anna [16] proved the global existence of solutions to the system (1.1) with (ρ−1

0 − 1, u0,∇d0) ∈
L∞(R3)× Ḃ

3
p−1
p,r (R3)× Ḃ

3
p−1
p,r (R3) (1 < p < 3, 1 < r < ∞) and ρ0 bounded away from vacuum and

infinity, while complete well-posedness (including uniqueness) required the additional regularity.
Building on the works of [1] and [11], Zhai, Li and Yan [30] established local well-posedness for the

system when the initial data satisfy (ρ−1
0 −1, u0,d0−d̄0) ∈ Ḃ

3/2
2,1 (R3)×Ḃ

1/2
2,1 (R3)×Ḃ

3/2
2,1 (R3), where

the orientation vector d asymptotically tends to d̄0 as |x| → ∞. Furthermore, they extended this
to global existence and uniqueness by imposing some additional conditions on the initial data that

C∥ρ−1
0 − 1∥

Ḃ
3/2
2,1 (R3)

(1 + ∥u3
0∥Ḃ1/2

2,1 (R3)
+ (∥uh

0∥Ḃ1/2
2,1 (R3)

+ ∥d0∥2Ḃ3/2
2,1 (R3))

) ≤ 1,

C(∥uh
0∥Ḃ1/2

2,1 (R3)
+ ∥d0∥Ḃ3/2

2,1 (R3)
(1 + (∥uh

0∥Ḃ1/2
2,1 (R3)

+ ∥d0∥Ḃ3/2
2,1 (R3)

)2

+ ∥u3
0∥Ḃ1/2

2,1 (R3)
+ (∥uh

0∥Ḃ1/2
2,1 (R3)

+ ∥d0∥Ḃ3/2
2,1 (R3)

)1/2) ≤ 1.

Building upon the techniques introduced in [13], Hu and Liu [20] successfully generalized the
results to system (1.1). In the recent works [8], Chen, Liang, and Ye [8] resolved an open ques-
tion from [16] by establishing global existence and uniqueness of weak solutions to system (1.1)
in the critical Besov space. Their result require 0 < c0 ≤ ρ0 ≤ C0 < +∞ and small norm
∥(u0,∇d0)∥Ḃ1/2

2,1 (R3)
. Subsequently, Chen et al. [9] relaxed the regularity assumptions, proving well-

posedness for (u0,∇d0) ∈ Ḣ1/2(R3)× (Ḣ1/2(R3) ∩ Ḃ
1/2

2, 43
(R3) and the initial density ρ0 ∈ L∞(R3)

permitting vacuum states. This extension substantially improves their earlier result in [8], par-
ticularly in handling density degeneracy. Notable progress on critical space methods for nematic
liquid crystal systems includes the fundamental contributions of [7, 18, 23, 24, 25, 26].



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 3

Inspired by [15], we prove the existence of unique global solutions to the three-dimensional
inhomogeneous incompressible nematic liquid crystal system under the regularity assumptions
that the initial density ρ0 satisfies ∥ρ0 − 1∥L∞(R3) ≤ c for sufficiently small constant c > 0 with
no additional regularity required and the initial data (u0,∇d0) belongs to the critical Besov space

Ḃ
−1+ 3

p

p,1 (R3) for 1 < p ≤ 2 and 2 < p < 3. Now we can formulate our main results.

Theorem 1.1. Let p ∈ (1, 3) and q ∈ (1,∞) such that 3
p + 2

q = 3. There exist a positive constant

c such that if the initial density ρ0 satisfies

∥ρ0 − 1∥L∞(R3) < c, (1.4)

and if the initial data (u0,∇d0) satisfies

(u0,∇d0) ∈ (Ḃ
−1+ 3

p

p,1 (R3))2 (1 < p ≤ 2),

(u0,∇d0) ∈ (Ḃ
−1+ 3

p

p,1 (R3) ∩ L2(R3))2 (2 < p < 3),
(1.5)

with
∥(u0,∇d0)∥

Ḃ
−1+ 3

p
p,1 (R3)

< c, (1.6)

then system (1.1) has a unique global-in-time weak solution (ρ, u,∇P,d) with

∇P ∈ Lq,1(R+;Lp(R3)) and (u,∇d) ∈ (Ẇ 2,1
p,(q,1)(R

+ × R3))2. (1.7)

If p > 2, then the following estimate holds:

∥ρ− 1∥L∞(R+×R3) = ∥ρ0 − 1∥L∞(R3) < c, (1.8)

and, furthermore, the following properties hold:

• ∇u, ∇2d ∈ L1(R+;L∞(R3)) and u,∇d ∈ L2(R+;L∞(R3));

• tu, t∇d ∈ W 2,1
m,(s,1)(R

+;Lm(R3)) and t∇P ∈ Ls,1(R+;Lm(R3)) for 3 < m < ∞ and

q < s < ∞ such that 3
m + 2

s = 1;

• tDtu, tDt∇d ∈ Ẇ 2,1
p,(q,1)(R

+ × R3);

• u,∇d, tDtu, tDt∇d ∈ Ls,1(R+;Lm(R3)).

Remark 1.2. The work in [29] established the existence of a unique global solution for the three-
dimensional compressible liquid crystal flow without imposing smallness conditions. It is our belief
that this result can be extended to certain classes of large initial data via a decomposition into
vertical and horizontal components. A rigorous consideration of this extension will be discussed
in our future research.

Remark 1.3. Remarkably, by Besov embeddings Ḃs
p1,r1(R

d) ↪→ Ḃ
s−d( 1

p1
− 1

p2
)

p2,r2 (Rd) for 1 ≤ p1 ≤
p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞, Theorem 1.1 in [8] imposes milder conditions than our Theorem
1.1 if 1 < p ≤ 2 and consequently obtains sharper results. Nevertheless, our work achieves greater
generality by extending the admissible integrability range from p = 2 to 1 < p < 3, while [8]

restricts initial data to critical Besov spaces Ḃ
1/2
2,1 (R3).

Remark 1.4. The perturbation condition (1.4) is introduced to reformulate the momentum equa-
tion into a homogeneous liquid crystal system. This transformation enables us to exploit the
smallness of (1.4) during energy estimates, allowing its terms to be absorbed by the left hand.

The following theorem states the uniqueness of the solutions to system (1.1).

Theorem 1.5. Let T > 0 and consider two solutions (ρ1, u1, P1,d1) and (ρ2, u2, P2,d2) of system
(1.1) on [0, T ]× R3 sharing identical initial data. Furthermore, we assume that

• √
ρ
1
(u1 − u2),∇d1 −∇d2 ∈ L∞(0, T ;L2(R3));

• ∇u1 −∇u2,∇2d1 −∇2d2 ∈ L2(0, T ;L2(R3));
• ∇u2,∇2d2,∇2d1 ∈ L1(0, T ;L∞(R3));
• ∇d1,∇d2 ∈ L2(0, T ;L∞(R3));
• tDtu2 ∈ L2(0, T ;L∞(R3));



4 X. WU, X. LIANG EJDE-2025/101

• t∇Dtu2 ∈ L2(0, T ;L3(R3));

• ∇d1 ∈ Ḃ
−1+ 3

p

p,1 (R3).

