Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 75, pp. 1–28.

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

DOI: 10.58997/ejde.2024.75

SOLVABILITY OF AN ATTRACTION-REPULSION

CHEMOTAXIS NAVIER-STOKES SYSTEM WITH ARBITRARY

POROUS MEDIUM DIFFUSION

YADHAVAN KARUPPUSAMY, SHANGERGANESH LINGESHWARAN,

MANIMARAN JEYARAJ

Abstract. In this work, we proposed a model that describes the influence of

two chemically opposed stimuli in the movement of species living in a fluid
environment. We investigated the well-posedness of a system that models the

attraction-repulsion chemotaxis Navier-Stokes system with nonlinear diffusion.

We validate the existence of a global three-dimensional weak solution. Fur-
thermore, with some restrictions on the nonlinear exponent and degradation

coefficients of the chemical signal, we established the existence of a three-

dimensional global bounded weak solutions for the system.

1. Introduction

Chemotaxis is a fascinating natural occurrence that involves organisms moving
in response to chemical signals in their surroundings. It is a trait possessed by all
motile organisms, ranging from single-celled bacteria to complex multicellular be-
ings. This phenomenon plays significant importance in several biological processes,
such as immune responses, wound healing, and the movement of microorganisms
in their respective habitats. The chemotaxis system was modeled by Keller and
Segel [19], and their model has provided a framework for studying this fascinating
phenomenon as

ut = ∆u−∇ · (χu∇v),
vt = ∆v − αv + βu

(1.1)

where u refers to species density, v refers to concentration of signal and χ refers
to sensitivity function. The global weak solution for the quasilinear degenerate
Keller-Segel system of the parabolic-parabolic type was established in [15]. The
global weak solutions and their decay properties were discussed for the degenerate
Keller-Segel model in [27]. Different results about the singularity, boundedness,
and global existence of solutions were obtained for a simplified form of (1.1) in
[12, 13, 26].

2020 Mathematics Subject Classification. 35Q30, 35K57, 35D30, 92C17.

Key words and phrases. Attraction-repulsion; chemotaxis; Navier-Stokes; weak solution.
©2024. This work is licensed under a CC BY 4.0 license.
Submitted May 24, 2024. Published November 25, 2024.

1



2 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

A model that describes the behavior of bacteria living near solid-air-water con-
tact lines was developed in [31] as

ut + z · ∇u = ∆u−∇ · (χ(v)u∇v),
vt + z · ∇v = ∆v − k(v)u,

zt + τ(z · ∇)z +∇p = ∆z − u∇ϕ,
div z = 0

(1.2)

where u refers to cell density, v refers to the concentration of oxygen and z, p
refer to the velocity field and pressure of the fluid, respectively. When τ = 1, the
system (1.2) is called Keller-Segel-Navier-Stokes system (KS-NS) whereas τ = 0
is called as Keller-Segel-Stokes (KSS) system. The global existence of classical
solutions of KS-NS near-constant states in R3 and the existence of a global weak
solution in R2 under some assumptions were studied in [8]. A numerical method
for solving KS-NS was introduced in [5] with a full exploration of its dynamics.
The global-in-time existence of weak solutions of KSS under some assumptions was
established in [24]. The local-in-time smooth solution for the Keller-Segel-Navier-
Stokes (KS-NS) system under minimal assumptions on the sensitivity function and
oxygen consumption rate was established in [1]. Additionally, it was demonstrated
that the local and global existence of solutions for the KS-NS system when the
oxygen concentration follows either a hyperbolic or parabolic type, as detailed in [3].
The global weak solutions for KSS with regularity in initial data and boundedness
of large of large-data solutions were investigated in [32]. Global-in-time classical
solution of KSS value was established in [21] for specific initial. Results on weak
and bounded weak solutions, mild solutions, and boundedness for KS-NS and KSS
under some assumption on χ, k, and diffusion property were established in [4, 14,
17, 18, 20].

Suppose the diffusion of the cell is considered as migration in a porous medium
then (1.2) becomes

ut + z · ∇u = ∆u1+α −∇ · (χu∇v),
vt + z · ∇v = ∆v − k(v)u,

zt + τ(z · ∇)z +∇p = ∆z − u∇ϕ,
div z = 0.

(1.3)

The existence of global-in-time solution for (1.3) with τ = 0 in R2 established in
[11], when α ∈ ( 12 , 1

]
and the same have been established for α = 1

3 in [22]. Also,
global existence of weak solutions for (1.3) was established in [11] for a bounded
domain in R3 with α ∈ [0.8, 1] and τ = 1. The global existence of weak solutions of
(1.3) with α > 1

7 and τ = 0 was discussed in [30] under the assumption that initial
data are sufficiently regular and positive. The global existence of bounded weak
solutions of (1.3) with α > 0 and τ = 0 studied in [29] under some assumptions on
initial data. The weak and bounded weak solutions were established globally in [7]
under some assumptions on χ, k, and diffusive exponent.

The movement of organisms in biological processes is greatly impacted by chem-
ical stimuli, which can either attract or repel them. Chemoattraction represents an
organism moving towards an increasing signal concentration, and chemorepulsion
means that an organism moves away from an increasing signal concentration, which
are both important factors in chemical migration. A recent study in [25] focused



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 3

on modeling a system that highlights the chemotactic response of microglial cells
as

ut = ∆u−∇ · (χ(v)u∇v) +∇ · (ξ(w)u∇w),
vt = ∆v + βu− γv,

wt = ∆w + δu− ηw

(1.4)

where u denotes species density, v denotes concentration of chemoattractant, and w
denotes concentration of chemorepellant. Global solvability, the existence of steady
states and blow up of (1.4)were discussed in [28] with the assumption of γ = η.
Also, the same generalized for γ, η > 0 in [10]. The classical solutions and steady
states of (1.4) were studied globally in [16] for one dimension. The existence of a
global bounded classical solution for (1.4) with logistic source under the assumption
of limitation on growth in the logistic source was established in [33].

One interesting phenomenon in biology is the migration of species, which is
thought to be influenced by attraction-repulsion chemotaxis signals in fluid. As
a result, equation (1.4) can be coupled with the Navier-Stokes equation to better
understand the dynamics of these phenomena as

ut = ∆u−∇ · (χ(v)u∇v) +∇ · (ξ(w)u∇w),
vt = ∆v + βu− γv,

wt = ∆w + δu− ηw,

zt + τ(z · ∇)z +∇p = ∆z − u∇ϕ

(1.5)

where u denotes density of bacteria, v denotes concentration of attraction signal, w
refers concentration of repulsive signal, and z refers velocity field of the fluid. The
global-in-time classical solution for (1.5) with logistic source under the assumption
that initial data are sufficiently regular and non-negative was discussed in [34].
The uniqueness and existence of mild global solutions were discussed in [9] in the
Besov-Morrey type of (1.5).

However, there is no paper in the literature to study the attraction-repulsion
chemotaxis-Navier-Stokes equation with nonlinear diffusion exponent 1+α. In this
paper, we considered (1.5) with the diffusion of the bacteria assumed as migration
in a porous medium, in R3 × [0, T ) with T > 0, as

ut + z · ∇u = ∆u1+α −∇ · (χ(v)u∇v) +∇ · (ξ(w)u∇w),
vt + z · ∇v = ∆v + βu− γv,

wt + z · ∇w = ∆w + δu− ηw,

zt + τ(z · ∇)z +∇p = ∆z − u∇ϕ,
div z = 0,

(1.6)

where u refers species density, v refers concentration of attraction signal, w refers
concentration of repulsive signal, z refers velocity field of the fluid, p refers hydro-
static pressure, ϕ refers gravitational potential, ν refers outward normal vector, χ
and ξ are non-negative sensitivity functions and α, β, γ, δ, η are positive constants.
Therefore, in this work, we have attempted to study the existence of weak and
bounded weak solutions of the proposed model for α > 0 and α > 1/8 respectively
with some assumption in the chemotactic sensitivity function.

This paper is organized as follows. In section 2, we defined the weak solution
and suitable approximation problem of the model (1.6) with τ = 1. Also, a priori
estimate is derived using the approximation problem and weak solution established



4 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

globally using the priori estimate. In section 3, we defined the bounded weak
solution and suitable approximation for the proposed model (1.6) with τ = 0. We
demonstrated the existence of global bounded weak solution.

2. Existence of weak solutions

We considered the attraction-repulsion chemotaxis Navier-Stokes system, that
is, (1.6) with τ = 1 in this secion, as follows:

ut + z · ∇u = ∆u1+α −∇ · (χ(v)u∇v) +∇ · (ξ(w)u∇w),
vt + z · ∇v = ∆v + βu− γv,

wt + z · ∇w = ∆w + δu− ηw,

zt + (z · ∇)z +∇p = ∆z − u∇ϕ,
div z = 0.

(2.1)

This section first defines the weak solution for (2.1). Then, we introduce the suitable
approximation problem for (2.1). Before demonstrating our main finding, namely
weak solutions to (2.1) in R3, we first establish some key lemma used to substantiate
our main findings.

Definition 2.1. For α > 0 and T ∈ (0,∞), the unknown functions u, v, w ≥ 0
and z represents a vector function on R3 × (0, T ), (u, v, w, z) is termed as a weak
solution of (2.1) if

(1) u(1 + |x|+ | log u|) ∈ L∞(0, T ;L1(R3)),

(2) Given p ∈ [1, 1 + α], u ∈ L∞(0, T ;Lp(R3)), and ∇u
p+α

2 ∈ L2(0, T ;L2(R3)),
(3) v, w ∈ L∞(0, T ;H1(R3)) ∩ L2(0, T ;H2(R3)),
(4) v, w ∈ L∞(R3 × [0, T )),
(5) z ∈ L∞(0, T ;L2(R3)),
(6) ∇z ∈ L2(0, T ;L2(R3)),
(7) Given a test function φ ∈ C∞

0 (R3 × [0, T )), it holds∫ t

0

∫
R3

(−uφt − zu · ∇φ+∇u1+α · ∇φ− uχ(v)∇v · ∇φ+ uξ(w)∇w · ∇φ) dx dt

=

∫
R3

u0φ(·, 0)dx,∫ t

0

∫
R3

(−vφt − zv · ∇φ+∇v · ∇φ− βuφ+ γvφ) dx dt =

∫
R3

v0φ(·, 0)dx,∫ t

0

∫
R3

(−wφt − zw · ∇φ+∇w · ∇φ− δuφ+ ηwφ) dx dt =

∫
R3

w0φ(·, 0)dx,

(8) Given a test function ψ ∈ C∞
0 (R3 × [0, T ),R3) with ∇ · ψ = 0, it holds∫ t

0

∫
R3

(−z · ψt +∇z · ∇ψ + ((z · ∇)z) · ψ + u∇ϕ · ψ) dx dt =
∫
R3

z0 · ψ(·, 0)dx.

