DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018249DYNAMICAL SYSTEMS SERIES BVolume 24, Number 4, April 2019 pp. 1919–1942

CONVERGENCE RATES OF SOLUTIONS FOR A TWO-SPECIES

CHEMOTAXIS-NAVIER-STOKES SYTSTEM WITH

COMPETITIVE KINETICS

Hai-Yang Jin

Department of Mathematics, South China University of Technology

Guangzhou 510640, China

Tian Xiang∗

Institute for Mathematical Sciences, Renmin University of China

Beijing 100872, China

(Communicated by Michael Winkler)

Abstract. We study the convergence rates of solutions to the two-species

chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics:

(n1)t + u · ∇n1 = ∆n1 − χ1∇ · (n1∇c) + µ1n1(1− n1 − a1n2),

x ∈ Ω, t > 0,

(n2)t + u · ∇n2 = ∆n2 − χ2∇ · (n2∇c) + µ2n2(1− a2n1 − n2),

x ∈ Ω, t > 0,

ct + u · ∇c = ∆c− (αn1 + βn2)c, x ∈ Ω, t > 0,

ut + κ(u · ∇)u = ∆u+∇P + (γn1 + δn2)∇φ, ∇ · u = 0,

x ∈ Ω, t > 0

under homogeneous Neumann boundary conditions for n1, n2, c and no-slip

boundary condition for u in a bounded domain Ω ⊂ Rd(d ∈ 2, 3) with smooth

boundary. The global existence, boundedness and stabilization of solutions

have been obtained in 2-D [8] and 3-D for κ = 0 andmaxχ1,χ2minµ1,µ2

‖c0‖L∞(Ω)

being sufficiently small [4]. Here, we examine further convergence and derive

the explicit rates of convergence for any supposedly given global bounded clas-sical solution (n1, n2, c, u); more specifically, in L∞-topology, we show that

(n1(·, t), n2(·, t), u(·, t)) t→∞→

( 1−a11−a1a2

, 1−a21−a1a2

, 0) exponentially,

if a1, a2 ∈ (0, 1),

(0, 1, 0) exponentially, if a1 > 1 > a2,

(0, 1, 0) algebraically, if a1 = 1 > a2,

(1, , 0, 0) exponentially, if a2 > 1 > a1,

(1, 0, 0) algebraically, if a2 = 1 > a1.

In either cases, the c-solution component converges exponentially to 0.

Moreover, it is shown that only the rate of convergence for u is expressed

2010 Mathematics Subject Classification. Primary: 35B40, 35K55, 35B44, 35K57; Secondary:35Q92, 92C17.

Key words and phrases. Chemotaxis-fluid system, boundedness, exponential convergence, al-gebraic convergence, convergence rates.∗ Corresponding author.

1919

1920 HAI-YANG JIN AND TIAN XIANG

in terms of the model parameters and the first eigenvalue of −∆ in Ω underhomogeneous Dirichlet boundary conditions, and all other rates of convergence

are explicitly expressed only in terms of the model parameters ai, µi, α and β

and the space dimension d.

1. Introduction. We consider the following two-species chemotaxis-fluid systemwith competitive terms:

(n1)t + u · ∇n1 = ∆n1 − χ1∇ · (n1∇c) + µ1n1(1− n1 − a1n2), x ∈ Ω, t > 0,

(n2)t + u · ∇n2 = ∆n2 − χ2∇ · (n2∇c) + µ2n2(1− a2n1 − n2), x ∈ Ω, t > 0,

ct + u · ∇c = ∆c− (αn1 + βn2)c, x ∈ Ω, t > 0,

ut + κ(u · ∇)u = ∆u+∇P + (γn1 + δn2)∇φ, ∇ · u = 0, x ∈ Ω, t > 0,

∂νn1 = ∂νn2 = ∂νc = 0, u = 0, x ∈ ∂Ω, t > 0,

ni(x, 0) = ni,0(x), c(x, 0) = c0(x), u(x, 0) = u0(x), x ∈ Ω, i = 1, 2,

(1.1)where Ω ⊂ Rd (throughout the whole paper, d ∈ 2, 3) is a bounded domain withsmooth boundary ∂Ω and ∂ν denotes differentiation with respect to the outwardnormal of ∂Ω; κ ∈ R, χ1, χ2, a1, a2 ≥ 0 and µ1, µ2, α, β, γ, δ > 0 are constants;n1,0, n2,0, c0, u0, φ are known functions satisfying

0 < n1,0, n2,0 ∈ C(Ω), 0 < c0 ∈W 1,q(Ω), u0 ∈ D(Aϑ), (1.2)

φ ∈ C1+η(Ω) (1.3)

for some q > d, ϑ ∈(

34 , 1), η > 0 and A is the Stokes operator.

The system (1.1), an extension of the chemotaxis-fluid system introduced byTuval et al. [28], depicts the evolution of two competing species which react ona single chemoattractant in a liquid surrounding environment. Here, n1 and n2

denote densities of species, c means the chemical concentration, and finally, u andP represent the fluid velocity field and its associated pressure. So, it is the mixedcombination of the complex interaction between chemotaxis, the Lotka-Volterrakinetics and fluid.

Even in the absence of chemotaxis and fluid in (1.1), the reduced Lotka-Volterracompetition system has been extensively studied. It is now well-known that itspositive equilibrium (N1, N2) with

N1 :=1− a1

1− a1a2, N2 :=

1− a2

1− a1a2(1.4)

is globally asymptotically stable in weak competition case a1 < 1 < 1a2

[5, 6]. while,

in the strong competition case a1 > 1 > 1a2

, the positive equilibrium (N1, N2) isunstable and the system admits non-constant steady states when the system pa-rameters and the domain geometry are properly balanced [11, 15, 16]. In the onecomponent chemotaxis-only context (n2 ≡ 0 ≡ u), the global existence, bound-edness and stabilization of solutions to (1.1) have also been studied widely, cf.[23, 24, 10, 14]; see also [30, 32, 34, 36] for classical Keller-Segel models with signalproduction instead of consumption, i.e., −αn1c replaced with −c+αn1 in the thirdequation.

In one-species (n2 ≡ 0), global existence of weak (and/or eventual smoothness ofweak solutions) and classical solutions and asymptotic behavior have been investi-gated, e.g., in [31, 33, 35] without logistic source (µ1 = 0) and also the convergencerate has been explored [37] and in [13, 25, 27] with logistic source.

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1921

In two-species context, related studies first begin with fluid-free systems with sig-nal production (in which the asymptotic stability usually depends on some smallnesscondition on the chemo-sensitivities) to understand the influence of chemotaxis andthe Lotka-Volterra kinetics [1, 2, 18, 17, 20, 19, 22]. For the two-species chemotaxis-fluid system with competitive terms (1.1), the global existence, boundedness of clas-sical solutions and stabilization to equilibria were very recently studied by Hirataet al. [8] in the 2-D setting and by Cao et al. [4] in the 3-D setting for κ = 0. Forκ = 1, the global existence of weak solutions to the above system, and their eventualsmoothness and stabilization were studied in [9]. We now state the bounded andclassical solutions established in [8, 4] as follows:

(B2) (Boundedness in 2-D [8]) In the case that Ω ⊂ R2 is a bounded domainwith smooth boundary, let χ1, χ2, a1, a2 ≥ 0, µ1, µ2, α, β, γ, δ > 0 and let (1.2)and (1.3) hold. The the IBVP (1.1) with κ = 1 possesses a unique classicalsolution (n1, n2, c, u, P ), up to addition of spatially constant functions to P ,such that

n1, n2 ∈ C(Ω× [0,∞)) ∩ C2,1(Ω× (0,∞)),

c ∈ C(Ω× [0,∞)) ∩ C2,1(Ω× (0,∞)) ∩ L∞loc([0,∞);W 1,q(Ω)),

u ∈ C(Ω× [0,∞)) ∩ C2,1(Ω× (0,∞)) ∩ L∞loc([0,∞);D(Aϑ)),

P ∈ C1,0(Ω× (0,∞)).

Moreover, there exists a constant C > 0 such that for all t > 0

‖n1(·, t)‖L∞(Ω) + ‖n2(·, t)‖L∞(Ω) + ‖c(·, t)‖W 1,q(Ω) + ‖u(·, t)‖L∞(Ω) ≤ C. (1.5)

(B3) (Boundedness in 3-D [4]) In the case that Ω ⊂ R3 is a bounded domainwith smooth boundary and that κ = 0, there exists a constant ξ0 > 0 such thatwhenever maxχ1, χ2‖c0‖L∞(Ω) < ξ0 minµ1, µ2, the statements of bound-edness in (B2) hold.