Then (ρ1, u1, P1,d1) = (ρ2, u2, P2,d2) on [0, T ]× R3.

Remark 1.6. Unlike in [15], the proof of uniqueness via energy estimates here requires condition

∇d1 ∈ Ḃ
−1+ 3

p

p,1 (R3), as specified in (5.2) and (5.3).

The rest of this article is organized as follows. In Section 2, we present some information on
the Besov spaces and Lorentz spaces. Section 3 states the a priori estimates and gives the proof
of Theorem 1.1. The existence and uniqueness of solutions are established in Sections 4 and 5.

We conclude this section by establishing some notation that will be used throughout this article.
The material derivative operator is defined as Dt := ∂t + u · ∇. Given a Banach space X and q ∈
[1,+∞), the Banach space Lq(0, T ;X) consists of all strongly measurable functions f : (0, T ) → X
such that the mapping t 7→ ∥f(t)∥X belongs to Lq(0, T ;X). The space is equipped with the norm:

∥f∥Lq(0,T ;X) :=
( ∫ T

0
∥f(t)∥qX dt

)1/q
. For any pair of measurable functions (f, g) on X, we define

the joint norm ∥(f, g)∥X := ∥f∥X + ∥g∥X .

2. Preliminaries

In this section, we present the definitions of Besov spaces, Lorentz spaces, and key lemmas,
which can be found in the references [5, 6, 14, 15, 17, 22].

First, we introduce the dyadic decomposition and the homogeneous Besov spaces. Given a
tempered distribution u ∈ S ′

h, we define

∀j ∈ Z, ∆̇ju := φ(2−j ·)u, Ṡju =
∑

j′≤j−1

∆̇j′u = χ(2−j · u),

where χ(τ) and φ(τ) are smooth functions such that

suppφ ⊂ C and ∀τ > 0,
∑
j∈Z

φ(2−jτ) = 1,

suppχ ⊂ B and ∀τ > 0, χ(τ) + sumj≥0φ(2
−jτ) = 1.

Here C = {τ ∈ Rd| 34 ≤ |τ | ≤ 8
3}, B = {τ ∈ Rd||τ | ≤ 4

3}.

Definition 2.1 ([5]). Let (p, r) be in [1,∞]2 and s be in R. Assume that u ∈ S ′
h(Rd), which

means that u is in S ′(Rd) and satisfies limj→−∞ ∥Sju∥L∞ = 0. We set

∥u∥Ḃs
p,r

:= ∥(2js∥∆̇ju∥Lp)j∈Z∥lr(Z).

(i) If s < d
p (or s =

d
p if r = 1), we define Ḃs

p,r := {u ∈ S ′
h(Rd)|∥u∥Ḃs

p,r
< ∞}.

(ii) If k ∈ N and if s < d
p (or s=

d
p if r = 1), then we define Ḃs

p,r(Rd) as the subset of u in S ′
h(Rd)

such that ∂βu belongs to Ḃs−k
p,r (Rd) whenever |β| = k.

Lemma 2.2 (Besov embedding [5]). (1) For any (p, q) in [1,∞]2 such that p ≤ q, we have

Ḃ
d
p−

d
q

p,1 (Rd) ↪→ Lq(Rd).

(2) Let 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞. Then, for any real number s,

Ḃs
p1,r1(R

d) ↪→ Ḃ
s−d( 1

p1
− 1

p2
)

p2,r2 (Rd).

Let us define Lorentz spaces and recall some useful properties.

Definition 2.3 ([17, 22]). Given f a measurable function on a measure space (X, ν) and 1 ≤
p, r ≤ ∞, we define

∥f∥Lp,r(X,ν) =

{( ∫∞
0

(t1/pf∗(t))r dt
t

)1/r
, if r < ∞,

supt>0 t
1/pf∗(t), if r = ∞.

(2.1)



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 5

where f∗(t) := inf{s ≥ 0|ν(|f | > s) ≤ t}. The set of all f with ∥f∥Lp,r(X,ν) is called the Lorentz
space with indices p and r.

The following properties of Lorentz spaces will play an important part in the proof of uniqueness.

Lemma 2.4 ([17]). Let Ω be a measurable set, R+ be the positive real set and Rd be the d-
dimensional Euclidean space. Then we have the following properties.

(1) Embedding: Lp,r1(Ω) ↪→ Lp,r2(Ω) if r1 ≤ r2, and Lp,p(Ω) = Lp(Ω);
(2) the Hölder inequality: For 1 < p, p1, p2 < ∞ and 1 ≤ r, r1, r2 ≤ ∞, it holds

∥fg∥Lp,r(Ω) ≤ C∥f∥Lp1,r1 (Ω)∥g∥Lp2,r2 (Ω)

with 1
p = 1

p1
+ 1

p2
and 1

r = 1
r1

+ 1
r2
. This inequality still holds for the pairs (1, 1) and

(∞,∞) with the convention L1,1(Ω) = L1(Ω) and L∞,∞(Ω) = L∞(Ω);
(3) For any α > 0 and nonnegative measurable function f , one has ∥fα∥Lp,r(Ω) = ∥f∥αLpα,rα(Ω);

(4) For any k > 0, one has ∥x−k1R+∥
L

1
k

,∞(Ω)
= 1.

Next, we introduce the Besov interpolation.

Lemma 2.5 ([15, Proposition A.4]). Let 1 ≤ q < ∞, 1 ≤ p < r ≤ ∞, and θ ∈ (0, 1) such that

1

r
+

1

d
− 2θ

dq
=

1

p
.

Then, there exists C such that

∥∇u∥Lr(Rd) ≤ C∥∇2u∥θLp(Rd)∥u∥
1−θ

Ḃ
2− 2

q
p,∞ (Rd)

.

The following two maximal regularity estimates for Navier-Stokes equations play a pivotal role
in the proof of our main results.

Lemma 2.6 ([15, Proposition A.5]). Let 1 < p, q < ∞ and 1 ≤ r ≤ ∞. Then for any initial data

u0 ∈ Ḃ
2− 2

q
p,r (Rd) with div u0 = 0, and any external force f ∈ Lq,r(0, T ;Lp(Rd)), the Stokes system

∂tu−∆u+∇P = f, (t, x) ∈ R+ × Rd,

div u = 0,

u|t=0 = u0.

in

Ẇ 2,1
p,(q,r)((0, T )× Rd) := {u ∈ C([0, T ]; Ḃ

2− 2
q

p,r (Rd)) : ut,∇2u ∈ Lq,r(0, T ;Lp(Rd))}
admits a unique solution (u,∇P ) satisfying ∇P ∈ Lq,r(0, T ;Lp(Rd)) and the maximal regularity
estimate

∥u∥
L∞(0,T ;Ḃ

2− 2
q

p,r (Rd))
+ ∥(ut,∇2u,∇P )∥Lq,r(0,T ;Lp(Rd))

≤ C
(
∥u0∥

Ḃ
2− 2

q
p,r (Rd)

+ ∥f∥Lq,r(0,T ;Lp(Rd))

)
,

where the constant C > 0 is independent of T .

Next we have a maximal regularity estimate for heat equation.

