Stabilization in a two dimensional two-species aerotaxis-Navier–Stokes system

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Abstract

This paper deals with the two-species aerotaxis-Navier–Stokes equations with Lotka–Volterra competitive kinetics in a two dimensional domain ΩR2. We consider the general chemotactic sensitivity functions and oxygen(chemical) consumption rate functions. We obtain the stabilization of the solution to the system. For the specific conditions on the chemotactic sensitivity and initial data, we obtain the global stabilization when the competition between two species is stronger than that of each own species.

Introduction

In this paper, we study the interaction between reaction and chemotaxis in the following system of the two species aerotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics: tn1+(u)n1Δn1=(χ1(c)n1c)+μ1n1(1n1a1n2),tn2+(u)n2Δn2=(χ2(c)n2c)+μ2n2(1a2n1n2),tc+(u)cΔc=(κ1(c)n1+κ2(c)n2),tu+(u)uΔu+P=(γn1+δn2)Φ,u=0, in (x,t)Ω×(0,)νn1=νn2=νc=0,u=0, on Ω×(0,),n1(x,0)=n1,0(x),n2(x,0)=n2,0,c(x,0)=c0,u(x,0)=u0(x), in Ω,where Ω is a smooth bounded domain in R2 and ν denotes normal derivative; ai0(i=1,2) denotes competitive degradation rates of ni, which denotes each distinct bacteria’s density. Nonnegative chemotactic sensitivity function for each species ni is denoted by χi(c). Also we denote c for oxygen concentration and κi(c) for nonnegative oxygen consumption rate for each species ni. We note that γ and δ are positive constants. We denote u, P and Φ for fluid velocity, scalar pressure and gravitational potential, respectively. Throughout this paper, we assume the constants and the functions satisfy the following two conditions: ΦC2(Ω),γ,δ>0, and μi>0,ai0 for i=1,2,and κi,χiC2([0,)),κi(0)=0,χi0, and κi>0 for i=1,2.The chemotactic dynamics of bacteria density and chemical concentration was first initiated by Keller–Segel [1], [2] and Patlak [3]. Many interesting features of solutions to the system – related with blow-ups and global well-posedness issue – have been received many attentions by the authors (see [4], [5], [6], [7], [8] and references therein).

In [9], Winkler considered the global existence and the stabilization of solutions to the singular sensitivity and signal absorption model in a planar domain (for the high dimensional study, please see [10]): tn=Δnncc,tc=Δccn.On the other hand, consideration of the logistic source of bacteria density or Lotka–Volterra competition has been believed to tend to prevent the blow-up of solutions to the chemotaxis equations. Global well-posedness and stabilization have been studied in many literatures (see [11], [12], [13], [14] and references therein).

Especially, in [15], L. Wang et al. considered the following competitive signal absorbing model in n dimensional bounded domain tn1d1Δn1=(χ1(c)n1c)+μ1n1(1n1a1n2),tn2d2Δn2=(χ2(c)n2c)+μ2n2(1a2n1n2),tcd3Δc=(α1n1+α2n2)c.They obtained global existence and stabilization (only when at least one of ai is less than or equal to 1) in [15] (see also [16], [17]).

In [18], the following model has been considered for describing the dynamics of the Bacillus Subtilis in a water droplet tn+(u)nΔn=(χ(c)nc),tc+(u)cΔc=κ(c)n, in (x,t)Ω×(0,)tu+(u)uΔu+P=γnΦ,u=0.When Ω is a two dimensional domain, the global existence of bounded solution and the stabilization have been obtained with some assumptions on χ(c) and κ(c) (see [19], [20], [21] and references therein).

In the case of single species (n20 in (1.1)) with logistic term, Tao and Winkler [22] obtained global well-posedness and stabilization of solutions to (1.1) in a two-dimensional domain and Lankeit [23] obtained eventual smoothness and asymptotic behaviour of solutions to (1.1) in three-dimensional domain. Also two-species model has been studied mathematically in many literatures (see [24], [25], [26], [27] and references therein).

For two-species aerotaxis-Navier–Stokes equations in a two-dimensional domain, Hirata, Kurima, Mizukami, and Yokota [28] obtained global-in-time existence and stabilization of bounded solution to (1.1) in the case that χi(c)χi and κi(c)βic for i=1,2 and positive constants βi. Cao, Kurima and Mizukami [29], [30] have obtained eventual smoothness and asymptotic behaviour for solution to (1.1) in a three-dimensional domain with the same conditions of the coefficients as in [28].

In this paper, we intend to consider more generalized coefficient functions χi(c) and κi(c) satisfying (1.2). We note that the microbial competition between B. Subtilis and other species has been discovered in many biological situations (for example, we refer [31] for competition between B. Subtilis and S. Aureus).