(UC) (Uniform Convergence [8, 4]) Let (n1, n2, c, u, P ) be the solution of (1.1)obtained from (B2) or (B3). Then it fulfills the following convergence prop-erties:(i) Assume that a1, a2 ∈ (0, 1). Then, with (N1, N2) defined in (1.4),

n1(·, t)→ N1, n2(·, t)→ N2, c(·, t)→ 0, u(·, t)→ 0 in L∞(Ω) as t→∞,

(ii) Assume that a1 ≥ 1 > a2. Then

n1(·, t)→ 0, n2(·, t)→ 1, c(·, t)→ 0, u(·, t)→ 0 in L∞(Ω) as t→∞.

In this paper, we study dynamical properties for any supposedly global-in-timeand bounded classical solution to (1.1), with particular focus on the model of con-vergence as well as their explicit rates of convergence. Before proceeding to ourmain results, let us observe that, the n1- and n2-equations in (1.1) are symmetric.Thus, we should naturally have a result about stabilization to (0, 1, 0, 0). This wasnot mentioned in related works, cf. [1, 8, 17]. Now, let λP denote the Poincareconstant, cf. (4.19), and, finally, let

ξ =1

2min

(1− a1a2)µ1 min

12 ,

a1(1+a1a2)a2

2 max 1

N1, a1µ1

a2µ2N2

, (αN1 + βN2)

, (1.6)

1922 HAI-YANG JIN AND TIAN XIANG

ν =1

4min

(1− a2)µ2 min

1

2,

a2

1 + a2

, 2β, (a1 − 1)µ1

, (1.7)

µ =1

4min(1− a1)µ1 min

1

2,

a1

1 + a1

, 2α, (a2 − 1)µ2. (1.8)

Then we are at the position to state our main results on exponential and algebraicconvergence of global-in-time bounded solutions to model (1.1).

Theorem 1.1. Let Ω ⊂ Rd(d ∈ 2, 3) be a bounded and smooth domain, κ = 0if d = 3, and let (1.2) and (1.3) be in force, finally, let (n1, n2, c, u, P ) be a givenglobal- and bounded-in time classical solution of (1.1). Then it enjoys the followingdecay properties.

(I) When a1, a2 ∈ (0, 1), (n1, n2, u) converges exponentially to (N1, N2, 0):‖n1(·, t)−N1‖L∞(Ω) ≤ m1e

− ξd+2 t, ∀t ≥ 0,

‖n2(·, t)−N2‖L∞(Ω) ≤ m2e− ξd+2 t, ∀t ≥ 0,

‖u(·, t)‖L∞(Ω) ≤ m3e− ε

2(d+2)minλP ,ξt, ∀t ≥ 0.

(1.9)

(II) When a1 = 1 > a2, (n1, n2, u) converge algebraically to (0, 1, 0):‖n1(·, t)‖L∞(Ω) ≤ m4(t+ 1)−

1d+1 , ∀t ≥ 0,

‖n2(·, t)− 1‖L∞(Ω) ≤ m5(t+ 1)−1d+2 , ∀t ≥ 0,

‖u(·, t)‖L∞(Ω) ≤ m6(t+ 1)−εd+2 , ∀t ≥ 0.

(1.10)

(III) When a1 > 1 > a2, (n1, n2, u) converges exponentially to (0, 1, 0):‖n1(·, t)‖L∞(Ω) ≤ m7e

− (a1−1)µ12(d+1)

t, ∀t ≥ 0,

‖n2(·, t)− 1‖L∞(Ω) ≤ m8e− νd+2 t, ∀t ≥ 0,

‖u(·, t)‖L∞(Ω) ≤ m9e− ε

2(d+2)minλP ,ν, ∀t ≥ 0.

(1.11)

(II′) When a2 = 1 > a1, (n1, n2, u) converge algebraically to (1, 0, 0):‖n1(·, t)− 1‖L∞(Ω) ≤ m10(t+ 1)−

1d+2 , ∀t ≥ 0,

‖n2(·, t)‖L∞(Ω) ≤ m11(t+ 1)−1d+1 , ∀t ≥ 0,

‖u(·, t)‖L∞(Ω) ≤ m12(t+ 1)−εd+2 , ∀t ≥ 0.

(1.12)

(III′) When a2 > 1 > a1, (n1, n2, u) converge exponentially to (1, 0, 0):‖n1(·, t)− 1‖L∞(Ω) ≤ m13e

− µd+2 t, ∀t ≥ 0,

‖n2(·, t)‖L∞(Ω) ≤ m14e− (a2−1)µ2

2(d+1)t, ∀t ≥ 0,

‖u(·, t)‖L∞(Ω) ≤ m15e− ε

2(d+2)minλP ,µ, ∀t ≥ 0.

(1.13)

(IV) In either cases, the c-solution component converges exponentially to 0:

‖c(·, t)‖L∞(Ω) ≤ m16e− (αN1+βN2)

2 t, ∀t ≥ 0, (1.14)

where (N1, N2) = (N1, N2) in Case (I), (N1, N2) = (0, 1) in cases (II) and

(II′), and (N1, N2) = (1, 0) in cases (III) and (III′).

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1923

Here, ε ∈ (0, 1) is arbitrarily given, only m3,m6,m9 and m15 depend on ε; allmi(i = 1, 2, 3, · · · , 16) are suitably large constants depending on the initial datan1,0, n2,0, c0, u0, φ, the model parameters and Sobolev embedding constants but noton time t, see Section 4. Moreover, we have the following lower estimates:

mi = O(1), i = 1, 2, 7, 14, m3 ≥ O(1)(1 + (1− a1a2)−εd+2 ),

m4 ≥ O(1)(

1 + (1− a2)−1d+1

), m5 ≥ O(1)

(1 + (1− a2)−

1d+2

),

m6 ≥ O(1)(

1 + (1− a2)−εd+2

), m8 ≥ O(1)(1 + (a1 − 1)−

1d+2 + (1− a2)−

1d+2 ),

m9 ≥ O(1)(1 + (a1 − 1)−2εd+2 + (1− a2)−

2εd+2 ), m10 ≥ O(1)

(1 + (1− a1)−

1d+2

),

and

m11 ≥ O(1)(

1 + (1− a1)−1d+1

), m13 ≥ O(1)(1 + (1− a1)−

1d+2 + (a2 − 1)−

1d+2 ),

as well as

m12 ≥ O(1)(

1 + (1− a1)−εd+2

), m15 ≥ O(1)(1 + (a1 − 1)−

2εd+2 + (1− a2)−

2εd+2 ).

From these estimates, we see the conditions that 1 − a1a2 > 0, 1 − a2 > 0,1− a1 > 0 and a1 > 1 > a2 etc are very crucial in their respective cases.

With certain regularity and dissipation properties of global bounded solutions,the idea for the proof of convergence is quite known and developed, cf. [1, 4, 8, 17, 25,26, 27, 33] for example. The strategy for obtaining the explicit rates of convergenceas stated in Theorem 1.1 consists mainly of four steps. In the first step, we presentstronger regularity properties, e.g., W 1,∞-regularity for ni, W

1,p-regularity for uwith any finite p, and W 2,∞-regularity for c, for any bounded solution of (1.1) thanthose shown in [8, 4]; these are done in Section 2. In the crucial second step donein Section 3, we use refined computations to make those widely known Lyapunovfunctionals (cf. eg. [1, 4, 8, 17]) explicit, which will enable us to derive the explicitrates of convergence. Armed with the information provided by Step two, we thenmove on to calculate precisely the rates of convergence in L1-and L2-norm for theconsidered bounded solution, and related necessary estimates are also studied ingreat details. These constitute our Step three and are conducted in Section 4.Finally, thanks to the improved regularities, we apply the well-known Gagliardo-Nirenberg interpolation inequality to pass the obtained L1- and L2-convergence tothe L∞-convergence; these are our Step four and are also conducted in Section 4.

2. Regularity properties of bounded solutions. Let (n1, n2, c, u, P ) be a sup-posedly given global-in-time and bounded classical solution to (1.1) in the senseof (1.5). In this section, we provide more strong regularity properties for any suchbounded solution than those shown in [8, 4], which are needed to achieve our desiredrates of convergence in L∞-norm. We start with the regularity of u and c.

Lemma 2.1. Let Ω ⊂ Rd be a bounded and smooth domain and κ = 0 if d = 3.For d < p <∞, there exists a constant C > 0 such that

‖u(·, t)‖W 1,p ≤ C, ∀t > 1 (2.1)

and

‖c(·, t)‖W 1,∞ + ‖∆c(·, t)‖L∞ ≤ C, ∀t > 1. (2.2)

1924 HAI-YANG JIN AND TIAN XIANG

Proof. Using the essentially same argument as in [33, Lemma 6.3], we obtain (2.1)for d = 2. When d = 3, we can obtain (2.1) for k = 0 by using the Stokes semigroupand noting the Lp-boundedness of γn1 + δn2∇φ for p > 3. The W 1,∞-boundednessof c can be seen in [8, Lemma 3.9] and [27, Lemma 3.12]. With these and theL∞-boundedness of n1, n2 and u, an direct application of the standard parabolicschauder theory (cf. [12, 21]) to the third equation in (1.1) yields (2.2).