Lemma 2.7 ([14, Proposition 2.1]). Let 1 < p, q < ∞ and 1 ≤ r ≤ ∞. Then for any initial data

u0 ∈ Ḃ
2− 2

q
p,r (Rd) and any source term f ∈ Lq,r(0, T ;Lp(Rd)), the heat equation

∂tu−∆u = f, (t, x) ∈ R+ × Rd,

u|t=0 = u0.

in

Ẇ 2,1
p,(q,r)(R

+ × Rd) := {u ∈ Cb(R+; Ḃ
2− 2

q
p,r (Rd)) : ut,∇2u ∈ Lq,r(R+;Lp(Rd))}

admits a unique solution u satisfying the maximal regularity estimate

∥u∥
L∞(R+;Ḃ

2− 2
q

p,r (Rd))
+ ∥(ut,∇2u)∥Lq,r(R+;Lp(Rd)) ≤ C

(
∥u0∥

Ḃ
2− 2

q
p,r (Rd)

+ ∥f∥Lq,r(R+;Lp(Rd))

)
.



6 X. WU, X. LIANG EJDE-2025/101

3. Proof of Theorem 1.1

A priori estimates.

Proposition 3.1. Let (ρ, u,d) be a smooth solution of system (1.1) on [0, T ] × R3, with u and
∇d sufficiently decaying at infinity and ρ satisfying

sup
t∈[0,T ]

∥ρ(t)− 1∥L∞(R3) ≤ c << 1, (3.1)

Then, for all 1 < m, p, q, s < ∞ such that

3

p
+

2

q
= 3 and

3

m
+

2

s
= 1 with p < m < ∞ and q < s < ∞, (3.2)

it follows that

∥(u,∇d)∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥(ut, Dtu,∇2u,∇P,∇dt, Dt∇d,∇3d)∥Lq,1(0,T ;Lp(R3))

+ ∥(u,∇d)∥Ls,1(0,T ;Lm(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
,

(3.3)

and
∥(u,∇d)∥L2(0,T ;L∞(R3)) ≤ C∥(u0,∇d0)∥

Ḃ
−1+ 3

p
p,1 (R3)

. (3.4)

Proof. First, we can rewrite (1.1)2 as

ut −∆u+∇P = (1− ρ)ut − ρu · ∇u− div(∇d⊙∇d). (3.5)

From the Hölder inequality in Lorentz spaces and the maximal regularity estimates for the Navier-
Stokes equations, it follows that

∥u∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥(ut,∇2u,∇P )∥Lq,1(0,T ;Lp(R3)) + ∥u∥Ls,1(0,T ;Lp(R3))

≤ C(∥u0∥
Ḃ

−1+ 3
p

p,1 (R3)
+ ∥(1− ρ)ut∥Lq,1(0,T ;Lp(R3)) + ∥ρu · ∇u∥Lq,1(0,T ;Lp(R3))

+ ∥div(∇d⊙∇d)∥Lq,1(0,T ;Lp(R3)))

≤ C(∥u0∥
Ḃ

−1+ 3
p

p,1 (R3)
+ ∥ρ− 1∥L∞(0,T ;L∞(R3))∥ut∥Lq,1(0,T ;Lp(R3))

+ ∥ρ∥L∞(0,T ;L∞(R3))∥u · ∇u∥Lq,1(0,T ;Lp(R3)) + ∥ div(∇d⊙∇d)∥Lq,1(0,T ;Lp(R3))).

It follows from condition (3.1) that the second term on the right-hand side of the above inequality
can be absorbed by the left-hand side, and ∥ρ∥L∞(0,T ;L∞(R3)) is a bounded quantity.
• Estimate for ∥u · ∇u∥Lq,1(0,T ;Lp(R3)). By the embedding

Ḃ
−1+ 3

p

p,1 (R3) ↪→ L3(R3) (3.6)

and

Ẇ 1,p(R3) ↪→ Lp∗(R3) with
1

p∗
=

1

p
− 1

3
, (3.7)

one has

∥u · ∇u∥Lq,1(0,T ;Lp(R3)) ≤ C∥u∥L∞(0,T ;L3(R3))∥∇u∥Lq,1(0,T ;Lp∗ (R3))

≤ C∥u∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇2u∥Lq,1(0,T ;Lp(R3)).

• Estimate for ∥ div(∇d⊙∇d)∥Lq,1(0,T ;Lp(R3))

∥ div(∇d⊙∇d)∥Lq,1(0,T ;Lp(R3)) ≤ C∥∇d∥L∞(0,T ;L3(R3))∥∇2d∥Lq,1(0,T ;Lp∗ (R3))

≤ C∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3)).

Applying ∇ to (1.1)3, we obtain

∇dt +∇uT∇d + u · ∇(∇d) = ∆∇d +∇(|∇d|2)d + |∇d|2∇d,



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 7

which can be written as

∇dt −∆∇d = −∇uT∇d− u · ∇(∇d) +∇(|∇d|2)d + |∇d|2∇d. (3.8)

On the other hand, the maximal regularity estimate for the heat equation yields

∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥(∇dt,∇3d)∥Lq,1(0,T ;Lp(R3)) + ∥∇d∥Ls,1(0,T ;Lp(R3))

≤ C(∥∇d0∥
Ḃ

−1+ 3
p

p,1 (R3)
+ ∥∇uT∇d∥Lq,1(0,T ;Lp(R3)) + ∥u · ∇(∇d)∥Lq,1(0,T ;Lp(R3))

+ ∥∇(|∇d|2)d∥Lq,1(0,T ;Lp(R3)) + ∥|∇d|2∇d∥Lq,1(0,T ;Lp(R3))).

• Estimate for ∥∇uT∇d∥Lq,1(0,T ;Lp(R3)).

∥∇uT∇d∥Lq,1(0,T ;Lp(R3)) ≤ C∥∇d∥L∞(0,T ;L3(R3))∥∇u∥Lq,1(0,T ;Lp∗ (R3))

≤ C∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇2u∥Lq,1(0,T ;Lp(R3)).

• Estimate for ∥u · ∇(∇d)∥Lq,1(0,T ;Lp(R3)).

∥u · ∇(∇d)∥Lq,1(0,T ;Lp(R3)) ≤ C∥u∥L∞(0,T ;L3(R3))∥∇2d∥Lq,1(0,T ;Lp∗ (R3))

≤ C∥u∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3)).

• Estimate for ∥∇(|∇d|2)d∥Lq,1(0,T ;Lp(R3)).

∥∇(|∇d|2)d∥Lq,1(0,T ;Lp(R3)) ≤ C∥∇d∥L∞(0,T ;L3(R3))∥∇2d∥Lq,1(0,T ;Lp∗ (R3))∥d∥L∞(0,T ;L∞(R3))

≤ C∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3)).

• Estimate for ∥|∇d|2∇d∥Lq,1(0,T ;Lp(R3)). From the Gagliardo-Nirenberg inequality, we have

∥|∇d|2∇d∥Lp(R3) ≤ C∥∇d∥3L3p(R3) ≤ C(∥∇d∥
2
3

L3(R3)∥∇
3d∥

1
3

Lp(R3))
3

≤ C∥∇d∥2L3(R3)∥∇
3d∥Lp(R3).

