Stabilization in a two dimensional two-species aerotaxis-Navier–Stokes system
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: where is a smooth bounded domain in and denotes normal derivative; denotes competitive degradation rates of , which denotes each distinct bacteria’s density. Nonnegative chemotactic sensitivity function for each species is denoted by . Also we denote for oxygen concentration and for nonnegative oxygen consumption rate for each species . We note that and are positive constants. We denote , and for fluid velocity, scalar pressure and gravitational potential, respectively. Throughout this paper, we assume the constants and the functions satisfy the following two conditions: and 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]): 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 dimensional bounded domain They obtained global existence and stabilization (only when at least one of 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 When is a two dimensional domain, the global existence of bounded solution and the stabilization have been obtained with some assumptions on and (see [19], [20], [21] and references therein).
In the case of single species ( 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 and for and positive constants . 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 and 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 where denotes the realization of the Stokes operator in the solenoidal subspace of , with domain .
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 be a bounded domain with smooth boundary. Assume that , , and satisfy (1.2)–(1.3) and also that , , , and satisfy (1.5). Then there exist functions which solve (1.1) classically in . Moreover the solution is bounded in the sense that there exists satisfying
In this paper, we are interested in the “strong” competition case with . 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 and are locally stable. Kishimoto and Weinberger [32] showed that there is no positive constant stable equilibrium if 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 . We will show that if chemotactic sensitivities and 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 . If , , , , and satisfy , , and for all , then the solution to (1.1) satisfies
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: Assume that . Then Assume that . Then
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 is zero. If both degradation rates 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 be a unique global classical solution to (1.1) obtained in Theorem 1. Then, there exist and satisfying
Lemma 5 Let satisfy
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 .
References (33)
- et al.
Initiation of slide mold aggregation viewd as an instability
J. Theoret. Biol.
(1970) - et al.
Model for chemotaxis
J. Theoret. Biol.
(1971) Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model
J. Differential Equations
(2010)Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption
J. Differential Equations
(2018)Chemotaxis with logistic source: very weak global solutions and their boundedness properties
J. Math. Anal. Appl.
(2008)- et al.
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant
J. Differential Equations
(2018) - et al.
Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption
J. Differential Equations
(2020) - et al.
Boundedness and stabilization in a two-species chemotaxis system with signal absorption
Comput. Math. Appl.
(2019) Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source
J. Differential Equations
(2015)- et al.
Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion
J. Differential Equations
(2016)
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier–Stokes system with competitive kinetics
J. Differential Equations
The spatial homogeneity of stable equilibria of some reaction–diffusion systems on convex domains
J. Differential Equations
Random walk with persistence and external bias
Bull. Math. Biol. Biophys.
A blow-up mechanism for chemotaxis model
Ann. Sc. Norm. Super. Pisa
From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I
Jahresber. Dtsch. Math.-Ver.
From 1970 until present: The Keller–Segel model in chemotaxis and its consequences II
Jahresber. Dtsch. Math.-Ver.
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