The problem we are dealing with here is quite complex due to the strong pres-
ence of degeneracy in the diffusion terms. Hence, we introduced an approximation
problem to the proposed model (2.1) which allows us to overcome the degeneracy
problem and make progress in our analysis. By considering the regularized problem



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 5

of (2.1) as below, we have made significant strides in understanding the underlying
dynamics and behavior of the system.

uϵt + zϵ · ∇uϵ = ∆(uϵ + ϵ)1+α −∇ · (χ(vϵ)uϵ∇vϵ) +∇ · (ξ(wϵ)uϵ∇wϵ),

vϵt + zϵ · ∇vϵ = ∆vϵ + βuϵ − γvϵ,

wϵt + zϵ · ∇wϵ = ∆wϵ + δuϵ − ηwϵ,

zϵt + (zϵ · ∇)zϵ +∇pϵ = ∆zϵ − uϵ∇ϕ,
div zϵ = 0

(2.2)

with initial conditions

u0ϵ = ϕϵ ∗ u0, v0ϵ = ϕϵ ∗ v0, w0ϵ = ϕϵ ∗ w0, z0ϵ = ϕϵ ∗ z0,

where ϕϵ is a usual mollifier with ϵ ∈ (0, 1). Hereafter, throughout the paper we use
all unknown (uϵ, vϵ, wϵ, zϵ) as (u, v, w, z) for notation simplicity. According to the
standard theory of existence and regularity, for every ϵ > 0, equation (2.2) admits
a local-in-time classical solution. The proof for this assertion is not provided here
because it follows a similar methodology as in [2, 23].

Consider the functional

E(t) :=

∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22 (2.3)

and

D(t) := ∥∇u
1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22, (2.4)

where ⟨x⟩ = (1 + |x|2) 1
2 . Next, we present two lemmas with some assumptions on

sensitivity functions and nonlinear exponent, which play a major role in establishing
the main result of the paper.

Lemma 2.2. Let (u, v, w, z) be a classical solution of (2.2), for all ϵ ∈ (0, 1) and
initial data (u0ϵ , v0ϵ , w0ϵ , z0ϵ) satisfies the following independent of ϵ:

(1) u0ϵ(1 + |x|+ | log u0ϵ |) ∈ L1(R3),
(2) u0ϵ ∈ L1+α(R3),
(3) v0ϵ , w0ϵ ∈ L∞(R3) ∩H1(R3),
(4) z0ϵ ∈ L2(R3).

Assume that

α >
1

6
, χ′, ξ′ ∈ L∞

loc, ϕ ∈W 2,∞(R3). (2.5)

Then, given t ∈ (0, T ]),

sup
0≤τ≤t

E(τ) +

∫ t

0

D(τ)dτ < C, (2.6)

where C > 0 is a constant solely depends on initial data.

Proof. By integrating (2.2)1, we can obtain ∥u(t)∥1 ≡ ∥u0∥1, which leads us to
the conclusion that the total mass of u conserved. Additionally, by applying the
maximal principle to (2.2)2 and (2.2)3, we obtain ∥v∥L∞(R3×[0,T )) ≤ ∥v0∥∞ and
∥w∥L∞(R3×[0,T )) ≤ ∥w0∥∞.



6 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

Case (i): 1/6 < α ≤ 1/3. First, multiply (2.2)1 by log u and integrating to obtain

d

dt

∫
R3

u log u dx+

∫
R3

∇ log u · ∇(u+ ϵ)1+α dx

=

∫
R3

∇u · (χ∇v) dx−
∫
R3

∇u · (ξ∇w) dx.
(2.7)

The second term in LHS of above is evaluated as∫
R3

∇ log u · ∇(u+ ϵ)1+α dx ≥
∫
R3

∇ log u · (1 + α)uα∇u dx

=
4

1 + α
∥∇u

1+α
2 ∥22.

(2.8)

The first term in RHS of (2.7) is evaluated using Young’s inequality as∫
R3

∇u · (χ∇v) dx ≤ 2χ

1 + α

(
ϵ1∥∇u

1+α
2 ∥22 + C(ϵ1)

∫
R3

u1−α|∇v|2 dx
)
, (2.9)

where we used that |∇u| = 2
1+αu

1−α
2

∣∣∇u 1+α
2

∣∣ and χ := supR3×[0,T ) |χ(v)|.
The last term in RHS of (2.7) is evaluated using Young’s inequality as∫

R3

u1−α|∇v|2 dx

=

∫
R3

u1−α∇v · ∇v dx

≤ C1

(∫
R3

|∇u1−α||∇v| dx+

∫
R3

u1−α|∆v| dx
)

≤ C1

(∫
R3

C2u
1−3α

2

∣∣∇u 1+α
2

∣∣|∇v| dx+

∫
R3

u1−α|∆v| dx
)

≤ C1C2

∫
R3

ϵ2
∣∣∇u 1+α

2

∣∣2 + C(ϵ2)u
1−3α|∇v|2 dx+ C1

∫
R3

u1−α|∆v| dx,

where we used that
∣∣∇u1−α

∣∣ = 2(1−α)
(1+α) u

1−3α
2

∣∣∇u 1+α
2

∣∣. Using the above in (2.9), we

obtain ∫
R3

∇u · (χ∇v) dx

≤ C ′′
1 ∥∇u

1+α
2 ∥22 + C ′′

2

∫
R3

u1−3α|∇v|2 dx+ C ′′
3

∫
R3

u1−α|∆v| dx,
(2.10)

where C ′′
i =

2χC′
i

1+α for i ∈ {1, 2, 3}, C ′
1 = ϵ1 + C1C2C(ϵ1)ϵ2, C

′
2 = C1C2C(ϵ1)C(ϵ2)

and C ′
3 = C1C(ϵ1). Following the same procedure as above, we obtain

−
∫
R3

∇u · (ξ∇w) dx

≤ C ′
4∥∇u

1+α
2 ∥22 + C ′

5

∫
R3

u1−3α|∇w|2 dx+ C ′
6

∫
R3

u1−α|∆w| dx,
(2.11)



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 7

where C ′
i, i ∈ {4, 5, 6} are positive constants. Using (2.8), (2.10) and (2.11) in (2.7),

we obtain
d

dt

∫
R3

u log u dx+ C ′∥∇u
1+α
2 ∥22

≤ C ′′
2

∫
R3

u1−3α|∇v|2 dx+ C ′′
3

∫
R3

u1−α|∆v| dx

+ C ′
5

∫
R3

u1−3α|∇w|2 dx+ C ′
6

∫
R3

u1−α|∆w| dx,

(2.12)

where C ′ = 4
1+α − C ′′

1 − C ′
4. Multiply (2.2)1 by ⟨x⟩ and integrate over R3. Then

using Young’s inequality along with simple algebraic calculations (See [22, (3.6) ]
and [6, (2.13)]) leads to

d

dt

∫
R3

⟨x⟩u dx =

∫
R3

uz · ∇⟨x⟩ dx+

∫
R3

(u+ ϵ)1+α∆⟨x⟩ dx

+

∫
R3

∇⟨x⟩ · uχ∇v dx−
∫
R3

∇⟨x⟩ · uξ∇w dx

≤ C7

(
1 + ∥z∥22 + ∥∇v∥22 + ∥∇w∥22

)
+

(
C(ϵ) + ϵ∥∇u

1+α
2 ∥22

)
.

(2.13)

Multiply (2.2)1 by uα and integrating to obtain

1

1 + α

d

dt
∥u∥1+α

1+α +

∫
R3

∇uα · ∇(u+ ϵ)1+α dx

=

∫
R3

∇uα · u(χ∇v) dx−
∫
R3

∇uα · u(ξ∇w) dx.
(2.14)

Using |∇uα| = αu
α−1
2 |∇u|, |u1+α| = (1 + α)u

α
2 |∇u|, and

|∇u
1+2α

2 |2 =
(1 + 2α)2

4
u

2α−1
2 |∇u|2,

we obtain ∫
R3

∇uα · ∇(u+ ϵ)1+α dx ≥ 4α(1 + α)

(1 + 2α)2
∥∇u

1+2α
2 ∥22. (2.15)

The first term in RHS of (2.14) evaluated using Young’s inequality as∫
R3

∇uα · u(χ∇v) dx ≤ Cχ

∫
R3

∣∣∇u 1+2α
2

∣∣(u 1
2 |∇v|) dx

≤ Cχϵ3∥∇u
1+2α

2 ∥22 + CχC(ϵ3)

∫
R3

u|∇v|2 dx.
(2.16)

The last term in above evaluated using Young’s inequality as∫
R3

u|∇v|2 dx

≤ C9

∫
R3

|∇u||∇v|+ u|∆v| dx

≤ 2C9

1 + 2α

∫
R3

u
1−2α

2

∣∣∇u 1+2α
2

∣∣|∇v| dx+ C9

∫
R3

u|∆v| dx

≤ 2C9ϵ4
1 + 2α

∥∇u
1+2α

2 ∥22 +
2C9C(ϵ4)

1 + 2α

∫
R3

u1−2α|∇v|2 dx+ C9

∫
R3

u|∆v| dx.



8 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

Using the above inequality in (2.16), we obtain∫
R3

∇uα · u(χ∇v) dx

≤ C ′
10∥∇u

1+2α
2 ∥22 + C ′

11

∫
R3

u1−2α|∇v|2 dx+ C ′
12

∫
R3

u|∆v| dx,
(2.17)

where χ := supR3×[0,T ) |χ(v)|, Cχ = 2αχ
1+α , C

′
10 = Cχϵ3 +

2CχC9C(ϵ3)ϵ4
1+2α , C ′

11 =
2C9CχC(ϵ3)C(ϵ4)

1+2α and C ′
12 = CχC9C(ϵ3). Following the same procedure as above,

we obtain

−
∫
R3

∇uα · u(ξ∇w) dx

≤ C ′
13∥∇u

1+2α
2 ∥22 + C ′

14

∫
R3

u1−2α|∇w|2 dx+ C ′
15

∫
R3

u|∆w| dx,
(2.18)

where C ′
i, i ∈ {13, 14, 15} are positive constants. Using (2.15), (2.17) and (2.18) in

(2.14), we have

d

dt
∥u∥1+α

1+α + C15∥∇u
1+2α

2 ∥22

≤ C ′
11

∫
R3

u1−2α|∇v|2 dx+ C ′
14

∫
R3

u1−2α|∇w|2 dx+ C ′
12

∫
R3

u|∆v|dx

+ C ′
15

∫
R3

u|∆w|dx,

(2.19)

where C15 = 4α(1+α)2

(1+2α)2 − C ′
10 − C ′

13.