Then, employing the Hölder inequality in Lorentz spaces, it holds

∥|∇d|2∇d∥Lq,1(0,T ;Lp(R3)) ≤ C∥∇d∥2L∞(0,T ;L3(R3))∥∇
3d∥Lq,1(0,T ;Lp(R3))

≤ C∥∇d∥2
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3)).

Next, we denote

E0 := ∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
, T ∗ := sup

{
t < T : ∥(u,∇d)∥

L∞(0,t;Ḃ
−1+ 3

p
P,1 (R3))

≤ c1

}
,

E1(t) := ∥(u,∇d)∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥(ut, Dtu,∇2u,∇P,∇dt, Dt∇d,∇3d)∥Lq,1(0,T ;Lp(R3))

+ ∥(u,∇d)∥Ls,1(0,T ;Lm(R3)).

We conclude that E1(t) ≤ CE0 + Cc1E1(t). By selecting c1 sufficiently small to ensure Cc1 ≤ 1
2 ,

we obtain

E1(t) ≤ 2CE0.

By taking ε0 sufficiently small to ensure 2Cε0 ≤ c1/2, then from the standard continuous argument,
we can find that

E1(t) ≤ CE0.

On the other hand, by the Hölder inequality in Lorentz spaces, the Gagliardo-Nirenberg inequality
and the embedding, one has

∥(Dtu,Dt∇d)∥Lq,1(0,T ;Lp(R3))

≤ C∥(ut,∇dt)∥Lq,1(0,T ;Lp(R3)) + C∥u∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥(∇2u,∇3d)∥Lq,1(0,T ;Lp(R3))



8 X. WU, X. LIANG EJDE-2025/101

≤ CE1(t) + C(E1(t))
2

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

Finally, by using the Gagliardo-Nirenberg inequality and the embedding

∥f∥L∞(R3) ≤ C∥f∥1−
q
2

L3(R3)∥∇
2f∥q/2Lp(R3) ≤ C∥f∥1−

q
2

Ḃ
−1+ 3

p
p,1 (R3)

∥∇2f∥q/2Lp(R3),

we have∫ T

0

∥(u,∇d)∥2L∞(R3) dτ ≤ C

∫ T

0

∥(u,∇d)∥2−q

Ḃ
−1+ 3

p
p,1 (R3)

∥(∇2u,∇3d)∥qLp(R3) dτ

≤ C∥(u,∇d)∥2−q

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

∥(∇2u,∇3d)∥qLq,1(0,T ;Lp(R3))

≤ C(E1(t))
2 ≤ C∥(u0,∇d0)∥2

Ḃ
−1+ 3

p
p,1 (R3)

. □

Proposition 3.2. Under the hypotheses of Proposition 3.1, we obtain

∥t(u,∇d)∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
+ ∥((tu)t,∇2(tu),∇(tP ), (t∇d)t,∇3(td))∥Ls,1(0,T ;Lm(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

(3.9)

Furthermore, it follows that∫ T

0

∥(∇u,∇2d)∥L∞(R3)dt+
(∫ T

0

t∥(∇u,∇2d)∥2L∞(R3)dt
)1/2

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

Proof. Multiplying both sides of (3.5) by time t yields

(tu)t −∆(tu) +∇(tP ) = (1− ρ)(tu)t + ρu− tρu · ∇u− tdiv(∇d⊙∇d).

From the Hölder inequality in Lorentz spaces and the maximal regularity estimates for the Navier-
Stokes equations, it follows that

∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
+ ∥((tu)t,∇2(tu),∇(tP ))∥Ls,1(0,T ;Lm(R3))

≤ C∥((1− ρ)(tu)t + ρu− tρu · ∇u− tdiv(∇d⊙∇d))∥Ls,1(0,T ;Lm(R3))

≤ C∥ρ− 1∥L∞(0,T ;L∞(R3))∥(tu)t∥Ls,1(0,T ;Lm(R3))

+ C∥ρ∥L∞(0,T ;L∞(R3))

(
∥u∥Ls,1(0,T ;Lm(R3)) + ∥tu · ∇u∥Ls,1(0,T ;Lm(R3))

)
+ C∥tdiv(∇d⊙∇d)∥Ls,1(0,T ;Lm(R3)).

(3.10)

It follows from condition (3.1) that the first term on the right-hand side of the above inequality
can be absorbed by the left-hand side, and ∥ρ∥L∞(0,T ;L∞(R3)) is a bounded quantity.
• Estimate for ∥tu · ∇u∥Ls,1(0,T ;Lm(R3)). By the Hölder inequality in Lorentz spaces and the
embedding

Ḃ
3/m
m,1 (R

3) ↪→ L∞(R3), (3.11)

one has

∥tu · ∇u∥Ls,1(0,T ;Lm(R3)) ≤ C∥u∥Ls,1(0,T ;Lm(R3))∥t∇u∥L∞(0,T ;L∞(R3))

≤ C∥u∥Ls,1(0,T ;Lm(R3))∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
.

• Estimate for ∥tdiv(∇d⊙∇d)∥Ls,1(0,T ;Lm(R3)). Similarly, one has

∥tdiv(∇d⊙∇d)∥Ls,1(0,T ;Lm(R3)) ≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇2d∥L∞(0,T ;L∞(R3))

≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
.

On the other hand, multiplying both sides of (3.8) by time t yields

(t∇d)t − t∆∇d = ∇d− t∇uT∇d− tu · ∇(∇d) + t∇(|∇d|2)d + t|∇d|2∇d.



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 9

The maximal regularity estimate for the heat equation yields

∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
+ ∥((t∇d)t,∇3(t∇d))∥Ls,1(0,T ;Lm(R3))

≤ C∥∇d∥Ls,1(0,T ;Lm(R3)) + C∥t∇uT∇d∥Ls,1(0,T ;Lm(R3))

+ C∥tu · ∇(∇d)∥Ls,1(0,T ;Lm(R3)) + C∥t∇(|∇d|2)d∥Ls,1(0,T ;Lm(R3))

+ C∥t|∇d|2∇d∥Ls,1(0,T ;Lm(R3)).

(3.12)

• Estimate for ∥t∇uT∇d∥Ls,1(0,T ;Lm(R3)). By the Hölder inequality in Lorentz spaces and the
embedding (3.11), one has

∥t∇uT∇d∥Ls,1(0,T ;Lm(R3)) ≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇u∥L∞(0,T ;L∞(R3))

≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
.

• Estimate for ∥tu · ∇(∇d)∥Ls,1(0,T ;Lm(R3)). Similarly, one has

∥tu · ∇(∇d)∥Ls,1(0,T ;Lm(R3)) ≤ C∥u∥Ls,1(0,T ;Lm(R3))∥t∇2d∥L∞(0,T ;L∞(R3))

≤ C∥u∥Ls,1(0,T ;Lm(R3))∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
.

• Estimate for ∥t∇(|∇d|2)d∥Ls,1(0,T ;Lm(R3)). Similarly, one has

∥t∇(|∇d|2)d∥Ls,1(0,T ;Lm(R3)) ≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇2d∥L∞(0,T ;L∞(R3))∥d∥L∞(0,T ;L∞(R3))

≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
.

