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Journal of Function Spaces
Volume 2018, Article ID 4051812, 5 pages
https://doi.org/10.1155/2018/4051812
Research Article

A Result on the Existence and Uniqueness of Stationary Solutions for a Bioconvective Flow Model

GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile

Correspondence should be addressed to Aníbal Coronel; lc.oiboibu@lenoroca

Received 8 December 2017; Accepted 25 March 2018; Published 30 April 2018

Academic Editor: Juan Martinez-Moreno

Copyright © 2018 Aníbal Coronel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this note, we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem. The bioconvective model is a boundary value problem for a system of four equations: the nonlinear Stokes equation, the incompressibility equation, and two transport equations. The unknowns of the model are the velocity of the fluid, the pressure of the fluid, the local concentration of microorganisms, and the oxygen concentration. We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.

1. Introduction

Bioconvection is an important process in the biological treatment and in the life of some microorganisms. In a broad sense, biconvection originates from the concentration of upward swimming microorganisms in a culture fluid. It is well known that, under some physical assumptions, the process can be described by mathematical models which are called bioconvective flow models. The first model of this kind was derived by Moribe [1] and independently by Levandowsky et al. [2] (see also [3] for the mathematical analysis). In that model the unknowns are the velocity of the fluid, the pressure of the fluid, and the local concentration of microorganisms. More recently, Tuval et al. [4] have introduced a new bioconvective flow model considering an additional unknown variable, the oxygen concentration. Some advances in mathematical analysis and some numerical results for this new model are presented in [5] and [6], respectively.

In this paper, we are interested in the existence and uniqueness of solutions for the stationary problem associated with the bioconvective system given in [4] when the physical domain is a three-dimensional chamber [6] (a parallelepiped). Thus, the stationary bioconvective flow problem to be analyzed is formulated as follows. Given the external force , the source functions , , and the dimensionless function , find the velocity of the fluid , the fluid pressure , the local concentration of bacteria , and the local concentration of oxygen satisfying the boundary value problem:Here is the unit external normal to ; is a gravitational field with constant acceleration ; and ,  ,  ,  , and are some physical parameters defined as follows: with being the fluid viscosity, the diffusion constant for bacteria, the diffusion constant for oxygen, the fluid density, the bacterial density, the bacterial volume, a characteristic cell density, a characteristic length, the chemotactic sensitivity, the oxygen concentration above the fluid, and the oxygen consumption rate.

We consider the standard notation of the Lebesgue and Sobolev spaces which are used in the analysis of Navier-Stokes and related equations of fluid mechanics; see [711] for details and specific definitions. In particular, we use the following rather common spaces notation: where denotes the completion of in . Also, we consider the notation for the applications , , , and , which are defined as where is the standard inner product in or . It is well known that and are bilinear coercive forms and and are well defined trilinear forms with the following properties:for all and . Moreover, we need to introduce some notation related to some useful Sobolev inequalities and estimates for and . There exist , , and depending only on such that for all , , and . For details on Poincaré and trace inequalities, we refer to [8] and for the estimates of and consult [11].

The main result of the paper is the existence and uniqueness of weak solutions for (1)–(6). Indeed, let us introduce some appropriate notation:such that the result is precised as follows.

Theorem 1. Let us consider that , and , the average of on , are given. Also consider notations (12)–(15). If we assume that the following assumptions,are satisfied, there is satisfying (1)–(6). Moreover, if we consider that additionally and the following inequalities,are satisfied, the weak solution is unique.

It should be noted that existence and uniqueness results are derived in [12, 13] for the bioconvection problem, when the concentration of oxygen is assumed to be constant. In the case of [12], the proof is based on the application of the Galerkin approximation and in [13] on the application of the Gossez theorem. Moreover, other related results are given in [3, 5]. In particular, in [5], a well detailed discussion of some particular models derived from (1)–(6) is given.

2. Proof of Theorem 1

2.1. Variational Formulation

By standard arguments, the variational formulation of (1)–(6) is given byWe notice that if and is a solution of (1)-(2) with , we have that is a solution of (19). However, does not describe the bioconvective flow problem and we need to study the variational problem when the total local concentration of bacteria and the total local concentration of oxygen are some given strictly positive constants, that is, and . Thus, by considering the change of variable and , we can rewrite (19) as follows:

2.2. Some A Priori Estimates for , and

Proposition 2. Consider that the assumptions for the existence result of Theorem 1 are satisfied. If we assume that is a solution of (20)–(24), then with defined on (13). Furthermore, the following estimates are valid:

Proof. In order to prove the estimates, we select the test functions in (21)–(23). From (21) and (10), we deduce thatNow, by the trace inequality and integration by parts, we have that which implies thatHere, we have used the fact that , as a consequence of the assumption (16). Then, by integration by parts we get the boundFrom (22), using the properties (10) and the inequality (29), we have that or equivalently, we get the following estimate for :with being defined in (12). Similarly, from (23) and (28) with instead of , we deduce thatwhere is given in (12). Now, replacing the estimate (32) in (31) and applying (16), we deduce the existence of defined in (13) such that . We notice that the second and third relation in (16) imply that , , and , respectively, that is, under (16). Moreover, from (26) and (31), we deduce the estimates given in (25) for and , concluding the proof of the Proposition.

2.3. Proof of Theorem 1

To prove the existence, we can apply the Gossez theorem [9, 14]. Let us first define the mapping by the relation with denoting the duality pairing between and and , , and are positive fixed constant. From (10), (12), and (29), we then have that Now, selecting , , and such that we can prove that is positive for all such that . Moreover, we notice that it is straightforward to deduce that is continuous between the weak topologies of and . Thus, there is such that , concluding the proof of existence.

To prove the uniqueness we consider that there are two solutions , , satisfying (21)–(23). Then, subtracting, selecting the test functions , using (10), (16), (17), and applying Proposition 2, we getwith being defined in (13)–(15). From (38), Proposition 2, and the first inequality in (18), we have thatThen, replacing (39) in (37), using Proposition 2 to estimate , we obtain the bound with being defined in (18). Now, using this estimate in (36), we get that . Thus using the fact that we deduce that on , which also implies that and on , concluding the uniqueness proof.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors are supported by the project DIUBB 172409 GI/C at Universidad del Bío-Bío, Chile. Aníbal Coronel is supported by the projects DIUBB 183309 4/R and FAPEI at U. del Bío-Bío, Chile. Alex Tello and Ian Hess are supported by Conicyt-Chile through the grants program “Becas de Doctorado”.

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