Throughout this paper, we assume that initial data satisfies n1,0C0(Ω¯),n1,00 in Ω¯, and n1,00,n2,0C0(Ω¯),n2,00 in Ω¯, and n2,00,c0W1,(Ω),c00 in Ω¯, and u0D(Aγ)for some γ(12,1),where A denotes the realization of the Stokes operator in the solenoidal subspace Lσ2(Ω) of L2(Ω), with domain D(A)=W2,2(Ω)W01,2(Ω)Lσ2(Ω).

We state the global-in-time existence of smooth solution to (1.1) without the proof. The proof of the following Theorem is rather standard and direct consequence of the very slight modifications of the proofs in [28, theorem 1.1].

Theorem 1

Let ΩR2 be a bounded domain with smooth boundary. Assume that Φ, κi, and χi satisfy (1.2)(1.3) and also that n1,0, n2,0, c0, and u0 satisfy (1.5). Then there exist functions n1,n2C0(Ω¯×(0,))C2,1(Ω¯×(0,)),cC0(Ω¯×(0,))C2,1(Ω¯×(0,)),uC0(Ω¯×(0,))C2,1(Ω¯×(0,)),andPC1,0(Ω¯×(0,)),which solve (1.1) classically in Ω×(0,). Moreover the solution is bounded in the sense that there exists C>0 satisfying n1(,t)L(Ω)+n2(,t)L(Ω)+c(,t)L(Ω)+u(,t)L(Ω)C for all t0.

In this paper, we are interested in the “strong” competition case with a1,a21. Even if we consider the Lotka–Volterra model with diffusion and strong competition, without chemotaxis and fluids(“Lotka–Volterra strong competition with diffusion”), then still the global stabilization problem of solutions is widely open. It is known that the equilibria (0,1) and (1,0) are locally stable. Kishimoto and Weinberger [32] showed that there is no positive constant stable equilibrium if Ω is convex. For more references, see [33] and references therein.

For the global stabilization of the “strong” competition of the solution to (1.1), the main difficulty lies in the prevention of the comparison of the solutions due to the nonlinearity of chemotactic sensitivity χi(c). We will show that if chemotactic sensitivities χ1 and χ2 are identical and the initial data satisfies some pointwise comparison between densities of two species, then we can have comparison of solutions to (1.1) global-in-time. Hence we can prove the following stabilization under some restrictions of initial data and coefficients, which is our main result.

Theorem 2

Let the assumptions of Theorem 1 hold. We also assume that χ1(c)χ2(c)χ(c). If μ1, μ2, a1, a2, n1,0(x) and n2,0(x) satisfy 1μ2μ1<a1, 1a2, and (a1μ1μ2)n2,0(x)>(a2μ2μ1)n1,0(x) for all xΩ¯, then the solution to (1.1) satisfies (n1(,t),n2(,t),c(,t),u(,t))(0,1,0,0) as t.

Our last result is the stabilization of solution to (1.1) for the “not strong” competition case.

Theorem 3

Let the assumptions of Theorem 1 hold. Then the solution to (1.1) satisfies the followings:

  • (i)

    Assume that a1,a2[0,1). Then (n1(,t),n2(,t),c(,t),u(,t))(1a11a1a2,1a21a1a2,0,0) in L(Ω) as t.

  • (ii)

    Assume that 0a1<1a2. Then (n1(,t),n2(,t),c(,t),u(,t))(1,0,0,0) in L(Ω) as t.

Remark 1

In Theorem 3(i), we extended the results in [28, Theorem 1.2]. In (ii) of Theorem 3, we extended the stabilization of the solution to (1.1) when one of the degradation rate ai is zero. If both degradation rates ai are greater than 1(“strong” competition), then Theorem 2 is the first result for the two species aerotaxis-fluid model, but the stabilization of the solution in a general setting is left open.

The rest of this paper consists as follows: In Section 2, we provide the proof of Theorem 2 by proving two comparison lemmas and also obtain ordinary differential inequalities. As a consequence of the ordinary differential inequalities, we prove Theorem 2 and also Theorem 3.

Section snippets

Stabilization: Proofs of Theorems 2 and 3

In this section, we prove Theorem 2, Theorem 3. Before presenting the stabilization properties, let us recall two basic lemmas. The proofs of Lemmas 4–5 are parallel to those in many literatures (see, e.g., [11, Lemma 3.14, 3.17]). We skip the proofs and present the statements.

Lemma 4

Let (n1,n2,c,u,P) be a unique global classical solution to (1.1) obtained in Theorem 1. Then, there exist C1>0 and θ1>0 satisfying niCθ1,θ12(Ω¯×[t,t+1])C1 for all t1 and i=1,2.

Lemma 5

Let fC0(Ω¯×[0,)) satisfy fCθ,θ2(Ω¯×[

Acknowledgements

The authors thank anonymous referee for many valuable remarks. J. Kim and J. Lee’s work is supported by SSTF-BA1701-05 (Samsung Science & Technology Foundation). E. Jeong’s work is partially supported by Chung-Ang University Excellent Student Scholarship .

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