• Estimate for ∥t|∇d|2∇d∥Ls,1(0,T ;Lm(R3)). From the Gagliardo-Nirenberg inequality, we have

∥|∇d|2∇d∥Lm(R3) ≤ C∥∇d∥3L3m(R3) ≤ C(∥∇d∥
2
3

L3(R3)∥∇
3d∥

1
3

Lm(R3))
3

≤ C∥∇d∥2L3(R3)∥∇
3d∥Lm(R3).

Similarly, one has

∥t|∇d|2∇d∥Ls,1(0,T ;Lm(R3)) ≤ C∥∇d∥2L∞(0,T ;L3(R3))∥t∇
3d∥Ls,1(0,T ;Lm(R3))

≤ C∥∇d∥2
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥t∇3d∥Ls,1(0,T ;Lm(R3)).

Now let us denote

E2(t) := ∥t(u,∇d)∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
+ ∥((tu)t,∇2(tu),∇(tP ), (t∇d)t,∇3(td))∥Ls,1(0,T ;Lm(R3)).

Then we have
E2(t) ≤ C(1 + E2(t)) · (E1(t) + (E1(t))

2).

Applying the theory of bootstrapping again, one has

E2(t) ≤ CE0.

Bounding ∇u,∇2d relies on the following interpolation inequality (as (3.2) implies that p < 3 <
m),

∥f∥L∞(R3) ≤ ∥∇f∥
p(m−3)
3(m−p)

Lp(R3) ∥∇f∥
m(3−p)
3(m−p)

Lm(R3). (3.13)

Applying the Hölder inequality in Lorentz spaces with exponents:

(p1, r1) =
(3(m− p)

m(3− p)
,∞

)
, (p2, r2) =

(3q(m− p)

p(m− 3)
,
p2
q

)
, (p3, r3) =

(3s(m− p)

m(3− p)
,
p3
s

)
,

and using that t−α ∈ L
1
α ,∞(R+) with α = m(3−p)

3(m−p) , (3.3) and the first inequalitv of Proposition 3.1

we end up with∫ T

0

∥(∇u,∇2d)∥L∞(R3)dt ≤ C

∫ T

0

t−
m(3−p)
3(m−p) ∥(∇2u,∇3d)∥

p(m−3)
3(m−p)

Lp(R3) ∥t(∇
2u,∇3d)∥

m(3−p)
3(m−p)

Lm(R3)dt

≤ C∥(∇2u,∇3d)∥
p(m−3)
3(m−p)

Lq,1(0,T ;Lp(R3))∥t(∇
2u,∇3d)∥

m(3−p)
3(m−p)

Ls,1(0,T ;Lm(R3))



10 X. WU, X. LIANG EJDE-2025/101

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

Morever, from the Hölder inequality and the embedding (3.11), one has∫ T

0

t∥(∇u,∇2d)∥2L∞(R3)dt ≤
∫ T

0

t∥(∇u,∇2d)∥
Ḃ

3/m
m,1 (R3)

∥(∇u,∇2d)∥L∞(R3)dt

≤ ∥t(u,∇d)∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))

∫ T

0

∥(∇u,∇2d)∥L∞(R3)dt

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
. □

To prove the uniqueness, we need the following time weighted estimate.

Proposition 3.3. Under the hypotheses of Proposition 3.1, we obtain

∥t(Dtu,Dt∇d)∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))

+ ∥((tDtu)t, t∇2Dtu, (tDt∇d)t, t∇2Dt∇d)∥Lq,1(0,T ;Lp(R3))

+ ∥t(Dtu,Dt∇d)∥Ls,1(0,T ;Lm(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

(3.14)

Furthermore,

∥t(∇Dtu,∇Dt∇d)∥L2(0,T ;L3(R3)) + ∥t(Dtu,Dt∇d)∥L2(0,T ;L∞(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
. (3.15)

Proof. Applying Dt to (1.1)2 , we obtain

Dt(ρDtu)−Dt(∆u) +Dt(∇P ) = −Dt(div(∇d⊙∇d)).

By the definition of commutator, we have

[Dt;∇]f := −∇u · ∇f, [Dt; ∆]f := −∆u · ∇f − 2∇u · ∇2f.

Therefore,
Dt(ρDtu) = DtρDtu+ ρDtDtu = ρDtDtu = ρD2

t u,

ρD2
t u−∆Dtu+∇DtP = f,

(3.16)

where
f := −∆u · ∇u− 2∇u · ∇2u+∇u · ∇P −Dt(div(∇d⊙∇d)).

Multiplying both sides of (3.16) by time t yields

ρ(tDtu)t −∆(tDtu) +∇(tDtP ) = −tρu · ∇Dtu+ ρDtu+ tf.

Since div(tDtu) ̸= 0, we cannot directly apply the maximal regularity estimates for the Navier-
Stokes equations. However, the maximal regularity estimates for the heat equation [14, Proposition
2.1] need not div(tDtu) = 0. But we need to eliminate ∇(tDtP ). Now, we introduce the definition
of Helmholtz decomposition

P := Id +∇(−∆)−1div , Q := −∇(−∆)−1div ,

where P stands for Leray projector, while Q represents the orthogonal complement of P. Mean-
while, we observe that

∇(tDtP ) = Q
(
−tρu · ∇Dtu+ ρDtu+ tf − ρ(tDtu)t +∆(tDtu)

)
.

Since div u = 0, one has

divDtu =
∑

1≤i,j≤3

∂iu
j∂ju

i = Tr(∇u · ∇u),

Q(t∆Dtu) = t∇Tr(∇u · ∇u),

Q(u · ∇ut) = Q(ut · ∇u),



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 11

Q
(
ρ(tDtu)t

)
= Q

(
(ρ− 1)(tDtu)t +Dtu+ tu · ∇ut + tut · ∇u

)
,

and
(tDtu)t −∆(tDtu) = P[(1− ρ)(tDtu)t − tρu · ∇Dtu+ ρDtu+ tf ]

+Q(Dtu+ 2tut · ∇u)− t∇Tr(∇u · ∇u).
(3.17)

Applying the maximal regularity estimates for the heat equation [14, Proposition 2.1], the Hölder
inequality in Lorentz spaces and t that the Helmholtz projectors P and Q are bounded operators
on Lp(Rd), it follows that

∥tDtu∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥((tDtu)t, t∇2Dtu)∥Lq,1(0,T ;Lp(R3)) + ∥tDtu∥Ls,1(0,T ;Lm(R3))

≤ C(∥(1− ρ)(tDtu)t∥Lq,1(0,T ;Lp(R3)) + ∥tρu · ∇Dtu∥Lq,1(0,T ;Lp(R3))

+ ∥ρDtu∥Lq,1(0,T ;Lp(R3)) + ∥tf∥Lq,1(0,T ;Lp(R3)) + ∥Dtu∥Lq,1(0,T ;Lp(R3))

+ ∥tut · ∇u∥Lq,1(0,T ;Lp(R3)) + ∥t∇Tr(∇u · ∇u)∥Lq,1(0,T ;Lp(R3)))

≤ C(∥ρ− 1∥L∞(0,T ;L∞(R3))∥(tDtu)t∥Lq,1(0,T ;Lp(R3)) + ∥tρu · ∇Dtu∥Lq,1(0,T ;Lp(R3))

+ ∥ρDtu∥Lq,1(0,T ;Lp(R3)) + ∥tf∥Lq,1(0,T ;Lp(R3)) + ∥Dtu∥Lq,1(0,T ;Lp(R3))

+ ∥tut · ∇u∥Lq,1(0,T ;Lp(R3)) + ∥t∇Tr(∇u · ∇u)∥Lq,1(0,T ;Lp(R3))).