With the regularity properties in Lemma 2.1 at hand, we now utilize the quitecommonly used arguments (cf. [26, 27]) to show the following W 1,∞-regularity ofn1 and n2. To this end, we first establish the following L2-boundedness of thegradients of n1 and n2.

Lemma 2.2. Let the conditions be the same as Lemma 2.1. Then there exists aconstant C > 0 such that

‖∇n1(·, t)‖L2 + ‖∇n2(·, t)‖L2 ≤ C, ∀t > 1. (2.3)

Proof. Multiplying the first equation of system (1.1) by −∆n1, integrating by partsand using the boundedness of n1, n2, u and (2.2), we get

1

2

d

dt

∫Ω

|∇n1|2 +

∫Ω

|∆n1|2

=

∫Ω

u · ∇n1∆n1 + χ1

∫Ω

∇ · (n1∇c)∆n1 − µ1

∫Ω

n1(1− n1 − a1n2)∆n1

≤ 1

2

∫Ω

|∆n1|2 + c1

∫Ω

|∇n1|2 + c2,

which yieldsd

dt

∫Ω

|∇n1|2 +

∫Ω

|∆n1|2 ≤ 2c1

∫Ω

|∇n1|2 + 2c2. (2.4)

Here and below, unless otherwise stated, ci (numbered within lemmas) or C denotesome generic positive constants which may vary from line to line.

On the other hand, thanks to the boundedness of ‖n1‖L2 and the H2-ellipticestimate, we apply the Gagliardo-Nirenberg inequality in Ω ⊂ Rd to deduce

2c1‖∇n1‖2L2 ≤ c3‖D2n1‖L2‖n1‖L2 + c3‖n1‖2L2

≤ c4(‖∆n1‖L2 + ‖n1‖L2)‖n1‖L2 + c3‖n1‖2L2 ≤2

3‖∆n1‖2L2 + c5.

(2.5)

Substituting (2.5) into (2.4), one has

d

dt‖∇n1‖2L2 + c1‖∇n1‖2L2 ≤ 2c2 +

3

2c5,

which trivially gives

‖∇n1(·, t)‖2L2 ≤ ‖∇n1(·, 1)‖2L2e−c1(t−1) +2c2 + 3

2c5

c1≤ c6, ∀t > 1.

Applying the similar arguments to the n2-equation in (1.1), one can readily obtainthat ‖∇n2(·, t)‖2L2 ≤ c6 for all t > 1. We thus have shown the L2-estimate (2.3).

Lemma 2.3. There exists a constant C > 0 such that

‖n1(·, t)‖W 1,∞ + ‖n2(·, t)‖W 1,∞ ≤ C, ∀t > 1. (2.6)

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1925

Proof. First, we show there exists a constant c1 > 0 such that

‖n1(·, t)‖W 1,∞ ≤ c1, ∀t > 1. (2.7)

To this end, for any T > 2, we let

M(T ) := supt∈(2,T )

‖∇n1(·, t)‖L∞ .

Since clearly∇n1 is continuous on Ω×[0, T ], it follows thatM(T ) is finite. Moreover,since by our universal assumption n1 is bounded in L∞(Ω× (0,∞)), to prove (2.7),it is sufficient to derive the existence of c2 > 0 satisfying

M(T ) ≤ c2, ∀T > 2. (2.8)

To achieve (2.8), for any given t ∈ (2, T ), using the variation-of-constants formulato the first equation in (1.1), we get

n1(·, t) =e∆n1(·, t− 1)− χ∫ t

t−1

e(t−s)∆∇ · (n1(·, s)∇c(·, s))ds

−∫ t

t−1

e(t−s)∆u(·, s) · ∇n1(·, s)ds

+ µ1

∫ t

t−1

e(t−s)∆n1(·, s)(1− n1(·, s)− a1n2(·, s))ds,

which implies

‖∇n1(·, t)‖L∞ ≤‖∇e∆n1(·, t− 1)‖L∞ + χ

∫ t

t−1

‖∇e(t−s)∆∇ · (n1(·, s)∇c(·, s))‖L∞ds

+

∫ t

t−1

‖∇e(t−s)∆u(·, s) · ∇n1(·, s)‖L∞ds

+ µ1

∫ t

t−1

‖∇e(t−s)∆n1(·, s)(1− n1(·, s)− a1n2(·, s))‖L∞ds

=I1 + I2 + I3 + I4.

(2.9)

Next, we shall employ the widely known smoothing Lp-Lq type estimates of theNeumann heat semigroup et∆t≥0 in Ω (see [29, 3, 7] for instance) to estimateIi, i = 1, 2, 3, 4.

Thanks to the boundedness of n1, n2, u in Ω× (1,∞), (2.1) and (2.2), we employthose smoothing Neumann heat semigroup estimates to obtain that

I1 = ‖∇e∆n1(·, t− 1)‖L∞ ≤ c3‖n1(·, t− 1)‖L∞ ≤ c4 (2.10)

and that

I2 = χ

∫ t

t−1

‖∇e(t−s)∆∇ · (n1(·, s)∇c(·, s))‖L∞ds

≤ c5∫ t

t−1

[1 + (t− s)−

12−

d2p

]e−λ1(t−s)‖∇ · (n1(·, s)∇c(·, s))‖Lpds

≤ c5∫ t

t−1

[1 + (t− s)−

12−

d2p

]e−λ1(t−s)‖∇n1(·, s) · ∇c(·, s)‖Lpds

+ c5

∫ t

t−1

[1 + (t− s)−

12−

d2p

]e−λ1(t−s)‖n1(·, s)∆c(·, s)‖Lpds

1926 HAI-YANG JIN AND TIAN XIANG

≤ c6∫ t

t−1

[1 + (t− s)−

12−

d2p

]e−λ1(t−s)‖∇n1(·, s)‖Lpds+ c7, (2.11)

where λ1(> 0) is the first nonzero eigenvalue of −∆ under homogeneous boundarycondition and we have used the choice of p > d to ensure the finiteness of theGamma integral. Similarly, we can estimate I3 as follows:

I3 =

∫ t

t−1

‖∇e(t−s)∆u(·, s) · ∇n1(·, s)‖L∞ds

≤ c8∫ t

t−1

[1 + (t− s)−

12−

d2p

]e−λ1(t−s)‖∇n1(·, s)‖Lpds+ c9.

(2.12)

At last, using the boundedness of n1 and n2 again, one has

I4 =µ1

∫ t

t−1

‖∇e(t−s)∆n1(·, s)(1− n1(·, s)− a1n2(·, s))‖L∞ds

≤ c10

∫ t

t−1

[1 + (t− s)− 1

2

]e−λ1(t−s)ds

≤ c10

∫ 1

0

(1 + τ−12 )e−λ1τds ≤ (

1

λ1+ 2)c10.

(2.13)

Substituting (2.10), (2.11), (2.12) and (2.13) into (2.9), we infer that

‖∇n1(·, t)‖L∞ ≤ c11

∫ t

t−1

[1 + (t− s)−

12−

d2p

]e−λ1(t−s)‖∇n1(·, s)‖Lpds+ c12. (2.14)

Then, by the L2-boundedness of ∇n1(·, t) and ∇n2(·, t) in (2.3), the smoothness andhence boundedness of ∇n1 on Ω× [1, 2] and the definition of M(T ), we estimate

‖∇n1(·, s)‖Lp ≤ ‖∇n1(·, s)‖θL∞‖∇n1(·, s)‖1−θL2

≤ c13

(‖∇n1‖θL∞(Ω×[1,2])

+Mθ(T ))

≤ c14(1 +Mθ(T )), ∀s ∈ (1, T ),

(2.15)

where θ = p−2p ∈ (0, 1) due to p > 2.

Finally, since 12 + d

2p < 1, then a substitution of (2.15) into (2.14) entails

M(T ) ≤ c15Mθ(T ) + c16, ∀T > 2,

which upon a use of elementary inequality gives

M(T ) ≤ max2c16, (2c15)1

1−θ = max2c16, (2c15)p2 , ∀T > 2,

and hence (2.7) follows.The argument done for n1 can also be similarly applied to n2 to find that

‖n2(·, t)‖W 1,∞ ≤ c17, ∀t > 1.

This along with (2.7) yields simply (2.6), finishing the proof of the lemma.

3. Existence of explicit Lyapunov functionals. From boundedness to con-vergence, besides enough information on regularity, we still need some decayingestimates of bounded solutions under investigation. For the latter, the availabilityof a Lyapunov functional is crucial, see [1, 4, 8, 17] for instance. In this section,for our purpose, we particularize those known Lyapunov functionals used in thosepapers to obtain the explicit rates of convergence as stated in Theorem 1.1. Let us

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1927

start with the case of a1, a2 ∈ (0, 1). In this case, the explicit Lyapunov functionalthat we obtain for the chemotaxis-fluid system (1.1) reads as follows:

Lemma 3.1. Define

E1 :=

∫Ω

(n1 −N1 −N1 log

n1

N1

)+a1µ1

a2µ2

∫Ω

(n2 −N2 −N2 log

n2

N2

)

+1

2

(N1χ

21

4+a1µ1N2χ

22

4a2µ2+ 1

)∫Ω

c2

(3.1)

and

F1 :=

∫Ω

(n1 −N1)2 +

∫Ω

(n2 −N2)2.