Multiplying (2.2)2 by −∆v and integrating to obtain

d

dt
∥∇v∥22 + 2∥∆v∥22 ≤

∫
R3

∆v · (z · ∇v)dx− β

∫
R3

u|∆v|dx+ γ

∫
R3

v|∆v|dx. (2.20)

The first term in RHS of above is evaluated as∫
R3

∆v · (z · ∇v)dx =

∫
R3

∑
i,j

v ∂i∂jv∂izj dx ≤ C0∥∆v∥22∥∇z∥22.

Using the above in (2.20), we obtain

d

dt
∥∇v∥22 + 2∥∆v∥22 ≤ C0∥∆v∥22∥∇z∥22 − β

∫
R3

u|∆v| dx+ γ

∫
R3

v|∆v| dx. (2.21)

Following similar procedure as above, we obtain

d

dt
∥∇w∥22 + 2∥∆w∥22 ≤ C1∥∆w∥22∥∇z∥22 − δ

∫
R3

u|∆w|dx+ η

∫
R3

w|∆w|dx. (2.22)

Multiply (2.2)4 by z and integrating, we obtain

d

dt
∥z∥22 + 2∥∇z∥22 ≤ C ′

16

∫
R3

u|z| dx, (2.23)



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 9

where C ′
16 = ∥∇ϕ∥L∞(R3). Adding (2.12), (2.13), (2.19) and (2.21)-(2.23), we obtain

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C0

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C ′

0

(∫
R3

u1−3α|∇v|2 dx+

∫
R3

u1−3α|∇w|2 dx+

∫
R3

u1−2α|∇v|2 dx

+

∫
R3

u1−2α|∇w|2 dx+

∫
R3

u1−α|∆v|dx+

∫
R3

u1−α|∆w|dx+

∫
R3

u|∆v| dx

+

∫
R3

u|∆w| dx+

∫
R3

v|∆v| dx+

∫
R3

w|∆w| dx+

∫
R3

u|z| dx
)
.

(2.24)
To deduce (2.6) from above, we estimate the integrals in RHS as follows:

As 0 ≤ 1 − 3α < 2/3, using the Young’s, Hölder and Sobolev inequality, we
obtain∫

R3

u1−3α|∇v|2 dx ≤

{∫
R3

(
C(ϵ4) + ϵ4u

2/3
)
|∇v|2 dx if 1/6 < α < 1/3,

∥∇v∥22 if α = 1/3.

≤

{
C(ϵ4)∥∇v∥22 + ϵ4∥u0∥2/3∥∆v∥22 if 1/6 < α < 1/3,

∥∇v∥22 if α = 1/3.

(2.25)

Following the same procedure as above, we obtain∫
R3

u1−3α|∇w|2 dx ≤

{
C(ϵ5)∥∇w∥22 + ϵ5∥u0∥2/3∥∆w∥22 if 1/6 < α < 1/3,

∥∇w∥22 if α = 1/3,

(2.26)∫
R3

u1−2α|∇v|2 dx ≤ C(ϵ6)∥∇v∥22 + ϵ6∥u0∥2/3∥∆v∥22, (2.27)∫
R3

u1−2α|∇w|2 dx ≤ C(ϵ7)∥∇w∥22 + ϵ7∥u0∥2/3∥∆w∥22. (2.28)

Next, we estimate the integral
∫
R3 u

1−α|∆v| dx. Using the Hölder, Young and
Gagliardo-Nierenberg inequalities in the integral, we obtain∫

R3

u1−α|∆v| dx ≤ C(ϵ8)∥u∥2−α
2−α + ϵ8∥∆v∥22

≤ C(ϵ8)C16∥u0∥
1+4α
2+3α

1 ∥∇u
1+α
2 ∥

6−6α
2+3α

2 + ϵ8∥∆v∥22

≤ C(ϵ8)C16

(
C(ϵ9)∥u0∥

1+4α
2+3α

1 + ϵ9∥∇u
1+α
2 ∥22

)
+ ϵ8∥∆v∥22

= C17 + C18∥∇u
1+α
2 ∥22 + ϵ8∥∆v∥22,

(2.29)

where C17 = C16C(ϵ8)C(ϵ9)∥u0∥
1+4α
2+3α

1 and C18 = C16C(ϵ8)ϵ9. Here, we used that
4
3 ≤ 6−6α

2+3α < 2. Following the same procedure as above, we obtain∫
R3

u1−α|∆w| dx ≤ C19 + C20∥∇u
1+α
2 ∥22 + ϵ10∥∆w∥22. (2.30)



10 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

Next, we estimate the integral
∫
R3 u|∆v| dx. Using the Young and Gagliardo-

Nierenberg inequalities in the integral, we obtain∫
R3

u|∆v| dx ≤ C(ϵ11)∥u∥22 + ϵ11∥∆v∥22

≤ C(ϵ11)C21∥u0∥
1+6α
2+6α

1 ∥∇u
1+2α

2 ∥
6

2+6α

2 + ϵ11∥∆v∥22

≤ C(ϵ11)C21

(
C(ϵ12)∥u0∥

1+6α
2+3α

1 + ϵ12∥∇u
1+2α

2 ∥22
)
+ ϵ11∥∆v∥22

= C ′
21 + C22∥∇u

1+2α
2 ∥22 + ϵ11∥∆v∥22,

(2.31)

where C ′
21 = C21C(ϵ11)C(ϵ12)∥u0∥

1+6α
2+3α

1 and C22 = C21ϵ12C(ϵ11). Here, we used
that 3

2 ≤ 6
2+6α < 2. Following the same procedure as above, we obtain∫

R3

u|∆w| dx ≤ C23 + C24∥∇u
1+2α

2 ∥22 + ϵ13∥∆w∥22. (2.32)

Next we estimate the integrals
∫
R3 v|∆v| dx,

∫
R3 w|∆w| dx. Using the Young and

Gagliardo-Nierenberg inequalities in the integrals, we obtain∫
R3

v|∆v| dx ≤ C(ϵ′14)C0∥v0∥
8
7
1 ∥∆v∥

6
7
2 + ϵ′14∥∆v∥22, (2.33)∫

R3

w|∆w| dx ≤ C(ϵ′15)C1∥w0∥
8
7
1 ∥∆w∥

6
7
2 + ϵ′15∥∆w∥22. (2.34)

Next, we estimate the integral
∫
R3 u|z| dx. Using the Young, Gagliardo-Nierenberg

and Sobolev inequalities in the integral, we obtain∫
R3

u|z| dx ≤ C25∥u∥ 6
5
∥z∥6 ≤ ϵ16∥u∥26

5
+ C(ϵ16)∥∇z∥22

≤ C25C
′
25∥u0∥

3+10α
2+6α

1 ∥∇u
1+2α

2 ∥
2

2+6α

2 + C(ϵ16)∥∇z∥22

≤ C25C
′
25∥u0∥

3+10α
2+6α

1

(
ϵ17∥∇u

1+2α
2 ∥22 + C(ϵ17)

)
+ C(ϵ16)∥∇z∥22

= C26 + C27∥∇u
1+2α

2 ∥22 + C(ϵ16)∥∇z∥22,

(2.35)

where C26 = ϵ16C
′
25∥u0∥

3+10α
2+6α

1 C(ϵ17) and C27 = ϵ16C
′
25∥u0∥

3+10α
2+6α

1 ϵ17. Here we used
that 1 < 6

5 < 3 + 6α and 0 < 2
2+6α < 2. Substituting (2.25) − (2.35) in (2.24), we

have

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C0

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C

′
0

(
1 + ∥∇v∥22 + ∥∇w∥22

)
.

(2.36)

Integrating the above with respect to t, we obtain (2.6).

Case (ii): 1/3 < α ≤ 1. First, multiplying (2.2)1 by log u and integrating we
obtain

d

dt

∫
R3

u log u dx+
4

1 + α
∥∇u

1+α
2 ∥22 =

∫
R3

∇u·(χ∇v) dx−
∫
R3

∇u·(ξ∇w) dx. (2.37)



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 11

First, we evaluate the RHS of (2.37) as follows:∫
R3

∇u · (χ∇v) dx ≤ 2χ

1 + α

(
ϵ1∥∇u

1+α
2 ∥22 + C(ϵ1)

∫
R3

u1−α|∇v|2 dx
)

= C28∥∇u
1+α
2 ∥22 + C29

∫
R3

u1−α|∇v|2 dx,
(2.38)

where C28 = 2χϵ1
1+α and C29 = 2χC(ϵ1)

1+α . Following the same procedure as above, we
obtain

−
∫
R3

∇u · (ξ∇w) dx ≤ C30∥∇u
1+α
2 ∥22 + C31

∫
R3

u1−α|∇w|2 dx. (2.39)

Using (2.8), (2.38) and (2.39) in (2.37), we have

d

dt

∫
R3

u log u dx+ C32∥∇u
1+α
2 ∥22

≤ C29

∫
R3

u1−α|∇v|2dx+ C31

∫
R3

u1−α|∇w|2dx.
(2.40)

where C32 = 4
1+α − C28 + C30. Now, multiply (2.2)1 by uα and integrating we

obtain
1

1 + α

d

dt
∥u∥1+α

1+α +
4α(1 + α)

(1 + 2α)2
∥∇u

1+2α
2 ∥22

=

∫
R3

∇uα · u(χ∇v) dx−
∫
R3

∇uα · u(ξ∇w) dx.
(2.41)

The last term in RHS of (2.16) is evaluated as∫
R3

u|∇v|2 dx ≤ C9

∫
R3

|∇u||∇v|+ u|∆v| dx

≤ 2C9

1 + α

∫
R3

u
1−α
2

∣∣∇u 1+α
2

∣∣|∇v| dx+ C9

∫
R3

u|∆v| dx

≤ 2C9

1 + α

(
ϵ18∥∇u

1+α
2 ∥22 + C(ϵ18)

∫
R3

u1−α|∇v|2 dx
)

+ C9

∫
R3

u|∆v| dx.