It follows from condition (3.1) that the first term on the right-hand side of the above inequality
can be absorbed by the left-hand side, and ∥ρ∥L∞(0,T ;L∞(R3)) is a bounded quantity.
• Estimate for ∥tf∥Lq,1(0,T ;Lp(R3)) and ∥t∇Tr(∇u · ∇u)∥Lq,1(0,T ;Lp(R3)). By the Hölder inequality
in Lorentz spaces and (3.11), we have

∥ − t∆u · ∇u− 2t∇u · ∇2u+ t∇u · ∇P∥Lq,1(0,T ;Lp(R3)) + ∥t∇Tr(∇u · ∇u)∥Lq,1(0,T ;Lp(R3))

≤ C∥(∇2u,∇P )∥Lq,1(0,T ;Lp(R3))∥t∇u∥L∞(0,T ;L∞(R3))

≤ C∥(∇2u,∇P )∥Lq,1(0,T ;Lp(R3))∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
,

and

∥tDt(div(∇d⊙∇d))∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt∇d · ∇2d∥Lq,1(0,T ;Lp(R3)) + C∥tDt∇2d · ∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt∇d · ∇2d∥Lq,1(0,T ;Lp(R3)) + C∥t∇Dt∇d · ∇d∥Lq,1(0,T ;Lp(R3))

+ C∥t∇u · ∇(∇d) · ∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt∇d∥L∞(0,T ;L3(R3))∥∇2d∥Lq,1(0,T ;Lp∗(R3))

+ C∥∇d∥L∞(0,T ;L3(R3))∥t∇Dt∇d∥Lq,1(0,T ;Lp∗(R3))

+ C∥t∇2d∥L∞(0,T ;L∞(R3))∥∇u∥Lq,1(0,T ;Lp∗(R3))∥∇d∥L∞(0,T ;L3(R3))

≤ C∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3))

+ C∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥t∇2Dt∇d∥Lq,1(0,T ;Lp(R3))

+ C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇2u∥Lq,1(0,T ;Lp(R3))∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
+ C∥(u0,∇d0)∥3

Ḃ
−1+ 3

p
p,1 (R3)

+ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
∥t∇2Dt∇d∥Lq,1(0,T ;Lp(R3)).

• Estimate for ∥ρDtu∥Lq,1(0,T ;Lp(R3)) and ∥Dtu∥Lq,1(0,T ;Lp(R3)). From Proposition 3.1, we obtain

∥ρDtu∥Lq,1(0,T ;Lp(R3)) + ∥Dtu∥Lq,1(0,T ;Lp(R3)) ≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.



12 X. WU, X. LIANG EJDE-2025/101

• Estimate for ∥tρu·∇Dtu∥Lq,1(0,T ;Lp(R3)) and ∥tut ·∇u∥Lq,1(0,T ;Lp(R3)). From the Hölder inequality
in Lorentz spaces, Proposition 3.1-3.2, the embedding (3.6), (3.7) and (3.11), one has

∥tρu · ∇Dtu∥Lq,1(0,T ;Lp(R3)) + ∥tut · ∇u∥Lq,1(0,T ;Lp(R3))

≤ C∥ρ∥L∞(0,T ;L∞(R3))∥t∇Dtu∥Lq,1(0,T ;Lp∗(R3))∥u∥L∞(0,T ;L3(R3))

+ C∥ut∥Lq,1(0,T ;Lp(R3))∥t∇u∥L∞(0,T ;L∞(R3))

≤ C∥t∇2Dtu∥Lq,1(0,T ;Lp(R3))∥u∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))

+ C∥ut∥Lq,1(0,T ;Lp(R3))∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))

≤ C∥t∇2Dtu∥Lq,1(0,T ;Lp(R3))∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
+ C∥(u0,∇d0)∥2

Ḃ
−1+ 3

p
p,1 (R3)

.

(3.18)

In summary, we have

∥tDtu∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥((tDtu)t, t∇2Dtu)∥Lq,1(0,T ;Lp(R3)) + ∥tDtu∥Ls,1(0,T ;Lm(R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
+ C∥(u0,∇d0)∥3

Ḃ
−1+ 3

p
p,1 (R3)

+ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
(∥t∇2Dtu∥Lq,1(0,T ;Lp(R3)) + ∥t∇2Dt∇d∥Lq,1(0,T ;Lp(R3)) + 1).

(3.19)

Next, applying Dt to (1.1)3, we obtain

D2
t∇d−∆Dt∇d = −Dt(∇uT∇d) +Dt(∇(|∇d|2)d)−∆u · ∇(∇d)− 2∇u · ∇2(∇d). (3.20)

Multiplying both sides of (3.20) by time t yields

(tDt∇d)t − t∆Dt∇d = −tDt(∇uT∇d) + tDt(∇(|∇d|2)d)− tu · ∇Dt∇d

+Dt∇d− t∆u · ∇(∇d)− 2t∇u · ∇2(∇d).

By the maximal regularity estimate for the heat equation, one has

∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥((tDt∇d)t, t∇2Dt∇d)∥Lq,1(0,T ;Lp(R3))

+ ∥tDt∇d∥Ls,1(0,T ;Lm(R3))

≤ C(∥tDt(∇uT∇d)∥Lq,1(0,T ;Lp(R3)) + ∥tDt(∇(|∇d|2)d)∥Lq,1(0,T ;Lp(R3))

+ ∥tu · ∇Dt∇d∥Lq,1(0,T ;Lp(R3)) + ∥Dt∇d∥Lq,1(0,T ;Lp(R3))

+ ∥t∆u · ∇(∇d)∥Lq,1(0,T ;Lp(R3)) + ∥t∇u · ∇2(∇d)∥Lq,1(0,T ;Lp(R3))).

(3.21)

• Estimate for ∥tDt(∇uT∇d)∥Lq,1(0,T ;Lp(R3)). By the Hölder inequality in Lorentz spaces and
(3.11), we obtain

∥tDt(∇uT∇d)∥Lq,1(0,T ;Lp(R3))

≤ C∥t(∇Dtu−∇u · ∇u)∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥∇d∥L∞(0,T ;L3(R3))∥t∇Dtu∥Lq,1(0,T ;Lp∗(R3))

+ C∥t∇u∥L∞(0,T ;L∞(R3))∥∇d∥L∞(0,T ;L3(R3))∥∇u∥Lq,1(0,T ;Lp∗(R3))

≤ C∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥t∇2Dtu∥Lq,1(0,T ;Lp(R3))

+ C∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

∥∇2u∥Lq,1(0,T ;Lp(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
∥t∇2Dtu∥Lq,1(0,T ;Lp(R3)) + C∥(u0,∇d0)∥3

Ḃ
−1+ 3

p
p,1 (R3)

.