Then, in the case of a1, a2 ∈ (0, 1), the nonnegative functions E1 and F1 satisfy

d

dtE1(t) ≤ −(1− a1a2)µ1 min

1

2,

a1

(1 + a1a2)a2

F1(t) =: −τF1(t), ∀t > 0. (3.2)

Proof. By honest differentiation of E1 in (3.1) and using elementary Cauchy-Schwarzinequality, one can easily derive the dissipation estimate (3.2); or alternatively, inthe proof of [8, Lemma 4.1], by taking

k1 =a1µ1

a2µ2, l1 = (

N1χ21

4+a1µ1N2χ

22

4a2µ2+ 1), ε =

µ1

2(1− a1a2),

upon honest calculations, one can easily arrive at (3.2).

For the purpose of deriving our explicit Lyapunov functionals in Cases (II)-(IV),we wish to perform honest computations here. We illustrate it for the case thata1 ≥ 1 > a2. In this case, from (1.1), the fact that a1 ≥ 1 and the positivity ofn1, n2, we calculate that

d

dt

∫Ω

n1 = µ1

∫Ω

(1− n1 − a1n2)n1 ≤ −µ1

∫Ω

n21 − µ1

∫Ω

n1(n2 − 1),

d

dt

∫Ω

(n2 − 1− log n2) =−∫

Ω

|∇n2|2

n22

+ χ2

∫Ω

∇n2

n2· ∇c

− µ2

∫Ω

(n2 − 1)2 − a2µ2

∫Ω

n1(n2 − 1)

(3.3)

as well as1

2

d

dt

∫Ω

c2 = −∫

Ω

|∇c|2 −∫

Ω

(αn1 + βn2)c2.

Therefore, for any positive constants σ1, σ2 and η ∈ (0, 1), in view of the positivityof ni, α and β, a clear linear combination of the three estimates above shows

− d

dt

∫Ω

[n1 + σ1 (n2 − 1− log n2) +

σ2

2c2]

≥∫

Ω

[µ1n

21 + (µ1 + a2µ2σ1)n1(n2 − 1) + µ2σ1(n2 − 1)2

]+

∫Ω

[σ1|∇n2|2

n22

− χ2σ1∇n2

n2· ∇c+ σ2|∇c|2

]=

∫Ω

[√µ2σ1η(n2 − 1) +

(µ1 + a2µ2σ1)

2√µ2σ1η

n1

]2+[µ1 −

(µ1 + a2µ2σ1)2

4µ2σ1η

]n2

1

1928 HAI-YANG JIN AND TIAN XIANG

+ µ2σ1(1− η)

∫Ω

(n2 − 1)2 +

∫Ω

[(√σ1∇n2

n2−χ2√σ1

2∇c)2

+(σ2 −

χ22σ1

4

)|∇c|2

]≥ µ2σ1(1− η)

∫Ω

(n2 − 1)2 + [µ1 −(µ1 + a2µ2σ1)2

4µ2σ1η]

∫Ω

n21

+ (σ2 −χ2

2σ1

4)

∫Ω

|∇c|2.

(3.4)

With these calculations above, we obtain the next explicit decay property, which isa specification of [8, Lemma 4.3], see also [4, Section 4.2].

Lemma 3.2. Define

E2 :=

∫Ω

n1 +µ1

a2µ2

∫Ω

(n2 − 1− log n2) +µ1χ

22

8a2µ2

∫Ω

c2

and

F2 :=

∫Ω

n21 +

∫Ω

(n2 − 1)2.

Then, in the case of a1 ≥ 1 > a2, the nonnegative functions E2 and F2 satisfy

d

dtE2(t) ≤ −(1− a2)µ1 min

1

2a2,

1

1 + a2

F2(t) =: −σF2(t), ∀t > 0. (3.5)

Proof. The fact that a2 < 1 allows us to select

σ1 =µ1

a2µ2, σ2 =

µ1χ22

4a2µ2, η =

1 + a2

2∈ (0, 1),

then, upon a plain calculation from (3.4), we obtain the dissipation inequality (3.5).

By the symmetry of the n1-and n2-equations in (1.1), when a2 ≥ 1 > a1, usingsimilar arguments leading to Lemma 3.2, we have a dissipation inequality as follows:

Lemma 3.3. Define

E3 :=

∫Ω

(n1 − 1− log n1) +a1µ1

µ2

∫Ω

n2 +χ2

1

8

∫Ω

c2

and

F3 :=

∫Ω

(n1 − 1)2 +

∫Ω

n22.

Then, in the case of a2 ≥ 1 > a1, the nonnegative functions E3 and F3 satisfy

d

dtE3(t) ≤ −(1− a1)µ1 min

1

2,

a1

1 + a1

F3(t) =: ρF3(t), ∀t > 0. (3.6)

Proof. For any constants l1, l2 > 0 and ε ∈ (0, 1), using the fact that a2 ≥ 1 andsimilar computations to the ones leading to (3.4), we infer that

− d

dt

∫Ω

[(n1 − 1− log n1) + l1n2 +

l22c2]

≥ µ1(1− ε)∫

Ω

(n1 − 1)2 + [l1µ2 −(a1µ1 + l1µ2)2

4µ1ε]

∫Ω

n22

+ (l2 −χ2

1

4)

∫Ω

|∇c|2.

(3.7)

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1929

Now, thanks to a1 < 1, we set

l1 =a1µ1

µ2, l2 =

χ21

4, ε =

1 + a1

2∈ (0, 1),

and then we easily conclude (3.6) upon trivial computations from (3.7).

4. Convergence rates. Aided by those dissipation estimates as provided in Lem-mas 3.1, 3.2 and 3.3, even weaker regularity properties than those in Section 2, usingthe quite known arguments, cf. [1, 4, 8, 25, 26, 27, 33] for example, we know thatany global-in-time and bounded classical solution of (1.1) satisfies the convergenceproperties as follows:∥∥∥(n1(·, t), n2(·, t), c(·, t), u(·, t))− (N1, N2, 0, 0)

∥∥∥L∞(Ω)

→ 0 as t→∞. (4.1)

Here, (N1, N2) = (N1, N2) when a1, a2 ∈ (0, 1), (N1, N2) = (0, 1) when a1 ≥ 1 > a2,

and (N1, N2) = (1, 0) when a2 ≥ 1 > a1, where N1 and N2 are defined by (1.4). Inthis section, we derive the explicit rates of convergence as described in Theorem 1.1for any supposedly bounded and global-in-time classical solution to (1.1).

4.1. Exponential convergence rate of c. We first take up the convergence rateof c, which is based on a parabolic comparison argument.

Lemma 4.1. The c-solution component of any bounded solution of (1.1) stabilizesto zero exponentially, namely, there exists t0 > 1 such that

‖c(·, t)‖L∞ ≤ ‖c0‖L∞e−αN1+βN2

2 (t−t0), ∀t ≥ t0.

Proof. We show the proof only for the case that a1, a2 ∈ (0, 1). Firstly, it follows

from the convergence of (4.1), as t → ∞, that n1(·, t) → N1 and n2(·, t) → N2

uniformly in Ω. Henceforth, we can fix a t0 > 1 such that

N1

2≤ n1 ≤

3N1

2and

N2

2≤ n2 ≤

3N2

2on Ω× [t0,∞). (4.2)

This simply shows

αn1 + βn2 ≥αN1 + βN2

2on Ω× [t0,∞).

This along with the third equation in (1.1) and the positivity of c gives

ct ≤ ∆c− u · ∇c− αN1 + βN2

2c on Ω× [t0,∞). (4.3)

Let z(t) be the solution of the following associated ODE problem:z′(t) + αN1+βN2

2 z(t) = 0, t ≥ t0,

z(t0) = ‖c(·, t0)‖L∞ .

It is clear that z(t) satisfies (4.3) together with ∂νz = 0, and hence an applicationof the comparison principle and Hopf boundary point lemma immediately yields

c(x, t) ≤ z(t) = ‖c(·, t0)‖L∞e−αN1+βN2

2 (t−t0) for all x ∈ Ω, t ≥ t0.Using the basic fact that t → ‖c(·, t)‖L∞ is non-increasing again by comparisonprinciple, cf. [33, Lemma 2.1], one has

c(x, t) ≤ ‖c0‖L∞e−αN1+βN2

2 (t−t0) for all x ∈ Ω, t ≥ t0.

1930 HAI-YANG JIN AND TIAN XIANG

This completes the proof of Lemma 4.1 by noting the positivity of c.

4.2. Exponential convergence rates in Case I: a1, a2 ∈ (0, 1). In this case,we will show that the solution components (n1, n2, u) converge exponentially to(N1, N2, 0).