Using above in (2.16), the first term in RHS of (2.41) is evaluated as∫
R3

∇uα · u(χ∇v)dx ≤ C33∥∇u
1+2α

2 ∥22 + C34∥∇u
1+α
2 ∥22

+ C35

∫
R3

u1−α|∇v|2dx+ C36

∫
R3

u|∆v| dx,
(2.42)

where Cχ = 2αχ
1+α , C33 = CχC(ϵ3), C34 =

2Cχϵ3C9ϵ18
1+α , C35 =

2Cχϵ3C9C(ϵ18)
1+α and

C36 = Cχϵ3C9. Following the same procedure as above, we obtain

−
∫
R3

∇uα · u(ξ∇w) dx ≤ ϵ20∥∇u
1+2α

2 ∥22 + C37∥∇u
1+α
2 ∥22

+ C38

∫
R3

u1−α|∇w|2 dx+ C39

∫
R3

u|∆w| dx.
(2.43)



12 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

Using (2.42) and (2.43) in (2.41), we have

d

dt
∥u∥1+α

1+α + C40∥∇u
1+2α

2 ∥22 − C41∥∇u
1+α
2 ∥22

≤ C35

∫
R3

u1−α|∇v|2dx+ C36

∫
R3

u|∆v|dx

+ C38

∫
R3

u1−α|∇w|2dx+ C39

∫
R3

u|∆w|dx,

(2.44)

where C40 = 4α(1+α)2

(1+2α)2 −C33−ϵ20 and C41 = C37+C34. It is clear that (2.2)2, (2.2)3
and (2.2)4 are independent of α and therefore, adding (2.21)-(2.23) with (2.40),
(2.13), (2.44), we obtain

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C42

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C43

(∫
R3

u1−α|∇v|2 dx+

∫
R3

u1−α|∇w|2 dx+

∫
R3

u|∆v| dx

+

∫
R3

u|∆w| dx+

∫
R3

v|∆v| dx+

∫
R3

w|∆w| dx+

∫
R3

u|z| dx
)
.

(2.45)

As 0 ≤ 1− α < 2
3 , using the Young’s, Hölder and Sobolev inequality, we obtain∫

R3

u1−α|∇v|2 dx ≤

{∫
R3

(
C(ϵ21) + ϵ21u

2/3
)
|∇v|2 dx if 1/3 < α < 1,

∥∇v∥22 if α = 1.

≤

{
C(ϵ21)∥∇v∥22 + ϵ21∥u0∥2/3∥∆v∥22 if 1

3 < α < 1,

∥∇v∥22 if α = 1.

(2.46)

Following the same procedure as above, we obtain∫
R3

u1−α|∇w|2 dx ≤

{
C(ϵ22)∥∇w∥22 + ϵ22∥u0∥2/3∥∆w∥22 if 1/3 < α < 1,

∥∇w∥22 if α = 1.
(2.47)

As 0 < 6
2+6α < 2 and 1 < 6/5 < 3 + 6α, following the same procedure as (2.31),

(2.35) obtained and from previous case, we have∫
R3

u|∆v| dx ≤ C44 + C45∥∇u
1+2α

2 ∥22 + ϵ23∥∆v∥22,∫
R3

u|∆w| dx ≤ C46 + C47∥∇u
1+2α

2 ∥22 + ϵ24∥∆w∥22,∫
R3

v|∆v| dx ≤ C(ϵ′14)C0∥v0∥
8
7
1 ∥∆v∥

6
7
2 + ϵ′14∥∆v∥22,∫

R3

w|∆w| dx ≤ C(ϵ′15)C1∥w0∥
8
7
1 ∥∆w∥

6
7
2 + ϵ′15∥∆w∥22,∫

R3

u|z| dx ≤ C26 + C27∥∇u
1+2α

2 ∥22 + C(ϵ16)∥∇z∥22.



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 13

Substituting (2.46), (2.47) and the above estimates in (2.45), we obtain

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C42

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C43

(
1 + ∥∇v∥22 + ∥∇w∥22

)
.

(2.48)

Integrating above with respect to t, we obtain (2.6).

Case (iii): α > 1. First, multiplying (2.2)1 by log u and integrating we obtain

d

dt

∫
R3

u log u dx+
4

1 + α
∥∇u

1+α
2 ∥22 =

∫
R3

∇u·(χ∇v) dx−
∫
R3

∇u·(ξ∇w) dx. (2.49)

Using the assumption χ′, ξ′ ∈ L∞
loc, we obtain∫

R3

∇u · (χ∇v) dx ≤ C44

∫
R3

u|∇v|2 dx+ C44

∫
R3

u|∆v| dx

−
∫
R3

∇u · (ξ∇w) dx ≤ C45

∫
R3

u|∇w|2 dx+ C45

∫
R3

u|∆w| dx.

Using the above in (2.49), we obtain

d

dt

∫
R3

u log u dx+
4

1 + α
∥∇u

1+α
2 ∥22

≤ C44

∫
R3

u|∇v|2 dx+ C45

∫
R3

u|∇w|2 dx

+ C44

∫
R3

u|∆v|dx+ C45

∫
R3

u|∆w|dx.

(2.50)

Now, multiplying (2.2)1 by uα and integrating we obtain

1

1 + α

d

dt
∥u∥1+α

1+α +
4α(1 + α)

(1 + 2α)2
∥∇u

1+2α
2 ∥22

≤
∫
R3

∇uα · u(χ∇v) dx−
∫
R3

∇uα · u(ξ∇w) dx.
(2.51)

From (2.16), we obtain∫
R3

∇uα · u(χ∇v) dx ≤ C50∥∇u
1+2α

2 ∥22 + C51

∫
R3

u|∇v|2 dx, (2.52)

where C50 = Cχϵ3 and C51 = CχC(ϵ3). Following the same procedure as previous,
we obtain

−
∫
R3

∇uα · u(ξ∇w) dx ≤ C52∥∇u
1+2α

2 ∥22 + C53

∫
R3

u|∇w|2 dx. (2.53)

Substituting (2.52) and (2.53) in (2.51), we obtain

d

dt
∥u∥1+α

1+α + C ′
49∥∇u

1+2α
2 ∥22 ≤ C51

∫
R3

u|∇v|2 dx+ C53

∫
R3

u|∇w|2 dx, (2.54)



14 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

where C ′
49 = 4α(1+α)2

(1+2α)2 −C50 +C52. Adding (2.50), (2.13), (2.54) and (2.21)-(2.23),

we obtain

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C54

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C55

(∫
R3

u|∇v|2 dx+

∫
R3

u|∇w|2 dx+

∫
R3

u|∆v| dx+

∫
R3

u|∆w| dx

+

∫
R3

v|∆v| dx+

∫
R3

w|∆w| dx+

∫
R3

u|z| dx
)
.

(2.55)

Next, we estimate the first integral
∫
R3 u|∇v|2 dx of RHS. As 1+α

2 > 1, using Young’s

inequality, we obtain u ≤ ϵ26 + C(ϵ26)u
1+α
2 . Using the previous equation and

Young’s inequality, we obtain∫
R3

u|∇v|2dx

≤ ϵ26∥∇v∥22 + C(ϵ26)

∫
R3

u
1+α
2 |∇v|2 dx

= C(ϵ26)

∫
R3

u
1+α
2 ∇v · ∇v dx+ ϵ26∥∇v∥22

≤ C(ϵ26)C56

(∫
R3

∇u
1+α
2 · ∇v + u

1+α
2 ∆v dx

)
+ ϵ26∥∇v∥22

≤ C(ϵ26)C56

(
∥∇u

1+α
2 ∥22 + ∥∇v∥22 + ∥u∥1+α

1+α + ∥∆v∥22
)
+ ϵ26∥∇v∥22.

(2.56)

Now, we estimate the third term of RHS of above. Using the Gagliardo-Nierenberg
and Young inequality, we obtain

∥u∥1+α
1+α ≤ C57∥u0∥

2+2α
2+3α

1 ∥∇u
1+α
2 ∥

6α
2+3α

2 ≤ C57C(ϵ27) + ϵ27∥∇u
1+α
2 ∥22.

Here, we have used that 6α
2+3α < 2. Substituting the above in (2.56), we obtain∫

R3

u|∇v|2 dx ≤ C ′
54 + C ′

55∥∇u
1+α
2 ∥22 + C ′

56∥∇v∥22 + C ′
57∥∆v∥22, (2.57)

where C ′
54 = C(ϵ26)C(ϵ27)C56C57, C

′
55 = C(ϵ26)C56+C(ϵ26)C56ϵ27, C

′
56 = C(ϵ26)C56+

ϵ26 and C
′
57 = C(ϵ26)C56. Following the same procedure as in the previous estimate,

and from previous case, we obtain∫
R3

u|∇w|2 dx ≤ C ′
58 + C ′

59∥∇u
1+α
2 ∥22 + C ′

60∥∇w∥22 + C ′
61∥∆w∥22,∫

R3

u|∆v| dx ≤ C44 + C45∥∇u
1+2α

2 ∥22 + ϵ23∥∆v∥22,∫
R3

u|∆w| dx ≤ C46 + C47∥∇u
1+2α

2 ∥22 + ϵ24∥∆w∥22,∫
R3

v|∆v| dx ≤ C(ϵ′14)C0∥v0∥
8
7
1 ∥∆v∥

6
7
2 + ϵ′14∥∆v∥22,∫

R3

w|∆w| dx ≤ C(ϵ′15)C1∥w0∥
8
7
1 ∥∆w∥

6
7
2 + ϵ′15∥∆w∥22,



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 15∫
R3

u|z| dx ≤ C26 + C27∥∇u
1+2α

2 ∥22 + C(ϵ16)∥∇z∥22.

Substituting the above estimates and (2.57) in (2.55), we have

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C54

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C55

(
1 + ∥∇v∥22 + ∥∇w∥22

)
.

(2.58)

Integrating above with respect to t, we obtain (2.6). □

Lemma 2.3. Let (u, v, w, z) be a classical solution of (2.2), for ϵ ∈ (0, 1) and
initial data (u0ϵ , v0ϵ , w0ϵ , z0ϵ) satisfies the following conditions, independent of ϵ,

(1) u0(1 + |x|+ | log u0|) ∈ L1(R3),
(2) u0 ∈ L1+α(R3),
(3) v0, w0 ∈ L∞(R3) ∩H1(R3),
(4) z0 ∈ L2(R3).

Assume that

α > 0, ϕ ∈W 2,∞(R3), χ′, ξ′ ∈ L∞
loc with χ′(·) ≥ χ0, ξ

′(·) ≥ ξ0 (2.59)

for some constant χ0 > 0, ξ0 > 0. Then, given t ∈ (0, T ], we have

sup
0≤τ≤t

E(τ) +

∫ t

0

D(τ)dτ < C, (2.60)

where C > 0 is a constant solely depends on initial data.