• Estimate for ∥tDt(∇(|∇d|2)d)∥Lq,1(0,T ;Lp(R3)). Similarly, by the estimate for
∥tDt(div(∇d⊙∇d))∥Lq,1(0,T ;Lp(R3)), we obtain

∥tDt(∇(|∇d|2)d)∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt(∇2d · ∇d · d)∥Lq,1(0,T ;Lp(R3)) + C∥tDt(|∇d|2 · ∇d)∥Lq,1(0,T ;Lp(R3))



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 13

≤ C∥tDt∇2d · ∇d · d∥Lq,1(0,T ;Lp(R3)) + C∥t∇2d ·Dt∇d · d∥Lq,1(0,T ;Lp(R3))

+ C∥t∇2d · ∇d ·Dtd∥Lq,1(0,T ;Lp(R3)) + C∥tDt∇d · ∇d · ∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt∇2d · ∇d∥Lq,1(0,T ;Lp(R3))∥d∥L∞(0,T ;L∞(R3))

+ C∥t∇2d ·Dt∇d∥Lq,1(0,T ;Lp(R3))∥d∥L∞(0,T ;L∞(R3))

+ C∥t∇2d∥L∞(0,T ;L3(R3))∥∇d∥L∞(0,T ;L3(R3))∥Dtd∥Lq,1(0,T ;Lp∗(R3))

+ C∥tDt∇d∥L∞(0,T ;L3(R3))∥∇d · ∇d∥Lq,1(0,T ;Lp∗(R3))

≤ C∥tDt∇2d · ∇d∥Lq,1(0,T ;Lp(R3)) + C∥t∇2d ·Dt∇d∥Lq,1(0,T ;Lp(R3))

+ C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

∥∇Dtd∥Lq,1(0,T ;Lp(R3))

+ C∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇2d · ∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt∇2d · ∇d∥Lq,1(0,T ;Lp(R3)) + C∥t∇2d ·Dt∇d∥Lq,1(0,T ;Lp(R3))

+ C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

× (∥Dt∇d∥Lq,1(0,T ;Lp(R3)) + ∥∇u · ∇d∥Lq,1(0,T ;Lp(R3)))

+ C∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇2d · ∇d∥Lq,1(0,T ;Lp(R3)),

where

C∥tDt∇d · ∇2d∥Lq,1(0,T ;Lp(R3)) + C∥tDt∇2d · ∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
+ C∥(u0,∇d0)∥3

Ḃ
−1+ 3

p
p,1 (R3)

+ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
∥t∇2Dt∇d∥Lq,1(0,T ;Lp(R3)),

C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

× (∥Dt∇d∥Lq,1(0,T ;Lp(R3)) + ∥∇u · ∇d∥Lq,1(0,T ;Lp(R3)))

≤ C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

× (∥Dt∇d∥Lq,1(0,T ;Lp(R3)) + ∥∇u∥Lq,1(0,T ;Lp∗(R3))∥∇d∥L∞(0,T ;L3(R3)))

≤ C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

× (∥Dt∇d∥Lq,1(0,T ;Lp(R3)) + ∥∇2u∥Lq,1(0,T ;Lp(R3))∥∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
)

≤ C∥(u0,∇d0)∥3
Ḃ

−1+ 3
p

p,1 (R3)
,

and

C∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇2d · ∇d∥Lq,1(0,T ;Lp(R3))

≤ C∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇2d∥Lq,1(0,T ;Lp∗(R3))∥∇d∥L∞(0,T ;L3(R3))

≤ C∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3))∥∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
∥tDt∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

.

• Estimate for ∥tu · ∇Dt∇d∥Lq,1(0,T ;Lp(R3)). Similarly, we obtain

∥tu · ∇Dt∇d∥Lq,1(0,T ;Lp(R3)) ≤ C∥u∥L∞(0,T ;L3(R3))∥t∇Dtu∥Lq,1(0,T ;Lp∗(R3))

≤ C∥u∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥t∇2Dtu∥Lq,1(0,T ;Lp(R3))



14 X. WU, X. LIANG EJDE-2025/101

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
∥t∇2Dtu∥Lq,1(0,T ;Lp(R3)).

• Estimate for ∥Dt∇d∥Lq,1(0,T ;Lp(R3)). From Proposition 3.1, we obtain

∥Dt∇d∥Lq,1(0,T ;Lp(R3)) ≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

• Estimate for ∥t∆u · ∇(∇d)∥Lq,1(0,T ;Lp(R3)). Similarly, we obtain

∥t∆u · ∇(∇d)∥Lq,1(0,T ;Lp(R3)) ≤ C∥t∇2d∥L∞(0,T ;L∞(R3))∥∇2u∥Lq,1(0,T ;Lp(R3))

≤ C∥t∇d∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇2u∥Lq,1(0,T ;Lp(R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
.

• Estimate for ∥t∇u · ∇2(∇d)∥Lq,1(0,T ;Lp(R3)). Similarly, we obtain

∥t∇u · ∇2(∇d)∥Lq,1(0,T ;Lp(R3)) ≤ C∥t∇u∥L∞(0,T ;L∞(R3))∥∇3d∥Lq,1(0,T ;Lp(R3))

≤ C∥tu∥
L∞(0,T ;Ḃ

1+ 3
m

m,1 (R3))
∥∇3d∥Lq,1(0,T ;Lp(R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
.

In summary, we have

∥tDt∇d∥
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
+ ∥((tDt∇d)t, t∇2Dt∇d)∥Lq,1(0,T ;Lp(R3))

+ ∥tDt∇d∥Ls,1(0,T ;Lm(R3))

≤ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
+ C∥(u0,∇d0)∥3

Ḃ
−1+ 3

p
p,1 (R3)

+ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
(∥t∇2Dtu∥Lq,1(0,T ;Lp(R3)) + ∥t∇2Dt∇d∥Lq,1(0,T ;Lp(R3)) + 1)

+ C∥(u0,∇d0)∥2
Ḃ

−1+ 3
p

p,1 (R3)
∥tDt∇d∥

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

.

(3.22)

Thus, combining (3.19) with (3.22), since ∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
is small, we obtain (3.14).

In the end, as p ∈ ( 32 , 3) (which yields 1 < q < 2), we have

∥(Dtu,Dt∇d)∥L∞(R3) ≤ C∥(Dtu,Dt∇d)∥1−
q
2

L3(R3)
∥(∇2Dtu,∇2Dt∇d)∥q/2

Lp(R3)

≤ C∥(Dtu,Dt∇d)∥1−
q
2

Ḃ
−1+ 3

p
p,1 (R3)

∥(∇2Dtu,∇2Dt∇d)∥q/2Lp(R3).

Thus, one has

∥t(Dtu,Dt∇d)∥L2(0,T ;L∞(R3))

≤ C∥t(Dtu,Dt∇d)∥1−
q
2

L∞(0,T ;Ḃ
−1+ 3

p
p,1 (R3))

∥(∇2(tDtu),∇2(tDt∇d))∥q/2Lq(0,T ;Lp(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.