4.2.1. Convergence rates of n1 and n2 in Case I. In this subsection, we shall estab-lish the convergence rate of n1 and n2 on the basis of the convergence rate of c inLemma 4.1 and the regularity of n1 and n2 provided by Lemma 2.3. We first useLemmas 4.1 and 3.1 to obtain the exponential convergence rates of ‖n1 − N1‖L2

and ‖n2 −N2‖L2 .

Lemma 4.2. The n1- and n2- solution components of bounded solution of (1.1)verify

‖n1(·, t)−N1‖2L2 + ‖n2(·, t)−N2‖2L2 ≤ K1e−ξ(t−t0), ∀t ≥ t0, (4.4)

where K1 = K1(t0) = O(1) > 0 is defined by

K1 =

9[E1(t0) +

2(1−a1a2)µ1|Ω|‖c0‖2L∞ min

12 ,

a1(1+a1a2)a2

(N1χ

21

4 +a2µ2N2χ

22

4a1µ1+1)

min

2(αN1+βN2) max 1

N1,a1µ1a2µ2N2

, (1−a1a2)µ1 min 12 ,

a1(1+a1a2)a2

e

]2 min

1N1, a1µ1

a2µ2N2

(4.5)

and the exponential decay rate ξ is defined by (1.6).

Proof. Applying Taylor’s formula to the function ψ(z) = z−N1 log z at z = N1, weobtain

n1 −N1 −N1 logn1

N1= ψ(n1)− ψ(N1) =

ψ′′(ζ)

2(n1 −N1)2 =

N1

2ζ2(n1 −N1)2

(4.6)

for some ζ > 0 between n1 and N1. Then a combination of (4.2) and (4.6) gives

2

9N1(n1 −N1)2 ≤ n1 −N1 −N1 log

n1

N1≤ 2

N1(n1 −N1)2, ∀t ≥ t0,

and hence

2

9N1

∫Ω

(n1 −N1)2 ≤∫

Ω

(n1 −N1 −N1 log

n1

N1

)≤ 2

N1

∫Ω

(n1 −N1)2, ∀t ≥ t0.

(4.7)Similarly, one gets for all t ≥ t0 that

2

9N2

∫Ω

(n1 −N2)2 ≤∫

Ω

(n2 −N2 −N2 log

n2

N2

)≤ 2

N2

∫Ω

(n2 −N2)2. (4.8)

On the other hand, Lemma 4.1 quickly gives rise to∫Ω

c2 ≤ |Ω|‖c(·, t)‖2L∞ ≤ |Ω|‖c0‖2L∞e−(αN1+βN2)(t−t0), ∀t ≥ t0. (4.9)

A substitution of (4.7), (4.8) and (4.9) into the definition of E1 in (3.1) gives

E1(t) ≤ 2

N1

∫Ω

(n1 −N1)2 +2a1µ1

a2µ2N2

∫Ω

(n2 −N2)2 +mce−(αN1+βN2)(t−t0)

≤ θF1 +mce−(αN1+βN2)(t−t0), ∀t ≥ t0,

(4.10)

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1931

where

θ = 2 max 1

N1,a1µ1

a2µ2N2, mc = |Ω|‖c0‖2L∞(

N1χ21

4+a2µ2N2χ

22

4a1µ1+ 1). (4.11)

Hence, from (4.10) and the dissipation estimate (3.2), we derive that

d

dtE1 +

τ

θE1 ≤

τmc

θe−(αN1+βN2)(t−t0), ∀t ≥ t0,

and then, solving this Gronwall differential inequality, we readily get

E1(t) ≤ E1(t0)e−τθ (t−t0) +

τmc

θe(αN1+βN2)t0e−

τθ t

∫ t

t0

e[ τθ−(αN1+βN2)]sds

≤[E1(t0) +

2τmc

minαN1 + βN2,τθ eθ

]e−

minαN1+βN2,τθ

2 (t−t0), ∀t ≥ t0,

(4.12)

where we have used the following algebraic calculations:

e(αN1+βN2)t0e−τθ t

∫ t

t0

e[ τθ−(αN1+βN2)]sds

=

(t− t0)e−

τθ (t−t0), if τ

θ = (αN1 + βN2)

1τθ−(αN1+βN2)

[e−(αN1+βN2)(t−t0) − e− τθ (t−t0)

], if τ

θ 6= (αN1 + βN2)

≤ (t− t0)e−minαN1+βN2,τθ (t−t0)

≤ 2

minαN1 + βN2,τθ e

e−minαN1+βN2,

τθ

2 (t−t0).

(4.13)

By the definition of E1 in (3.1) and the estimates (4.7), (4.8), we see that

E1(t) ≥ 2

9min

1

N1,a1µ1

a2µ2N2

[∫Ω

(n1 −N1)2 +

∫Ω

(n2 −N2)2]. (4.14)

Joining (4.14) and (4.12) and substituting the definitions of τ , θ and mc in (3.2) and(4.11), we finally arrive at (4.4) with K1 and ξ given by (4.5) and (1.6), respectively.

Thanks to the regularity provided by Lemma 2.3, we employ the well-knownGagliardo-Nirenberg interpolation inequality to pass the L2-convergence of n1 andn2 in (4.4) to their L∞-convergence.

Lemma 4.3. Let Ω ⊂ Rd be a bounded and smooth domain. Then the n1- andn2- solution components of any bounded solution of (1.1) decay exponentially to(N1, N2):

‖n1(·, t)−N1‖L∞ + ‖n2(·, t)−N2‖L∞ ≤ Ce−ξd+2 (t−t0), ∀t ≥ t0 (4.15)

for some C > 0 independent of t. Here, the exponential decay rate ξ is defined by(1.6).

1932 HAI-YANG JIN AND TIAN XIANG

Proof. Due to the L2-convergence of n1, n2 in (4.4) and the uniform W 1,∞-bounded-ness of n1, n2 in (2.6), the Gagliardo-Nirenberg inequality enables us to concludethat

‖n1(·, t)−N1‖L∞ + ‖n2(·, t)−N2‖L∞

≤ c1(‖n1(·, t)‖

dd+2

W 1,∞‖n1(·, t)−N1‖2d+2

L2 + ‖n2(·, t)‖dd+2

W 1,∞‖n2(·, t)−N2‖2d+2

L2

)≤ c2

(‖n1(·, t)−N1‖

2d+2

L2 + ‖n2(·, t)−N2‖2d+2

L2

)≤ c3e−

ξd+2 (t−t0), ∀t ≥ t0.

(4.16)

This is nothing but the exponential decaying estimate (4.15).

4.2.2. Convergence rate of u in Case I. With the information gained from Lemmas4.2 and 2.1, we are now able to derive the convergence rate of u in L∞-norm. Toaccomplish this goal, we again begin with its convergence rate in L2-norm.

Lemma 4.4. The u-solution component of (1.1) fulfills

‖u(·, t)‖2L2 ≤(‖u(·, t0)‖2L2 +

2K1K1

minλP , ξe

)e−

minλP ,ξ2 (t−t0), ∀t ≥ t0, (4.17)

where K1, K1, ξ and λP are respectively defined by (4.5), (4.20), (1.6) and (4.19).

Proof. Recalling that ∇ · u = 0 in Ω and u|∂Ω = 0, we multiply the fourth equationin (1.1) by u and integrate it over Ω to obtain

1

2

d

dt

∫Ω

|u|2 +

∫Ω

|∇u|2 =

∫Ω

(γn1 + δn2)∇φ · u+ κ

∫Ω

|u|2∇ · u

= γ

∫Ω

(n1 −N1)∇φ · u+ δ

∫Ω

(n2 −N2)∇φ · u,(4.18)

where we also used the fact∫

Ω∇φ · u = 0. Therefore, we apply the Poincare

inequality:

λP

∫Ω

|u|2 ≤∫

Ω

|∇u|2 (4.19)

for the Poincare constant λP to (4.18) deduce that

d

dt

∫Ω

|u|2 + 2λP

∫Ω

|u|2

≤ 2γ

∫Ω

|n1 −N1||∇φ · u|+ 2δ

∫Ω

|n2 −N2||∇φ · u|

≤ λP∫

Ω

|u|2 +2γ2‖∇φ‖2L∞

λP

∫Ω

|n1 −N1|2 +2δ2‖∇φ‖2L∞

λP

∫Ω

|n2 −N2|2.

As a result, for

K1 = 2 max γ2

λP,

δ2

λP‖∇φ‖2L∞ , (4.20)

it follows that

d

dt

∫Ω

|u|2 + λP

∫Ω

|u|2 ≤ K1

(∫Ω

|n1 −N1|2 +

∫Ω

|n2 −N2|2). (4.21)

Substituting (4.4) into (4.21), we derive that

d

dt

∫Ω

|u|2 + λP

∫Ω

|u|2 ≤ K1K1e−ξ(t−t0), ∀t ≥ t0.