Proof. It is sufficient to prove only for 0 < α ≤ 1/6. Further, for α > 1/6 is already
established in the previous lemma. Multiplying (2.2)1 by log u and integrating we
obtain

d

dt

∫
R3

u log u dx+
4

1 + α
∥∇u

1+α
2 ∥22 ≤

∫
R3

∇u·(χ∇v) dx−
∫
R3

∇u·(ξ∇w) dx. (2.61)

The first term in RHS of the above is evaluated using our assumption and Young’s
inequality as∫

R3

∇u · (χ∇v) dx

= −
∫
R3

(χ′|∇v|2 + χ∆v)u dx

≤ −χ̂1

∫
R3

u|∇v|2 dx+ C58

∫
R3

|∇u||∇v| dx

≤ −χ0

∫
R3

u|∇v|2 dx+ C58C59

∫
R3

u
1−α
2

∣∣∇u 1+α
2

∣∣|∇v| dx
≤ −χ0

∫
R3

u|∇v|2 dx+ C60∥∇u
1+α
2 ∥22 + C61

∫
R3

u1−α|∇v|2 dx,

(2.62)



16 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

where C60 = C58C59ϵ26 and C61 = C58C59C(ϵ26). Here we used that |∇u| =

2u
1−α
2

1+α

∣∣∇u 1+α
2

∣∣. Similarly, we obtain

−
∫
R3

∇u · (ξ∇w)dx

≤ −ξ0
∫
R3

u|∇w|2 dx+ C ′
60∥∇u

1+α
2 ∥22 + C ′

61

∫
R3

u1−α|∇w|2 dx.
(2.63)

Using (2.62) and (2.63) in (2.61), we obtain

d

dt

∫
R3

u log u dx+ C62∥∇u
1+α
2 ∥22 + χ0

∫
R3

u|∇v|2dx+ ξ0

∫
R3

u|∇w|2 dx

≤ C61

∫
R3

u1−α|∇v|2 dx+ C ′
61

∫
R3

u1−α|∇w|2dx,
(2.64)

where C62 = 4
1+α − C60 − C ′

60. Now, multiplying (2.2)1 by uα and integrating we
obtain

1

1 + α

d

dt
∥u∥1+α

1+α +
4α(1 + α)

(1 + 2α)2
∥∇u

1+2α
2 ∥22

=

∫
R3

∇uα · u(χ∇v) dx−
∫
R3

∇uα · u(ξ∇w) dx.
(2.65)

Now, we estimate the first integral of RHS. Using Young’s inequality, we obtain∫
R3

∇uα · u(χ∇v) dx ≤ Cχ

∫
R3

∣∣∇u 1+2α
2

∣∣(u 1
2 |∇v|) dx

≤ C63∥∇u
1+2α

2 ∥22 + C ′
63

∫
R3

u|∇v|2 dx,
(2.66)

where χ := supR3 |χ(c)|, Cχ = 2αχ
1+α , C63 = Cχϵ27, and C

′
63 = CχC(ϵ27). Following

the procedure above, we obtain

−
∫
R3

∇uα · u(ξ∇w) dx ≤ C64∥∇u
1+2α

2 ∥22 + C ′
64

∫
R3

u|∇w|2 dx. (2.67)

Using (2.66) and (2.67) in (2.65), we obtain

d

dt
∥u∥1+α

1+α + C65∥∇u
1+2α

2 ∥22 ≤ C ′
63

∫
R3

u|∇v|2 dx+ C ′
64

∫
R3

u|∇w|2 dx, (2.68)

where C67 = 4α(1+α)2

(1+2α)2 − C63 − C64. As α does not affect (2.2)2, (2.2)3 and (2.2)4,

(2.23) in previous Lemma 2.2 holds. Adding (2.64), (2.13), (2.68) and (2.21)-(2.23),
we have

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C67

(∫
R3

u|∇v|2 + u|∇w|2dx+ ∥∇u
1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22

+ ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22
)

≤ C68

(∫
R3

u1−α|∇v|2dx+

∫
R3

u1−α|∇w|2dx−
∫
R3

u|∆v|dx

−
∫
R3

u|∆w|dx+

∫
R3

v|∆v|dx+

∫
R3

w|∆w|dx+

∫
R3

u|z| dx
)
.

(2.69)



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 17

Now, we estimate the first integral of the of RHs. Using u1−α ≤ C(ϵ28) + ϵ28u in
the integral and choosing sufficiently small ϵ28, we obtain∫

R3

u1−α|∇v|2 dx ≤ C69∥∇v∥22. (2.70)

Similarly, we obtain ∫
R3

u1−α|∇w|2 dx ≤ C ′
69∥∇w∥22. (2.71)

From the previous Lemma, we have∫
R3

v|∆v| dx ≤ C(ϵ′14)C0∥v0∥
8
7
1 ∥∆v∥

6
7
2 + ϵ′14∥∆v∥22,∫

R3

w|∆w| dx ≤ C(ϵ′15)C1∥w0∥
8
7
1 ∥∆w∥

6
7
2 + ϵ′15∥∆w∥22,∫

R3

u|z| dx ≤ C26 + C27∥∇u
1+2α

2 ∥22 + C(ϵ16)∥∇z∥22.

Using (2.70), (2.71) and the above estimates in (2.69), we obtain

d

dt

(∫
R3

u(log u+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+ C67

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
≤ C68

(
1 + ∥∇v∥22 + ∥∇w∥22

)
.

(2.72)

Integrating the above with respect to t, we obtain (2.60).. □

Now, we are ready to state the primary finding of this paper.

Theorem 2.4. Suppose that the initial data (u0, v0, w0, z0) satisfies the following:

(1) u0(1 + |x|+ | log u0|) ∈ L1(R3),
(2) u0 ∈ L1+α(R3),
(3) v0, w0 ∈ L∞(R3) ∩H1(R3),
(4) z0 ∈ L2(R3),

and either one of the assumptions (2.5) or (2.59) holds. Then for any T > 0, (2.2)
possesses a weak soluion (u, v, w, z) that satisfies

sup
0≤t≤T

(∫
R3

u(| log u|+ 2⟨x⟩) dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+

∫ T

0

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
dt < C,

(2.73)

where C is a constant that solely depends on initial data.

Proof. Recall that, the solutions of (2.2) in R3 × [0, T ) with initial conditions
(u0ϵ , v0ϵ , w0ϵ , z0ϵ) is given by

u0ϵ = ϕϵ ∗ u0, v0ϵ = ϕϵ ∗ v0, w0ϵ = ϕϵ ∗ w0, z0ϵ = ϕϵ ∗ z0,
where ϕϵ is a usual mollifier with ϵ ∈ (0, 1). The uniformity of the estimates
obtained in Lemma 2.2, regardless of the value of ϵ, is ensured by the convergence
of (u0ϵ , v0ϵ , w0ϵ , z0ϵ). In other words, it means that the constant C in (2.6) can be
selected without dependence on ϵ. In a similar way, we obtain constants such that
for q <∞,



18 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

(1) uϵ bounded in L∞((0, T )× R3),

(2) ∇u
q+α
2

ϵ bounded in L2((0, T )× R3),
(3) vϵ, wϵ, zϵ bounded in L∞(0, T ;W 1,q(R3)),
(4) vϵ, wϵ, zϵ bounded in Lq(0, T ;W 2,q(R3)),
(5) vϵt , wϵt , zϵt bounded in Lq(0, T ;Lq(R3)).

Our estimate allowed the local solution to be extended to arbitrary (0, T ) (as in [22,
27, 11, 7]). Let k ≥ 2+α be chosen. Then uϵt and u

k
ϵt belong to L

1(0, T ;W−2,2(R3))

(as in [22]), where the dual space of W 2,2(R3) is denoted by W−2,2(R3). Using the
Aubin-Lions compactness lemma, we have a weak limit (u, v, w, z) as ϵ → 0 which
is a weak solution. □

The above theorem can indeed be proven for a bounded domain with Neumann
boundary conditions for u, v, and w, as well as no-slip boundary conditions for z. To
be more specific, suppose we have a smooth boundary for the bounded domain Ω,
and we are considering the system (2.2) within Ω× [0, T ), with the aforementioned
boundary conditions as

∂u

∂ν
=
∂v

∂ν
=
∂w

∂ν
= 0, z = 0 on ∂Ω. (2.74)

Theorem 2.5. Suppose that the initial data (u0, v0, w0, z0) satisfies the following:

(1) u0ϵ ∈ L1(Ω) ∩ L1+α(Ω),
(2) v0ϵ and w0ϵ ∈ L∞(Ω) ∩H1(Ω),
(3) z0ϵ ∈ L2(Ω).

and either one of the assumptions (2.5) or (2.59) holds by replacing R3 by Ω. Then
for each T > 0, system (2.2) with boundary conditions (2.74) possesses a weak
soluion (u, v, w, z) that satisfies

sup
0≤t≤T

(∫
Ω

u| log u| dx+ ∥u∥1+α
1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22

)
+

∫ T

0

(
∥∇u

1+α
2 ∥22 + ∥∇u

1+2α
2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22

)
dt < C,

(2.75)

where C is a constant that solely depends on initial data.

Proof. We address only the modification to be done in above proof, as the proof is
similar. As ∥u∥L1(Ω) takes care the negative part of

∫
Ω
u log u, L1 estimate of u⟨x⟩

(2.13) is not needed. Also, (2.31) can be replaced by

∥u∥22 ≤ C0∥u0∥
1+6α
2+6α

1 ∥∇u
1+2α

2 ∥
6

2+6α

2 + C0∥u∥21. (2.76)

The rest of the proof is completed by employing similar ideas as those in Theorem
2.4, and hence omitted. □



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 19

3. Existence of bounded weak solutions

We consider the model the attraction-repulsion chemotaxis Stokes system, that
is, (1.6) with τ = 0, in this section as

ut + z · ∇u = ∆u1+α −∇ · (χ(v)u∇v) +∇ · (ξ(w)u∇w),
vt + z · ∇v = ∆v + βu− γv,

wt + z · ∇w = ∆w + δu− ηw,

zt +∇p = ∆z − u∇ϕ,
div z = 0.

(3.1)

This section first defines the bounded weak solution for (3.1). For the rationale
outlined in previous section, we introduce the suitable approximation problem for
(3.1). Before demonstrating our main finding, namely bounded weak solutions to
(3.1) in bounded domain with smooth boundary, we first establish a key lemma
used to substantiate our main findings.