Then, applying (3.14) gives the second part of (3.15).
To complete the proof of (3.15), it suffices to apply Lemma 2.5 with r = 3 to tDtu (keeping in

mind that −1 + 3
p = 2 − 2

q ), then from the Hölder inequality with respect to the time variable,

one has

∥t(∇Dtu,∇Dt∇d)∥L2(0,T ;L3(R3))

≤ C∥t(Dtu,Dt∇d)∥θ
L∞(0,T ;Ḃ

−1+ 3
p

p,1 (R3))
∥t(∇2Dtu,∇2Dt∇d)∥1−θ

Lq(0,T ;Lp(R3))

≤ C∥(u0,∇d0)∥
Ḃ

−1+ 3
p

p,1 (R3)
.



EJDE-2025/101 INHOMOGENEOUS INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEMS 15

where θ = 2p−3
3p−3 . Then, applying the first part (3.14) of proposition 3.3 gives the desired result.

The case 1 < p ≤ 3
2 reduces to the case we treated before since Ḃ

−1+ 3
p

p,1 (R3) ↪→ Ḃ
−1+ 3

p1
p1,1

(R3) for

some p1 ∈ ( 32 , 3). □

4. Existence part in Theorem 1.1

This section is dedicated to constructing global solutions under the stated conditions in Theorem
1.1. First, we smooth out the initial data (ρ0, u0,∇d0) to be

ρn0 := ρ0 ∗ jn, un
0 := u0 ∗ jn, dn0 := d0 ∗ jn,

where jn(|x|) = n2j(|x|/n) is the classical Friedrich’s mollifier. Following the classical theory for
inhomogeneous incompressible nematic liquid crystal systems (see [30] for instance), we construct
a sequence of smooth approximate solutions (ρn, un,∇dn)n∈N to system (1.1). By the a priori esti-
mates established in Proposition 3.1-3.3, we conclude that the solution sequence (ρn, un,∇dn)n∈N
is globally defined and uniformly bounded in the specified function spaces. To pass to the limit, we
employ compactness arguments. However, a significant difficulty emerges due to the non-reflexivity
of the Lorentz spaces, which prevents direct application of classical tools such as the Aubin-Lions
lemma. To resolve this issue, we analyze the approximate solutions within an expanded functional
framework

Ẇ 2,1
p,r (R+ × R3) := {u ∈ Cb(R+; Ḃ

2− 2
r

p,r (R3)) : ut,∇2u ∈ Lr(R+;Lp(R3)), 1 < r < ∞},
which is a slightly more inclusive reflexive space where standard compactness arguments become
applicable. The subsequent analysis follows the framework established in [15], where we incor-
porate the director field gradient ∇d into the velocity estimates while preserving the original
estimates for both u and ρ.

5. Uniqueness part in Theorem 1.1

This section is devoted to establishing the uniqueness of solutions constructed in Theorem 1.1.
The detailed procedure follows the framework established in [8]. For the subsequent analysis, we
recall several energy functionals and related estimates from [8].

X(t) = sup
τ∈(0,t]

τ−1∥δρ∥Ḣ−1(R3),

Y (t) = ( sup
τ∈(0,t]

∥√ρ1δu∥2L2(R3) + sup
τ∈(0,t]

∥∇δd∥2L2(R3) + ∥∇δu∥2L2(0,t;L2(R3)) + ∥∇2δd∥2L2(0,t;L2(R3)))
1/2,

with the following relationships

X(t) ≤ C(Y (t) +

∫ t

0

gXdτ), X(t) ≤ CY (t)eC
∫ t
0
gdτ . (5.1)

However, unlike [8], the term∫
R3

∇(|∇d1|2 · δd +∇δd · (∇d1 +∇d2) · d2) · ∇δd dx

requires modified estimates, which we now derive.
• Estimate for

∫
R3 ∇(|∇d1|2 · δd+∇δd · (∇d1 +∇d2) · d2) · ∇δd dx. Applying integration by parts

followed by the Hölder inequality and the Young inequality, we obtain∫
R3

∇(|∇d1|2 · δd +∇δd · ((∇d1 +∇d2) · d2)) · ∇δd dx

=

∫
R3

∇(|∇d1|2 · δd) · ∇δd−∇δd · (∇d1 +∇d2) · d2) ·∆δd dx

=

∫
R3

2∇2d1 · ∇d1 · δd · ∇δd dx+

∫
R3

|∇d1|2 · ∇δd · ∇δd dx

−
∫
R3

∇δd · (∇d1 +∇d2) · d2 ·∆δd dx



16 X. WU, X. LIANG EJDE-2025/101

≤ C∥∇2d1∥L∞(R3)∥∇d1∥L3(R3)∥δd∥L6(R3)∥∇δd∥L2(R3) + C∥∇d1∥2L∞(R3)∥∇δd∥2L2(R3)

+ C∥∇δd∥L2(R3)(∥∇d1∥L∞(R3) + ∥∇d2∥L∞(R3))∥d2∥L∞(R3)∥∆δd∥L2(R3)

≤ 1

4
∥∇2δd∥2L2(R3) + C∥∇d1∥

Ḃ
−1+ 3

p
p,1 (R3)

∥∇2d1∥L∞(R3)∥∇δd∥2L2(R3)

+ C∥∇d1∥2L∞(R3)∥∇δd∥2L2(R3) +
1

4
∥∇2δd∥2L2(R3)

+ C(∥∇d1∥L∞(R3) + ∥∇d2∥L∞(R3))
2∥∇δd∥2L2(R3).

By using estimates (84)-(90) from [8], we obtain

d

dt

∫
R3

|δu|2dx+

∫
R3

|∇δu|2dx+
d

dt

∫
R3

|∇δd|2dx+

∫
R3

|∇2δd|2dx

≤ C
∥∥(∇u2,∇2d2,∇2d1)∥L∞(R3)(1 + ∥∇d1∥

L∞(0,t;Ḃ
−1+ 3

p
p,1 (R3))

)

+ ∥(∇d1,∇d2)∥2L∞(R3)

)(
∥(δu,∇δd)∥2L2(R3)

)
+ C2X2(τ∥∇Dtu2∥L3(R3) + τ∥Dtu2∥L∞(R3))

2.

(5.2)

Integrating over the time interval [0, t] and applying the Gronwall inequality yields

Y 2(t) ≤ C(∥√ρ0δu0∥L2 + ∥∇δd0∥L2)e
C(∥g∥L1(0,t)(∥h∥L∞(0,t)+1)+∥f∥2

L2(0,t)
)

× e
C∥f∥2

L2(0,t)
e
4C(∥g∥

L1(0,t)
(∥h∥L∞(0,t)+1)+∥f∥2

L2(0,t)
)

,

(5.3)

where

f(t) = (t∥∇Dtu2∥L3(R3) + t∥Dtu2∥L∞(R3)) + ∥(∇d1,∇d2)∥L∞(R3),

g(t) = ∥(∇u2,∇2d2,∇2d1)∥L∞(R3), h(t) = ∥∇d1∥
Ḃ

−1+ 3
p

p,1 (R3)
.

Combining this with estimate (5.1) completes the proof of the uniqueness part in Theorem 1.1.

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Xinrui Wu

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

Email address: xinruiwu2001@163.com

Xingyu Liang

School of Mathematics, Nanjing University, Nanjing 210093, China
Email address: xingyuliang2000@163.com


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem ??
	A priori estimates

	4. Existence part in Theorem ??
	5. Uniqueness part in Theorem ??
	References