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1933

Solving this ODI and performing similar computations to (4.13), we readily obtain

‖u(·, t)‖2L2 ≤ ‖u(·, t0)‖2L2e−λP (t−t0) +K1K1eξt0e−λP t

∫ t

t0

e(λP−ξ)sds

≤(‖u(·, t0)‖2L2 +

2K1K1

minλP , ξe

)e−

minλP ,ξ2 (t−t0), ∀t ≥ t0,

which is precisely the desired exponential decay estimate (4.17).

Lemma 4.5. Let Ω ⊂ Rd be a bounded and smooth domain. Then, for any ε ∈(0, 1), there exists a constant

K1 ≥ O(1)(1 + (1− a1a2)−εd+2 ) (4.22)

such that‖u(·, t)‖L∞ ≤ K1e

− ε2(d+2)

minλP ,ξ(t−t0), ∀t ≥ t0, (4.23)

where the exponent rate ξ is defined by (1.6).

Proof. Thanks to the L2-convergence of u in (4.17) and the uniform-in-time W 1,p-boundedness of u in (2.1), the Gagliardo-Nirenberg inequality allows us to infer

‖u(·, t)‖L∞ ≤ c1‖u(·, t)‖dp

dp+2p−2d

W 1,p ‖u(·, t)‖2p−2d

dp+2p−2d

L2 ≤ c2e−(p−d)

dp+2p−2d minλP2 , ξ2(t−t0),

∀t ≥ t0,which implies (4.23) upon choosing p = [d+2(1−ε)]d/[(d+2)(1−ε)](> d). Noticing

that ξ = O(1)(1 − a1a2) by (1.6), then the lower bound for K1 in (4.22) followsfrom (4.17).

4.3. Algebraic convergence rates in Case II: a1 = 1 > a2. Here, we will showthat the solution components (n1, n2, u) converge at least algebraically to (0, 1, 0).

4.3.1. Convergence rates of n1 and n2 in Case II. Again, we start with the deriva-tion of the L1- and L2-convergence rates of n1 and n2.

Lemma 4.6. There exists t1 ≥ max1, t0 such that

‖n1(·, t)‖L1 + ‖n2(·, t)− 1‖2L2 ≤K2

t+ t1, ∀t ≥ t1, (4.24)

where

K2 =max

2t1E2(t1),

k1

(k1+√k21+2aσ

)σ

min1, 2µ1

9a2µ2

≥ O(1)(1 +1

σ) = O(1)(1 + (1− a2)−1)

(4.25)

with σ, k1, k2, k3 and a defined in (3.5), (4.29), (4.34) and (4.31), respectively.

Proof. Our proof makes use of the explicit Lyapunov functional provided by Lemma3.2. To proceed, we first apply the Holder inequality to find∫

Ω

n1 ≤ |Ω|12

(∫Ω

n21

) 12

. (4.26)

Next, since ‖n2(·, t)− 1‖L∞ → 0 as t→∞ and

limz→1

z − 1− log z

z − 1= 0,

1934 HAI-YANG JIN AND TIAN XIANG

we can take t0 > 0 such that |n2(x, t)− 1− log n2(x, t)| ≤ |n2(x, t)− 1| for all x ∈ Ωand t ≥ t0. Accordingly, we have∫

Ω

(n2 − 1− log n2) ≤∫

Ω

|n2 − 1| ≤ |Ω| 12(∫

Ω

(n2 − 1)2

) 12

, ∀t ≥ t0. (4.27)

In this case, (N1, N2) = (0, 1), so the exponential decay of c in Lemma 4.1 warrantsthat ∫

Ω

c2 ≤ |Ω|‖c‖2L∞ ≤ |Ω|‖c0‖2L∞e−β(t−t0), ∀t ≥ t0. (4.28)

From the definitions of E2 and F2 in Lemma 3.2, upon a combination of (4.26),

(4.27) and (4.28) and the fact that√A +√B ≤

√2(A+B) for A,B ≥ 0, we find

two constants

k1 = max1, µ1

a2µ2(2|Ω|) 1

2 , k2 =µ1χ

22

8a2µ2|Ω|‖c0‖2L∞ (4.29)

such that

E2(t) ≤ k1F12

2 (t) + k2e−β(t−t0), ∀t ≥ maxt0, t0,

which further gives us

E22(t) ≤ 2k2

1F2(t) + 2k22e−2β(t−t0), ∀t ≥ maxt0, t0.

A substitution of this into the dissipation inequality (3.5) entails

d

dtE2(t) +

σ

2k21

E22(t) ≤ k2

2σ

k21

e−2β(t−t0), t ≥ maxt0, t0. (4.30)

Now, to illustrate the algebraic decay (4.24), we first take t1 = maxt0, t0, 1, β

−1

so that

a =k2

2σ

k21

max

(t+ t1)2e−2β(t−t0) : t ≥ t0

=4k2

2t21σ

k21

e−2β(t1−t0); (4.31)

and then, for any

b ≥k1

(k1 +

√k2

1 + 2aσ)

σ, (4.32)

we put

y(t) =b

t+ t1, t ≥ 0. (4.33)

We use straightforward calculations from (4.33) and use (4.31) to see that

y′(t) +σ

2k21

y2(t)− k22σ

k21

e−2β(t−t0)

= (t+ t1)−2[ σ

2k21

b2 − b− k22σ

k21

(t+ t1)2e−2β(t−t0)]

≥ (t+ t1)−2(σ

2k21

b2 − b− a) ≥ 0, ∀t ≥ t1.

This, upon a clear choice of b in (4.32), an ODE comparison argument to (4.30)shows

E2(t) ≤max

2t1E2(t1),

k1

(k1+√k21+2aσ

)σ

t+ t1

=:k3

t+ t1, ∀t ≥ t1. (4.34)

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1935

Then since t1 ≥ t0, we infer from (4.8) and the definition of E2(t) in Lemma 3.2that

min1, 2µ1

9a2µ2(‖n1(·, t)‖L1 + ‖n2(·, t)− 1‖2L2

)≤ k3

t+ t1, ∀t ≥ t1.

This, upon a substitution of the respective definitions of k1, k2, k3 and a in (4.29),(4.34) and (4.31), proves our desired algebraic decay estimate (4.24).

Lemma 4.7. There exist two constants K3 and K4 independent of t fulfilling

K3 ≥ O(1)(

1 + (1− a2)−1d+1

), K4 ≥ O(1)

(1 + (1− a2)−

1d+2

)such that

‖n1(·, t)‖L∞ ≤K3

(t+ t1)1d+1

, ∀t ≥ t1 (4.35)

as well as

‖n2(·, t)− 1‖L∞ ≤K4

(t+ t1)1d+2

, ∀t ≥ t1. (4.36)

Proof. Equipped with the uniform W 1,∞-bounds of n1, n2 in Lemma 2.3, as before,by means of the Gagliardo-Nirenberg inequality, we readily infer, for all t ≥ t1,

‖n1(·, t)‖L∞ ≤ c1‖n1(·, t)‖dd+1

W 1,∞‖n1(·, t)‖1d+1

L1 ≤ c2‖n1(·, t)‖1d+1

L1 (4.37)

and

‖n2(·, t)− 1‖L∞ ≤ c3‖n2(·, t)‖dd+2

W 1,∞‖n2(·, t)− 1‖2d+2

L2 ≤ c4‖n2(·, t)− 1‖2d+2

L2 .

These along with the L1-and L2-convergence of n1, n2 in (4.24) and the bound forK2 in (4.25) yield immediately (4.35) and (4.36).

4.3.2. Convergence rate of u in Case II.

Lemma 4.8. The u-solution component of (1.1) fulfills

‖u(·, t)‖2L2 ≤K5

t+ t1, ∀t ≥ t1 (4.38)

for some positive constant K5 independent of t satisfying K5 ≥ O(1)(1+(1−a2)−1).

Proof. Thanks to the L1-and L2-convergence of n1, n2 in (4.24), we can easily adaptthe proof of Lemma 4.4 here. Indeed, from (4.18), the Poincare inequality (4.19)and the boundedness of u, we infer that

1

2

d

dt

∫Ω

|u|2+λP2

∫Ω

|u|2 ≤ γ‖∇φ‖L∞‖u‖L∞∫

Ω

n1+δ2‖∇φ‖2L∞

2λP

∫Ω

|n2−1|2. (4.39)

Thus, for

K5 = 2 maxγ‖u‖L∞(Ω×(0,∞)),δ2‖∇φ‖L∞

2λP‖∇φ‖L∞ <∞,

from (4.39) and (4.24), we obtain an ODI for ‖u‖2L2 as follows:

d

dt

∫Ω

|u|2 + λP

∫Ω

|u|2 ≤ K5

(∫Ω

n1 +

∫Ω

|n2 − 1|2)≤ K2K5

t+ t1, ∀t ≥ t1.