Definition 3.1. For α > 0 and T ∈ (0,∞), a weak solution (u, v, w, z) as intro-
duced in Definition 2.1 is termed as a bounded weak solution of (3.1) if

(1) Given p ∈ [1,∞), we have u ∈ L∞((0, T );R3) and∇u
p+α

2 ∈ L2(0, T ;L2(R3)),
(2) Given q ∈ [2,∞), we have v, w, z ∈ Lq(0, T ;W 2,q(R3)) and vt, wt, zt ∈

Lq(0, T ;Lq(R3)).

We define an approximation problem for the above system as

uϵt + zϵ · ∇uϵ = ∆(uϵ + ϵ)1+α −∇ · (χ(vϵ)uϵ∇vϵ) +∇ · (ξ(wϵ)uϵ∇wϵ),

vϵt + zϵ · ∇vϵ = ∆vϵ + βuϵ − γvϵ,

wϵt + zϵ · ∇wϵ = ∆wϵ + δuϵ − ηwϵ,

zϵt +∇pϵ = ∆zϵ − uϵ∇ϕ,
div zϵ = 0

(3.2)

with initial conditions

u0ϵ = ϕϵ ∗ u0, v0ϵ = ϕϵ ∗ v0, w0ϵ = ϕϵ ∗ w0, z0ϵ = ϕϵ ∗ z0,

where ϕϵ is a usual mollifier with ϵ ∈ (0, 1). According to the standard theory
of existence and regularity, for every ϵ > 0, equation (2.2) admits a local-in-time
classical solution. The proof for this assertion is not provided here because it
follows a similar methodology as in [2, 23]. Hereafter, we use the unknowns (uϵ,
vϵ, wϵ, and zϵ) as (u, v, w, z) for simplicity of notation. First, we deduce some
estimates, independent of ϵ, of the solution to (3.2) which are uniform in nature.
Using those estimates, local bounded weak solution is extended to arbitrary (0, T )
and we construct bounded weak solution of (3.1).

Lemma 3.2. Suppose that (u, v, w, z) is a classical solution of (3.2), for all ϵ ∈
(0, 1) and the initial data (u0ϵ , v0ϵ , w0ϵ , z0ϵ) satisfies (1)− (4) of Lemma 2.2 along
with u0ϵ ∈ L∞(R3) and v0ϵ , w0ϵ , z0ϵ ∈ W 1,q(R3) for any q ∈ [2,∞). Furthermore,
assume that

α >
1

8
, γ = η = 0, χ′, ξ′ ∈ L∞

loc with χ′(·) ≥ χ0, ξ
′(·) ≥ ξ0, (3.3)

for some positive constants χ0, ξ0. Then, for each t ∈ (0, T ], we have:



20 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

(1) For 1 ≤ p ≤ ∞,

u ∈ L∞(0, T ;Lp(R3)) and ∇u
p+α

2 ∈ L2(0, T ;L2(R3)), (3.4)

(2) For 2 ≤ q <∞,

v, w, z ∈ L∞(0, T ;W 1,q(R3)) ∩ Lq(0, T ;W 2,q(R3)), (3.5)

(3) For 2 ≤ q <∞,

vt, wt, zt ∈ Lq(0, T ;Lq(R3)). (3.6)

Proof. For 1 ≤ p ≤ 1 + α, by Lemma 2.2, (3.4) holds. It is sufficient to show that
for α > 1

8 , u satisfies (3.4), as (3.4)− (3.6) follows from that. Multiplying (3.2)1 by

up−1 and integrating it by parts, we have

1

p

d

dt
∥u∥pp +

∫
R3

∇up−1 · ∇(u+ ϵ)1+α dx

= −
∫
R3

up−1∇ · (χu∇v) dx+

∫
R3

up−1∇ · (ξu∇w) dx.
(3.7)

Also, we have∫
R3

∇up−1 · ∇(u+ ϵ)1+α dx ≥ 4(p− 1)(1 + α)

(p+ α)2
∥∇u

p+α
2 ∥22, (3.8)

by using that ∇up−1 · ∇u1+α = (p − 1)(1 + α)up−2+α|∇u|2 and |∇u
p+α

2 |22 =
(p+α)2

4 up+α−2|∇u|2. The first term in RHS of (3.7) is evaluated using Young’s
inequality as

−
∫
R3

up−1∇ · (χu∇v) dx

≤ 2χ

(p+ α)

∫
R3

|∇u
p+α

2 |u
p−α

2 |∇v| dx

≤ 2χ

(p+ α)

(
ϵ29∥∇u

p+α
2 ∥22 + C(ϵ29)

∫
R3

up−α|∇v|2dx
)

≤ C70∥∇u
p+α

2 ∥22 + C71

∫
R3

up−α|∇v|2 dx,

(3.9)

by using that |∇u| = 2
p+αu

p−α
2 |∇u

p+α
2 | where χ = supR3×[0,T ) χ(v), C70 = 2χϵ29

(p+α)

and C71 = 2χC(ϵ29)
(p+α) . Similarly, we have∫

R3

up−1∇ · (ξu∇w) dx ≤ C72∥∇u
p+α

2 ∥22 + C73

∫
R3

up−α|∇w|2 dx. (3.10)

Using (3.8)-(3.10) in (3.7), we have

1

p

d

dt
∥u∥pp + C74∥∇u

p+α
2 ∥22 ≤ C71

∫
R3

up−α|∇v|2 dx+ C73

∫
R3

up−α|∇w|2 dx (3.11)



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 21

where C74 = 4(p−1)(1+α)
(p+α)2 − C70 − C72. The first term in RHS of the above is

evaluated using the Hölder, Sobolev and Young inequalities as∫
R3

up−α|∇v|2 dx

≤ ∥up−α∥2 p
p−α

∥|∇v|2∥ p
α

≤ Cp1
∥u∥p−α

p ∥∆v∥2 6p
2p+3α

≤ Cp1

(α
p
+
p− α

p
∥u∥pp

)
∥∆v∥2 6p

2p+3α

.

(3.12)

Similarly, we have∫
R3

up−α|∇w|2 dx ≤ Cp2

(α
p
+
p− α

p
∥u∥pp

)
∥∆w∥2 6p

2p+3α

. (3.13)

Using (3.12) and (3.13) in (3.11), we have

1

p

d

dt
∥u∥pp + C74∥∇u

p+α
2 ∥22 ≤ C ′

71

(
∥∆v∥2 6p

2p+3α

+ ∥∆w∥2 6p
2p+3α

)
∥u∥pp

+ C ′
72

(
∥∆v∥2 6p

2p+3α

+ ∥∆w∥2 6p
2p+3α

)
.

Using the Gronwall inequality above, we have

∥u∥pp ≤ exp
(∫ t

0

∥∆v(s)∥2 6p
2p+3α

+ ∥∆w(s)∥2 6p
2p+3α

ds
)

×
∫ t

0

∥∆v(s)∥2 6p
2p+3α

+ ∥∆w(s)∥2 6p
2p+3α

ds+ ∥u0∥pp.
(3.14)

Hence, (3.4) holds, whenever the following holds∫ T

0

∥∆v(s)∥2 6p
2p+3α

+ ∥∆w(s)∥2 6p
2p+3α

ds <∞, 1 + α < p <∞. (3.15)

We prove the above statement in two case: (i) α > 1/3 and (ii) 1/8 < α ≤ 1/3.

Case (i): α > 1
3 . The first term in (3.15) is evaluated using the standard maximal

regularity estimate of heat equation. In (2.2)2, we have∫ T

0

∥∆v(s)∥2 6p
2p+3α

ds

≤ C75

(
∥∇v0∥2 6p

2p+3α

+

∫ T

0

∥u(s)∥2 6p
2p+3α

ds+

∫ T

0

∥z · ∇v∥2 6p
2p+3α

ds
)
.

(3.16)

For p > 1 + α, using an interpolation inequality, we have∫ T

0

∥u(s)∥2 6p
2p+3α

ds ≤ C76

∫ t

0

∥u(s)∥
2− (1+2α)(4p−3α)

2p(1+3α)

1 ∥u(s)∥
(1+2α)(4p−3α)

2p(1+3α)

3+6α ds

≤ C76

∫ T

0

∥∇u
1+2α

2 (s)∥
4

1+3α− 3α
p(1+3α)

2 ds.

(3.17)



22 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

The last term in RHS of (3.16) is evaluated using z ∈ L∞(0, T ;L6(R3)) and the
maximal regularity estimate for the heat equation in (3.2)2 as∫ T

0

∥z · ∇v∥2 6p
2p+3α

ds

≤
∫ T

0

∥z(s)∥26∥∇v(s)∥2 6p
p+3α

ds

≤ C

∫ T

0

∥∆v(s)∥22p
p+α

ds

≤ CC1

(
∥∇v0∥22p

p+α

+

∫ T

0

∥w(s)∥22p
p+α

ds+

∫ T

0

∥z · ∇v∥22p
p+α

ds
)

≤ C2 + C3

∫ T

0

∥∇v(s)∥2 6p
2p+3α

ds+ C4

∫ T

0

∥w(s)∥22p
p+α

ds

≤ C2 + C3

∫ T

0

∥∇v(s)∥2 6p
2p+3α

ds

+ C4C5

∫ T

0

∥w(s)∥
2− 3(1+2α)(p−α)

2(1+3α)p

1 ∥w(s)∥
3(1+2α)(p−α)

2(1+3α)p

3+6α ds

≤ C2 + C3

∫ T

0

∥∇v(s)∥2 6p
2p+3α

ds+ C6

∫ T

0

∥∇w
1+2α

2 (s)∥
3(p−α)
p(1+3α)

2 ds.

(3.18)

The above holds because α > 1/3. Similarly we obtain the same estimate for∫ T

0
∥∆w(s)∥2 6p

2p+3α

ds. Therefore (3.4) holds for p ∈ (max{1 + α, 3α},∞) and it

can be extended for p ∈ (1 + α,∞). Also it implies the following holds for every
1 ≤ p <∞: Given q <∞, we have

vt,∇2v, wt,∇2w, zt,∇2z ∈ Lq((0, T )× R3), ∇v,∇w ∈ L∞((0, T )× R3).

Using the above in (3.11) and an interpolation inequality, we have

d

dt
∥u∥pp ≤ C77∥u∥p−α

p−α ≤ C77∥u∥
α

p−1

1 ∥u∥
p(p−α−1)

p−1
p ≤ C77∥u∥

p(p−α−1)
p−1

p .

Using Gronwall’s inequality above, we have

∥u(t)∥p ≤ (C77p
2t)1/p + ∥u0∥p, t ≤ T.

Letting p→ ∞, we have u ∈ L∞(0, T ;Lp(R3)) for p ∈ (1 + α,∞).

Case (ii): 1
8 < α ≤ 1

3 . We prove (3.15) by showing that (3.4) holds, by deriving

for p ∈ [1, 1 + 4α). We estimate the first term in RHS of (3.11) using
∣∣∇up−α

∣∣ =
C0u

p−3α
2

∣∣∇u p+α
2

∣∣ and Young’s inequality as∫
R3

up−α|∇v|2dx =

∫
R3

up−α∇v · ∇v dx

≤ C ′
0

(∫
R3

|∇up−α||∇v| dx+

∫
R3

up−α|∆v| dx
)

≤ C ′
0

(∫
R3

C0u
p−3α

2

∣∣∇u p+α
2

∣∣|∇v| dx+

∫
R3

up−α|∆v| dx
)

≤ C78∥∇u
p+α

2 ∥2 + C79

∫
R3

up−3α|∇v|2dx+ C ′
0

∫
R3

up−α|∆v|dx.



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 23

Similarly, we have

∫
R3

up−α|∇w|2 dx

≤ C ′
78∥∇u

p+α
2 ∥2 + C ′

79

∫
R3

up−3α|∇w|2 dx+ C ′
1

∫
R3

up−α|∆w| dx.

Using the above estimates in (3.11), we have

d

dt
∥u∥pp + C ′∥∇u

p+α
2 ∥22 ≤ C80

∫
R3

up−3α|∇v|2 dx+ C81

∫
R3

up−α|∆v| dx

+ C82

∫
R3

up−3α|∇w|2 dx+ C83

∫
R3

up−α|∆w| dx.
(3.19)

Integrating the above with respect to t, we obtain

∥u∥pp + C ′
∫ t

0

∥∇u
p+α

2 ∥22ds

≤ C80

∫ t

0

∫
R3

up−3α|∇v|2 dx ds+ C82

∫ t

0

∫
R3

up−3α|∇w|2 dx ds

+ C ′
80

∫ t

0

∥u∥p−α+1
p−α+1ds+ C ′

81

∫ t

0

∥∆v∥p−α+1
p−α+1 + ∥∆w∥p−α+1

p−α+1ds+ ∥u0∥pp.

(3.20)

Evaluating first term of RHS using the Hölder inequality and a maximal regularity
estimate, we obtain

∫ t

0

∫
R3

up−3α|∇v|2dxds

≤ C84

∫ t

0

∥up−3α(s)∥ 1+α
p−3α

∥∇v(s)2∥ 1+α
1+4α−p

ds

≤ C84C85

∫ t

0

∥∆v(s)∥2 6+6α
5+14α−3p

≤ C84C85

(
∥∇v0∥2 6+6α

5+14α−3p

+

∫ t

0

∥u∥2 6+6α
5+14α−3p

ds+

∫ t

0

∥z · ∇v∥2 6+6α
5+14α−3p

ds
)

(3.21)

for all p satisfying 1 + α < p < 1 + 4α. Above, we used that 1/8 < α ≤ 1/3 and
p ∈ (1, 1 + 4α). As 1 +α < 6+6α

5+14α−3p < 3p+ 3α and using interpolation inequality,

the second term in RHS of (3.21) is evaluated as

∫ t

0

∥u(s)∥2 6+6α
5+14α−3p

ds ≤
∫ t

0

∥u∥2(1−t1)
1+α ∥u∥2t13p+3α ds ≤

∫ t

0

∥∇u
p+α

2 ∥
4t1
p+α

2 ds (3.22)



24 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

where t1 = (3p−14α+1)(p+α)
2(3p+2α−1) . The third term in RHS of (3.21) is estimated using

the Gagliardo-Nierenberg inequality and maximal regularity estimate as∫ t

0

∥z · ∇v∥2 6+6α
5+14α−3p

ds

≤ C ′
84

∫ t

0

∥∇v∥ 6+6α
4+13α−3p

≤ C ′
85

∫ t

0

∥∆v∥ 2+2α
2+5α−p

≤ C ′
85

(
∥∇v0∥2 2+2α

2+5α−p

+

∫ t

0

∥u∥2 2+2α
2+5α−p

ds+

∫ t

0

∥z · ∇v∥2 2+2α
2+5α−p

ds
)

≤ C ′
85∥∇v0∥2 2+2α

2+5α−p

+ C ′
85

∫ t

0

∥∇u
p+α

2 ∥
8(1+α)

(p+α)(2+5α−p)

2 ds

+ C86

∫ t

0

∥∇v∥2 6+6α
5+14α−3p

ds.

(3.23)

Substituting (3.22) amd (3.23) in (3.21), we have∫ t

0

∫
R3

up−3α|∇v|2dxds

≤ C87∥∇v0∥2 6+6α
5+14α−3p

+ C88∥∇v0∥2 2+2α
2+5α−p

+ C89

∫ t

0

∥∇u
p+α

2 ∥
4t1
p+α

2 ds

+ C90

∫ t

0

∥∇u
p+α

2 ∥
8(1+α)

(p+α)(2+5α−p)

2 ds+ C91

∫ t

0

∥∇v∥2 6+6α
5+14α−3p

ds.

(3.24)

Proceeding as above, we obtain∫ t

0

∫
R3

up−3α|∇w|2dxds

≤ C ′
87∥∇w0∥2 6+6α

5+14α−3p

+ C ′
88∥∇w0∥2 2+2α

2+5α−p

+ C ′
89

∫ t

0

∥∇u
p+α

2 ∥
4t1
p+α

2 ds

+ C ′
90

∫ t

0

∥∇u
p+α

2 ∥
8(1+α)

(p+α)(2+5α−p)

2 ds+ C ′
91

∫ t

0

∥∇w∥2 6+6α
5+14α−3p

ds.

(3.25)

Similarly, we deduce

C ′
80

∫ t

0

∥u∥p−α+1
p−α+1 ds+ C ′

81

∫ t

0

∥∆v∥p−α+1
p−α+1 + ∥∆w∥p−α+1

p−α+1 ds

≤ C91∥∇v0∥p−α+1
p−α+1 + C92∥∇w0∥p−α+1

p−α+1 + C93∥∇v0∥26(1+p−α)
7+p−α

+ C94∥∇w0∥26(1+p−α)
7+p−α

+ C95∥∇u
p+α

2 ∥
6(p−2α)
2α+3p−1

2

+ C96∥∇u
p+α

2 ∥
1+5α−5p
1−3α−3p

2 + C97∥∆v0∥
3(p−α−1)
2(p−α+1)

2 + C98∥∆w0∥
3(p−α−1)
2(p−α+1)

2 .

(3.26)

Using the above estimates in (3.19) and Young’s inequality, for p ∈ (1 + α, 1 + 4α)
we have

∥u∥pp + C ′
91

∫ t

0

∥∇u
p+α

2 ∥22ds ≤ C ′
92, 0 < t < T. (3.27)



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 25

Here we used the facts 4t1
p+α ∈ (0, 2) and 6+6α

5+14α−3p ∈ [2, 6] for 1/8 < α ≤ 1/3 and

1 + α < p < 1 + 4α. Hence, we have (3.4) holds for p ∈
(
2+11α

3 , 1 + 4α
)
, Also,

the above can be extended for p ∈ (1 + α, 1 + 4α). Choose s0 = 3
2 − 3α

4 . Then

from above, u ∈ L∞(0, T ;Ls0(R3)) as 1 ≤ s0 < 1 + 4α. To prove (3.15) holds for
α ∈ [1/8, 1/3), we choose s1 = 6p

2p+3α such that s1 ∈ (s0, 3s0 + 3α) for p > 1 + α.

Then by the standard maximal regularity estimate and using Hölder inequality, we
have ∫ T

0

∥∆v∥s1 ≤ C99

∫ T

0

∥u∥2s1 ds+
∫ T

0

∥z · ∇v∥2s1 ds. (3.28)

The first term in RHS of above evaluated using interpolation inequality as∫ T

0

∥u∥2s1 ds ≤ C91

∫ T

0

∥u∥
2− (s0+α)(6p−2ps0−3αs0)

2(2s0+3α)
s0 ∥u∥

(s0+α)(6p−2ps0−3αs0)

2(2s0+3α)

3s0+3α ds

≤ C91

∫ T

0

∥∇u
s0+α

2 ∥
p(12−4s0)−6αs0

p(2s0+3α)

2 ds

= C91

∫ T

0

∥∇u
s0+α

2 ∥
2− 2p(3α+4s−6)+6αs

p(3α+2s)

2 ds.

(3.29)

The second term in RHS of (3.28) evaluated using maximal regularity estimate as∫ T

0

∥z · ∇v∥2s1ds

≤ C ′
91

∫ T

0

∥∆v∥22p
p+α

ds

≤ C ′
91

(
∥∇v0∥22p

p+α

+

∫ T

0

∥u∥22p
p+α

ds+

∫ T

0

∥z · ∇v∥22p
p+α

ds
)

≤ C ′
92 + C ′

93

∫ T

0

∥u∥
(α+1)(3α(2α+1)+6αp+p)

(5α+2)p

1+α ∥u∥
3(2α+1)(α(α+1)+(α−1)p)

(5α+2)p

3+6α ds

≤ C ′
92 + C ′

94

∫ T

0

∥u
1+2α

2 ∥
6(α−1)(α−p)

(5α+2)p

1+α .

(3.30)

As (3.15) holds for α > 1
8 , we have u ∈ L∞(0, T ;Lp(R3)) for p ∈ (1 + α,∞). As

similar to the above, we have u is bounded in L∞-norm. □

Theorem 3.3. Suppose that the initial data (u0, v0, w0, z0) satisfies (1)-(4) of The-
orem 2.4 along with u0 ∈ L∞(R3) and v0, w0, z0 ∈ W 1,q(R3) for all q ∈ [2,∞).
Furthermore, assume that (3.3) holds. Then for each T > 0, system (3.2) possesses
a bounded weak solution (u, v, w, z) that satisfies

∥u∥L∞((0,T )×R3) + ∥∇u
p+α

2 ∥L2((0,T )×R3)

+ ∥v∥Lq(0,T ;W 2,q(R3)) + ∥w∥Lq(0,T ;W 2,q(R3)) + ∥z∥Lq(0,T ;W 2,q(R3))

+ ∥∂tv∥Lq(0,T ;Lq(R3)) + ∥∂tw∥Lq(0,T ;Lq(R3)) + ∥∂tz∥Lq(0,T ;Lq(R3)) < C,

where C is a constant depending on the initial data.