1936 HAI-YANG JIN AND TIAN XIANG

Solving this ODI, we end up with

‖u(·, t)‖2L2 ≤ ‖u(·, t1)‖2L2e−λP (t−t1) +K2K5e−λP t

∫ t

t1

eλP s

s+ t1ds

≤ ‖u(·, t1)‖2L2e−λP (t−t1) +K2K5K5

t+ t1

≤(‖u(·, t1)‖2L2e−λP t1K5 +K2K5K5

)(t+ t1)−1, ∀t ≥ t1,

from which (4.38) follows. Here, we have used the following facts

K5 = max(t+ t1)e−λP t : t ≥ t1 <∞and

K5 = max

(t+ t1)e−λP t∫ t

t1

eλP s

s+ t1ds : t ≥ t1

<∞.

The latter is due to

limt→∞

[(t+ t1)e−λP t

∫ t

t1

eλP s

s+ t1ds]

= limt→∞

∫ tt1eλP s

s+t1ds

(t+ t1)−1eλP t=

1

λp<∞.

With the L2-convergence of u in Lemma 4.8 at hand, the same argument as doneto Lemma 4.5 yields the following L∞-convergence of u.

Lemma 4.9. For any ε ∈ (0, 1), there exists a positive constant

K6 ≥ O(1)(

1 + (1− a2)−εd+2

)such that

‖u(·, t)‖L∞ ≤K6

(t+ t1)εd+2

, ∀t ≥ t1.

4.4. Algebraic convergence rates in Case (II′): a2 = 1 > a1. In this case, weshall show that (n1, n2, u) converges at least algebraically to (1, 0, 0). ComparingLemmas 3.2 and 3.3 and the n1-and n2-equations in (1.1), we see that this subsectionis fully parallel to Section 4.3, and so we simply write down their respective finaloutcomes.

Lemma 4.10. There exist t2 ≥ max1, t0 and positive constants K7 and K8

fulfilling

K7 ≥ O(1)(

1 + (1− a1)−1d+2

), K8 ≥ O(1)

(1 + (1− a1)−

1d+1

)such that

‖n1(·, t)− 1‖L∞ ≤K7

(t+ t2)1d+2

, ∀t ≥ t2

and

‖n2(·, t)‖L∞ ≤K8

(t+ t2)1d+1

, ∀t ≥ t2.

Lemma 4.11. For any ε ∈ (0, 1), there exists Cε ≥ O(1)(1 + (1 − a1)−εd+2 ) such

that

‖u(·, t)‖L∞ ≤Cε

(t+ t2)εd+2

, ∀t ≥ t2.

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1937

4.5. Exponential convergence rates in Case III: a1 > 1 > a2. In this section,we use the crucial fact that a1 > 1 to show that the bounded solution components(n1, n2, u) converge not only algebraically (as shown in Subsection 4.3) but alsoexponentially to (0, 1, 0) and we shall compute out explicitly their respective ratesof convergence. Armed with the knowledge from previous subsections, this sectioncan be short and thus we include their stabilization results in a single lemma.

Lemma 4.12. There exist t3 ≥ max1, t0 and C = O(1) > 0 such that the n1-solution component of (1.1) fulfills

‖n1(·, t)‖L∞ ≤ Ce−(a1−1)µ12(d+1)

(t−t3), ∀t ≥ t3; (4.40)

for some positive constant

K9 ≥ O(1)(1 + (a1 − 1)−1d+2 + (1− a2)−

1d+2 ), (4.41)

the n2-solution component satisfies

‖n2(·, t)− 1‖L∞ ≤ K9e− 1

4(d+2)min a2µ2σµ1

, 2β, (a1−1)µ1(t−t3), ∀t ≥ t3; (4.42)

finally, for any ε ∈ (0, 1), there exists

K10 ≥ O(1)(1 + (a1 − 1)−2εd+2 + (1− a2)−

2εd+2 ) (4.43)

such that the u-solution component satisfies

‖u(·, t)‖L∞ ≤ K10e− ε

8(d+2)min a2µ2σµ1

, 2β, (a1−1)µ1, 4λP (t−t3), ∀t ≥ t3. (4.44)

Here, σ is defined by (3.5) in Lemma 3.2.

Proof. A simple integration of the first equation in (1.1) shows

d

dt

∫Ω

n1 = − (a1 − 1)µ1

2

∫Ω

n1 − µ1

∫Ω

[a1n2 − 1 + n1 −

(a1 − 1)

2

]n1. (4.45)

The uniform convergence n1 → 0 and n2 → 1 (c.f. Subsection 4.3 or (4.1)) alongwith the fact a1 > 1 allows us to find t3 ≥ max1, t0 such that

a1n2 − 1 + n1 ≥(a1 − 1)

2on Ω× [t3,∞). (4.46)

Thus, when t > t3, based on (4.46) and (4.45), we derive a Gronwall inequality for‖n1‖L1 :

d

dt

∫Ω

n1 +(a1 − 1)µ1

2

∫Ω

n1 ≤ 0, ∀t ≥ t3,

which trivially yields the exponential decay of ‖n1‖L1 :

‖n1(·, t)‖L1 ≤ ‖n1(·, t3)‖L1e−(a1−1)µ1

2 (t−t3), ∀t ≥ t3. (4.47)

Then, by the uniformW 1,∞-bounds of n1 in Lemma 2.3 and the Gagliardo-Nirenberginequality, similar to (4.37), we readily obtain the exponential decay estimate (4.40).

Since (N1, N2) = (0, 1), the exponential convergence of c in Lemma 4.1 entails∫Ω

c2 ≤ |Ω|‖c(·, t)‖2L∞ ≤ |Ω|‖c0‖2L∞e−β(t−t0), ∀t ≥ t3. (4.48)

Using the essentially same argument leading to (4.8), we get

2

9

∫Ω

(n2 − 1)2 ≤∫

Ω

(n2 − 1− log n2) ≤ 2

∫Ω

(n2 − 1)2, ∀t ≥ t3. (4.49)

1938 HAI-YANG JIN AND TIAN XIANG

By (4.47), (4.48), (4.49) and the definitions of E2 and F2 in Lemma 3.2, we bound

E2 =

∫Ω

n1 +µ1

a2µ2

∫Ω

(n2 − 1− log n2) +µ1χ

22

8a2µ2

∫Ω

c2

≤ 2µ1

a2µ2

∫Ω

(n2 − 1)2 + ‖n1(·, t3)‖L1e−(a1−1)µ1

2 (t−t3) +µ1χ

22

8a2µ2|Ω|‖c0‖2L∞e−β(t−t0)

≤ 2µ1

a2µ2F2+

[‖n1(·, t3)‖L1 +

µ1χ22

8a2µ2|Ω|‖c0‖2L∞

]e−

12 min2β, (a1−1)µ1(t−t3)

=:2µ1

a2µ2F2 + K9e

− 12 min2β, (a1−1)µ1(t−t3), ∀t ≥ t3.

(4.50)

This along with the dissipation estimate (3.5) enables us to deduce that

d

dtE2(t) +

a2µ2σ

2µ1E2(t) ≤ a2µ2σ

2µ1K9e

− 12 min2β, (a1−1)µ1(t−t3), ∀t ≥ t3,

and then, solving this Gronwall differential inequality, we infer, for t ≥ t3, that

E2(t) ≤ E2(t3)e−a2µ2σ2µ1

(t−t3)

+a2µ2σ

2µ1K9e

12 min2β, (a1−1)µ1t3e−

a2µ2σ2µ1

t∫ t

t3

e[a2µ2σ2µ1

− 12 min2β, (a1−1)µ1]sds

≤[E2(t3) +

2K9a2µ2σµ1

mina2µ2σµ1

, 2β, (a1 − 1)µ1e

]e−

14 min a2µ2σµ1

, 2β, (a1−1)µ1(t−t3),

(4.51)

where we have used the following algebraic computations similar to (4.13):

e12 min2β, (a1−1)µ1t3e−

a2µ2σ2µ1

t∫ t

t3

e[a2µ2σ2µ1

− 12 min2β, (a1−1)µ1]sds

=

(t− t3)e−

a2µ2σ2µ1

(t−t3), if a2µ2σ2µ1

= 12 min2β, (a1 − 1)µ1[

e−12

min2β,(a1−1)µ1(t−t3)−e− a2µ2σ

2µ1(t−t3)

]a2µ2σ2µ1

− 12 min2β,(a1−1)µ1

, if a2µ2σ2µ1

6= 12 min2β, (a1 − 1)µ1

≤ (t− t3)e−12 min a2µ2σµ1

, 2β, (a1−1)µ1(t−t3)

≤ 4

mina2µ2σµ1

, 2β, (a1 − 1)µ1ee−

14 min a2µ2σµ1

, 2β, (a1−1)µ1(t−t3).