Proof. The existence of local weak solutions for model (3.1) can be obtained as for
model (2.1). Therefore, we omit the proof here for the sake simplicity. Now, it is
sufficient to prove that the system admits bounded weak solution. From Lemma



26 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

3.2 and using the Aubin-Lions compactness lemma we have a weak solution, which
indeed is a bounded weak solution for (3.1). □

The above theorem can be proven for a bounded domain Ω with Neumann bound-
ary conditions for u, v, and w, as well as no-slip boundary conditions for z as
specified in (2.74). Further, we have the result for bounded domain as corollary.

Corollary 3.4. Suppose that the initial data (u0, v0, w0, z0) satisfies (1)–(4) of
Theorem 2.4 when replacing R3 by Ω, along with u0 ∈ L∞(Ω) and v0, w0, z0 ∈
W 1,q(Ω) for any q ∈ [2,∞). Furthermore, assume that (3.3) holds when replacing
R3 by Ω. Then for each T > 0, system (3.2) with boundary conditions (2.74)
possesses a bounded weak solution (u, v, w, z) that satisfies

∥u∥L∞((0,T )×Ω) + ∥∇u
p+α

2 ∥L2((0,T )×Ω) + ∥v∥Lq(0,T ;W 2,q(Ω))

+ ∥w∥Lq(0,T ;W 2,q(Ω)) + ∥z∥Lq(0,T ;W 2,q(Ω)) + ∥∂tv∥Lq(0,T ;Lq(Ω))

+ ∥∂tw∥Lq(0,T ;Lq(Ω)) + ∥∂tz∥Lq(0,T ;Lq(Ω)) < C,

where C is a constant depending on the initial data.

References

[1] Myeongju Chae, Kyungkeun Kang, Jihoon Lee; Existence of smooth solutions to coupled

chemotaxis-fluid equations, Discret. Contin. Dyn. Syst. 33 (2013), no. 6, 2271–2297.
[2] Myeongju Chae, Kyungkeun Kang, Jihoon Lee; Existence of smooth solutions to coupled

chemotaxis-fluid equations, Discret. Contin. Dyn. Syst. 33 (2013), no. 6, 2271–2297.
[3] Myeongju Chae, Kyungkeun Kang, Jihoon Lee; Global Existence and Temporal Decay in

Keller-Segel Models Coupled to Fluid Equations, Commun. Partial Differ. Equations 39

(2014), no. 7, 1205–1235.
[4] Xiaoyu Chen, Jijie Zhao, Qian Zhang; Global existence of weak solutions for the 3D axisym-

metric chemotaxis-Navier-Stokes equations with nonlinear diffusion, Discret. Contin. Dyn.

Syst., 42 (2022), no. 9, 4489–4522.
[5] A. Chertock, K. Fellner, A. Kurganov, A. Lorz, P. A. Markowich; Sinking, merging and

stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach,

J. Fluid Mech. 694 (2012), 155–190.
[6] Yun-Sung Chung, Kyungkeun Kang; Existence of global solutions for a chemotaxis-fluid

system with nonlinear diffusion, J. Math. Phys. 57 (2016), no. 4, 041503.

[7] Yun-Sung Chung, Kyungkeun Kang, Jaewoo Kim; Global Existence of Weak Solutions for a
Keller-Segel-Fluid Model with Nonlinear Diffusion, J. Korean Soc. Math. 51 (2014), no. Issue

3, 635–654.

[8] Renjun Duan, Alexander Lorz, Peter Markowich; Global Solutions to the Coupled
Chemotaxis-Fluid Equations, Commun. Partial Differ. Equations 35 (2010), no. 9, 1635–
1673.

[9] Abelardo Duarte-Rodriguez, Lucas C. F. Ferreira, Elder J. Villamizar-Roa; Global Existence
for an Attraction–Repulsion Chemotaxis-Fluid System in a Framework of Besov–Morrey

type, J. Math. Fluid Mech. 22 (2020).
[10] Elio Espejo, Takashi Suzuki; Global existence and blow-up for a system describing the aggre-

gation of microglia, Appl. Math. Lett. 35 (2014), 29–34.
[11] Marco Di Francesco, Alexander Lorz, Peter A. Markowich; Chemotaxis-fluid coupled model

for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior,
Discret. Contin. Dyn. Syst. 28 (2010), no. 4, 1437–1453.

[12] Kentarou Fujie, Takasi Senba; Global existence and boundedness of radial solutions to a
two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity 29
(2016), no. 8, 2417.

[13] Miguel A Herrero, Juan J.L. Velazquez; Singularity patterns in a chemotaxis model., Math.
Ann. 306 (1996), no. 3, 583–624.



EJDE-2024/75 SOLVABILITY OF AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 27

[14] S. Ishida; Global existence and boundedness for chemotaxis-Navier-Stokes systems with

position-dependent sensitivity in 2D bounded domains, Discret. Contin. Dyn. Syst. 35 (2015),

no. 8, 3463–3482.
[15] Sachiko Ishida, Tomomi Yokota; Global existence of weak solutions to quasilinear degenerate

Keller–Segel systems of parabolic–parabolic type with small data, J. Differ. Equ. 252 (2012),

no. 3, 2469–2491.
[16] Liu Jia, Wang Zhi-An; Classical solutions and steady states of an attraction–repulsion chemo-

taxis in one dimension, J. Biol. Dyn. 6 (2012), no. sup1, 31–41, PMID: 22873673.

[17] Kyungkeun Kang, Hwa Kil Kim; Existence of Weak Solutions in Wasserstein Space for
a Chemotaxis Model Coupled to Fluid Equations, SIAM J. Math. Anal. 49 (2017), no. 4,

2965–3004.

[18] Kyungkeun Kang, Kyunghwa Kim, Changwook Yoon; Existence of weak and regular solutions
for Keller–Segel system with degradation coupled to fluid equations, J. Math. Anal. 485

(2020), no. 1, 123750.
[19] Evelyn F. Keller, Lee A. Segel; Model for chemotaxis, J. Theor. Biol. 30 (1971), no. 2,

225–234.

[20] Hideo Kozono, Masanari Miura, Yoshie Sugiyama; Existence and uniqueness theorem on mild
solutions to the Keller–Segel system coupled with the Navier–Stokes fluid, J. Funct. Anal.

270 (2016), no. 5, 1663–1683.

[21] Xie Li, Youjun Xiao; Global existence and boundedness in a 2D Keller-Segel–Stokes system,
Nonlinear Anal. Real World Appl. 37 (2017), 14–30.

[22] Jian-Guo Liu, Alexander Lorz; A coupled chemotaxis-fluid model: Global existence, Ann.

l’Institut Henri Poincare Anal. Non Lineaire 28 (2011), no. 5, 643–652.
[23] Jian-Guo Liu, Alexander Lorz; A coupled chemotaxis-fluid model: Global existence, Ann.

Inst. Henri Poincare (C) Anal. Non Lineaire 28 (2011), no. 5, 643–652.

[24] Alexander Lorz; A coupled Keller–Segel–Stokes model: global existence for small initial data
and blow-up delay, Commun. Math. Sci. 10 (2012), no. 2, 555–574.

[25] Magdalena Luca, Alexandra Chavez-Ross, Leah Edelstein-Keshet, Alex Mogilner; Chemotac-
tic signaling, microglia, and Alzheimer’s disease senile plaques: Is there a connection?, Bull.

Math. Biol. 65 (2003), no. 4, 693–730.

[26] Toshitaka Nagai; Global existence of solutions to a parabolic system for chemotaxis in two
space dimensions, Nonlinear Anal. Theory. Methods Appl. 30 (1997), no. 8, 5381–5388,

Proceedings of the Second World Congress of Nonlinear Analysts.

[27] Yoshie Sugiyama, Hiroko Kunii; Global existence and decay properties for a degenerate
Keller–Segel model with a power factor in drift term, J. Differ. Equ. 227 (2006), no. 1,

333–364.

[28] Youshan Tao, Zhi-an Wang; Competing effects of attraction vs. repulsion in chemotaxis,
Math. Models Methods Appl. Sci. 23 (2013), no. 01, 1–36.

[29] Youshan Tao, Michael Winkler; Global existence and boundedness in a Keller-Segel-Stokes

model with arbitrary porous medium diffusion, Discret. Contin. Dyn. Syst. 32 (2012), no. 5,
1901–1914.

[30] Youshan Tao, Michael Winkler; Locally bounded global solutions in a three-dimensional
chemotaxis-Stokes system with nonlinear diffusion, Ann. l’Institut Henri Poincare Anal. Non

Lineaire 30 (2013), no. 1, 157–178.

[31] Idan Tuval, Luis Cisneros, Christopher Dombrowski, Charles W. Wolgemuth, John O.
Kessler, Raymond E. Goldstein; Bacterial swimming and oxygen transport near contact lines,

Proc. Natl. Acad. Sci. USA. 102 (2005), no. 7, 2277–2282.
[32] Michael Winkler; Boundedness and large time behavior in a three-dimensional chemotaxis-

Stokes system with nonlinear diffusion and general sensitivity, J. Math. Fluid Mech. 54

(2015), no. 4, 3789–3828.

[33] Qingshan Zhang, Yuxiang Li; An attraction-repulsion chemotaxis system with logistic source,
Z Angew. Math. Mech. 96 (2016), no. 5, 570–584.

[34] Pan Zheng, Chunlai Mu, Xuegang Hu; Global dynamics for an attraction–repulsion chemo-
taxis–(Navier)–Stokes system with logistic source, Nonlinear Anal. Real World Appl. 45
(2019), 557–580.



28 Y. KARUPPUSAMY,S. LINGESHWARAN, M. JEYARAJ EJDE-2024/75

Yadhavan Karuppusamy

Department of Applied Sciences, National Institute of Technology Goa, Cuncolim -

403 703, Goa, India
Email address: yadhavan@nitgoa.ac.in

Shangerganesh Lingeshwaran
Department of Applied Sciences, National Institute of Technology Goa, Cuncolim -

403 703, Goa, India

Email address: shangerganesh@nitgoa.ac.in

Manimaran Jeyaraj

School of Advanced Sciences, Vellore Institute of Technology, Chennai- 600 127, Tamil

Nadu, India
Email address: manimaran.j@vit.ac.in


	1. Introduction
	2. Existence of weak solutions
	3. Existence of bounded weak solutions
	References