From the definition of E2 in Lemma 3.2 or alternatively (4.50) and the estimates(4.49) and (4.51), we conclude finally that

‖n2(·, t)− 1‖2L2 ≤ K9e− 1

4 min a2µ2σµ1, 2β, (a1−1)µ1(t−t3), ∀t ≥ t3, (4.52)

where

K9 =9a2µ2

2µ1

[E2(t3) +

4K9

mina2µ2σµ1

, 2β, (a1 − 1)µ1e

]. (4.53)

Then, as we did in (4.16) of Lemma 4.3, we use the uniform W 1,∞-boundedness ofn2 in (2.6) and the Gagliardo-Nirenberg inequality to improve the L2-convergenceof n2 in (4.52) to the L∞-convergence of n2 in (4.42). The lower bound for K9 in(4.41) follows from (4.53) and the fact that σ = O(1)(1− a2) by (3.5).

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1939

With the convergence rates of n1 and n2, we use the sprit of Lemma 4.4 to showfirst the L2-convergence of u. To start off, we utilize the L1- and L2-convergence ofn1 and n2 in (4.47) and (4.52) to bound the counterpart of (4.21) as follows:

d

dt

∫Ω

|u|2 + λP

∫Ω

|u|2

≤ K1

(∫Ω

n21 +

∫Ω

(n2 − 1)2)

≤ K1‖n1‖L∞(Ω×(0,∞))

∫Ω

n1 + K1

∫Ω

(n2 − 1)2

≤ K10e− (a1−1)µ1

2 (t−t3) + K10e− 1

4 min a2µ2σµ1, 2β, (a1−1)µ1(t−t3)

≤ 2K10e− 1

4 min a2µ2σµ1, 2β, (a1−1)µ1(t−t3) =: 2K10e

−ν(t−t3),

(4.54)

where, upon notice of the fact from (3.5) that ν = O(1) min(a1− 1), (1− a2) andthe use of (4.20) and (4.53),

K10 = maxK1‖n1‖L∞(Ω×(0,∞)), K1K9

= O(1)

(1 + (min(a1 − 1), (1− a2))−1

).

(4.55)

Solving the ordinary differential inequality (4.54) and performing similar computa-tions to the ones right after (4.51), we conclude from the fact that ν = O(1) min(a1−1), (1− a2) by (3.5) and (4.55) that for all t ≥ t3

‖u(·, t)‖2L2 ≤ ‖u(·, t3)‖2L2e−λP (t−t3) + 2K10eνt3e−λP t

∫ t

t3

e(λP−ν)sds

≤(‖u(·, t3)‖2L2 +

4K10

minλP , νe

)e−

minλP ,ν2 (t−t3)

≤ O(1)(

1 + (a1 − 1)−2 + (1− a2)−2)e−

minλP ,ν2 (t−t3).

(4.56)

With this L2-convergence and the uniform-in-time W 1,p-boundedness of u in (2.1),the Gagliardo-Nirenberg interpolation inequality allows us to deduce, as in Lemma4.5, the L∞-convergence (4.44). The lower bound for K10 in (4.43) can be seenfrom (4.56).

4.6. Exponential convergence rates in Case (III′): a2 > 1 > a1. Using theLyapunov functional provided by Lemma 3.3 and the arguments parallel to Subsec-tion 4.5, we find that any bounded solution (n1, n2, u) will converge exponentiallyto (1, 0, 0). We here shall omit the details and just write down their respective finalconvergence results.

Lemma 4.13. There exist t4 ≥ max1, t0 and a positive constant

K11 ≥ O(1)(1 + (1− a1)−1d+2 + (a2 − 1)−

1d+2 ),

such that the n1-solution component of (1.1) fulfills

‖n1(·, t)− 1‖L∞ ≤ K11e− 1

4(d+2)minρ, 2α, (a2−1)µ2(t−t4), ∀t ≥ t4;

the n2-solution component satisfies, for some C = O(1) > 0,

‖n2(·, t)‖L∞ ≤ Ce−(a2−1)µ22(d+1)

(t−t4), ∀t ≥ t4;

1940 HAI-YANG JIN AND TIAN XIANG

finally, for any ε ∈ (0, 1), there exists

K12 ≥ O(1)(1 + (a1 − 1)−2εd+2 + (1− a2)−

2εd+2 )

such that the u-solution component satisfies

‖u(·, t)‖L∞ ≤ K10e− ε

8(d+2)minρ, 2α, (a2−1)µ2, 4λP (t−t4), ∀t ≥ t4.

Here, ρ is defined by (3.6) in Lemma 3.3.

Proof of Theorem 1.1. Notice that ti ≥ maxt0, 1 > 1(i = 1, 2, 3, 4); the respectivedecay estimates and their decay rates asserted in Theorem 1.1 follow from somelemmas in this section with perhaps some large constants mi. More specifically, theexponential decay estimate (1.9) follows from Lemmas 4.3 and 4.5; the algebraicaldecay estimate (1.10) follows from Lemmas 4.7 and 4.9; the exponential decayestimate (1.11) follows from Lemma 4.12 and the decay rate ν in (1.7) follows froma substitution of the definition of σ in (3.5); the algebraical decay estimate (1.12)follows from Lemmas 4.10 and 4.11; the exponential decay estimate (1.13) followsfrom Lemma 4.13 and the decay rate µ in (1.8) follows from a substitution of thedefinition of ρ in (3.6); and, finally, the exponential decay estimate (1.14) followsfrom Lemma 4.1.

Acknowledgments. The authors are very grateful to the referee for his/her pos-itive and constructive comments/suggestions, which helped us greatly improve theexposition of the paper. The research of H.Y. Jin was supported by NSF of China(No. 11501218) and the Fundamental Research Funds for the Central Universities(No. 2017MS107), and the research of T. Xiang was supported by the NSF ofChina (No. 11601516 and 11571363) and the Research Fund of Renmin Universityof China (No. 2018030199).

REFERENCES

[1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with

competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553–583.

[2] T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-specieschemotaxis model, IMA J. Appl. Math., 81 (2016), 860–876.

[3] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under small-ness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891–1904.

[4] X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of clas-

sical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics,

arXiv:1703.01794, Math Meth Appl Sci., 41 (2018), 3138–3154.

[5] E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinearreaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1–16.

[6] P. De Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalizedVolterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648–663.

[7] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathemat-

ics, 840, Springer-Verlag, Berlin-New York, 1981.

[8] M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in atwo-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J.

Differential Equations, 263 (2017), 470–490.

[9] M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, Proceed-

ings of EQUADIFF 2017 Conference, (2017), 11–20.

[10] H. Jin and Z. Wang, Global stability of prey-taxis systems, J. Differential Equations, 262(2017), 1257–1290.

CONVERGENCE RATES IN THE CHEMOTAXIS-FLUID SYSTEM 1941

[11] Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193–221.

[12] O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of

Parabolic Type, AMS, Providence, RI, 1968.

[13] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math.

Models Methods Appl. Sci., 26 (2016), 2071–2109.

[14] J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a high-

dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst., 37 (2017),6099–6121.

[15] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equaations,

131 (1996), 79–131.

[16] M. Mimura, S. I. Ei and Q. Fang, Effect of domain-shape on coexistence problems in a

competition-diffusion system, J. Math. Biol., 29 (1991), 219–237.

[17] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition

model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B , 22 (2017), 2301–2319.

[18] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016),

2650–2669.

[19] M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with

non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592–1617.

[20] M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion,SIAM J. Math. Anal., 46 (2014), 3761–3781.

[21] M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear de-generate parabolic equations, J. Differential Equations, 103 (1993), 146–178.

[22] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis

model, J. Math. Biol., 68 (2014), 1607–1626.

[23] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math.

Anal. Appl., 381 (2011), 521–529.

[24] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a

three-dimensional chemotaxis system with consumption of chemoattractant, J. DifferentialEquations, 252 (2012), 2520–2543.

[25] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in athree-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555–2573.

[26] Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis

model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229–4250.

[27] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional

Keller–Segel-Navier–Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp.

[28] (10.1073/pnas.0406724102) I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O.

Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines,Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 2277–2282.

[29] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel

model, J. Differential Equations, 248 (2010), 2889–2905.

[30] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis systemwith logistic source, Comm. Partial Differential Equations, 35 (2010), 1516–1537.

[31] M. Winkler, Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling

cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319–351.

[32] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel

system, J. Math. Pures Appl., 100 (2013), 748–767.

[33] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Ra-tion. Mech. Anal., 211 (2014), 455–487.

[34] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxissystem with strong logistic dampening, J. Differential Equations, 257 (2014), 1056–1077.

1942 HAI-YANG JIN AND TIAN XIANG

[35] M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier–Stokessystem?, Trans. Amer. Math. Soc., 369 (2017), 3067–3125.

[36] T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel

chemotaxis system? J. Math. Anal. Appl., 459 (2018), 1172–1200.

[37] Q. Zhang and Y. Li, Convergence rates of solutions for a two-dimensional chemotaxis-Navier–

Stokes system, Discrete Contin. Dyn. Syst. Ser. B , 20 (2015), 2751–2759.

Received January 2018; revised April 2018.

E-mail address: mahyjin@scut.edu.cn

E-mail address: txiang@ruc.edu.cn