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

Authors: Aníbal Coronel; Luis Friz; Ian Hess; Alex Tello
Journal: Journal of Function Spaces
 (2018)

---

## 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.


---

## Body

## 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 F, the source functions fn, fc, and the dimensionless function r, find the velocity of the fluid u=u1,u2,u3t, the fluid pressure p, the local concentration of bacteria n, and the local concentration of oxygen c satisfying the boundary value problem:(1)-ScΔu+u·∇u+Sc∇p=γScng+F,inΩ≔∏i=130,Li,(2)div⁡u=0,inΩ,(3)-Δn+u·∇n+χdiv⁡nrc∇c=fn,inΩ,(4)-δΔc+u·∇c+βrcn=fc,inΩ,(5)∇c·ν=∇n·ν=0,u=0,on∂ΩLx3=0,(6)χnrc∇c·ν-∇n·ν=0,u=0,on∂ΩU≔∂Ω-∂ΩL.Here ν is the unit external normal to ∂Ω; g=(0,0,-g) is a gravitational field with constant acceleration g; and Sc,  γ,  α,  δ, and β are some physical parameters defined as follows: (7)Sc=ηDnρ,γ=Vbnrρb-ρL3ηDn,χ=χ¯cairDn,δ=DcDn,β=knrL2cairDn,with η being the fluid viscosity, Dn the diffusion constant for bacteria, Dc the diffusion constant for oxygen, ρ the fluid density, ρb the bacterial density, Vb>0 the bacterial volume, nr a characteristic cell density, L a characteristic length, χ¯ the chemotactic sensitivity, cair the oxygen concentration above the fluid, and k 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 [7–11] for details and specific definitions. In particular, we use the following rather common spaces notation:(8)HmΩ=Wm,2Ω,H~1Ω=f∈H1Ω:∫Ωfdx=0,H01Ω=C0∞Ω¯·H1Ω,C0,σ∞Ω=v∈C0∞Ω3:div⁡v=0,V=C0,σ∞Ω¯·H01Ω, where A¯·B denotes the completion of A in B. Also, we consider the notation for the applications a0:V×V→R, a:H1(Ω)×H1(Ω)→R, b0:V×V×V→R, and b:V×H1(Ω)×H1(Ω)→R, which are defined as (9)a0u,v=∇u,∇v,aϕ,ψ=∇ϕ,∇ψ,b0u,v,w=u·∇v,w,bu,ϕ,ψ=u·∇ϕ,ψ,where (·,·) is the standard inner product in L2(Ω) or L2(Ω). It is well known that a0 and a are bilinear coercive forms and b0 and b are well defined trilinear forms with the following properties:(10)b0u,v,w=-b0u,w,v,bu,ϕ,ψ=-bu,ψ,ϕ,b0u,v,v=0,bu,ϕ,ϕ=0,for all u,v,w∈V and ψ,ϕ∈H1(Ω). Moreover, we need to introduce some notation related to some useful Sobolev inequalities and estimates for b and b0. There exist Cpoi>0, Ctr>0, and C1 depending only on Ω such that (11)uL2Ω≤CpoiuV,cL2Ω≤CpoicH~1Ω,φL1∂Ω≤CtrφW1,1Ω,b0u,v,w≤C1uVvVwV,bu,c,n≤C1uVcH~1ΩnH~1Ω,for all u,v,w∈V, c,n∈H~1(Ω), and φ∈W1,1(Ω). For details on Poincaré and trace inequalities, we refer to [8] and for the estimates of b0 and b 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:(12)Θ1≔1-Ctr1-Ctr-2χrL1RCtrCpoi,Θ2≔1-Ctr1-Ctr-CtrCpoi,(13)Γ0=ΩΘ1CpoiΩ-χβα1rL∞R2Cpoi2Θ1Θ2χα1rL∞R2Θ2δΩfcL2Ω+fnL2Ω,(14)Γ1=γScgCpoiSc-C1CpoiγgΓ0+FL2Ω,Γ2=1-Ctr1-2rL1R1-Ctr+CtrCpoi,(15)Γ3=1-Ctrδ1-Ctr-CtrCpoi-C13rLipRΓ0,such that the result is precised as follows.Theorem 1.
Let us consider thatfc,fb∈L2(Ω), F∈L2(Ω) and n¯, the average of n on Ω, are given. Also consider notations (12)–(15). If we assume that the following assumptions,(16)r∈L∞R∩L1R,1-Ctr>CtrCpoimax⁡2χrL1R,1,1>χβn¯rL∞R2Cpoi2Θ1Θ2,are satisfied, there is (u,p,n,c)∈V×H1(Ω)×H1(Ω)×H1(Ω) satisfying (1)–(6). Moreover, if we consider that additionally r∈Lip(R) and the following inequalities,(17)Sc-C1CpoiγgΓ0+FL2Ω>0,δ1-Ctr-CtrCpoi-C13rL1RΓ0>0,(18)C1rLipRΓ0<1,Π=Γ1Γ2C1Γ0+rL∞RCΓ3Θ2Cpoiδ1-C1rLipRΓ0βCpoirL∞RΓ0+fcL2Ω<1,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 Theorem1

### 2.1. Variational Formulation

By standard arguments, the variational formulation of (1)–(6) is given by(19)Findu,n,c∈V×H1Ω×H1ΩsuchthatSca0u,v+b0u,u,v=γScng,v+F,v,∀v∈V,an,ϕ+bu,n,ϕ=χnrc∇c,∇ϕ+fn,ϕ,∀ϕ∈H1Ω,δac,φ+bu,c,φ=-βrcn,φ+δ∫∂ΩU∇c·νφdS+fc,φ,∀φ∈H1Ω.We notice that if fc=fn=0 and u0 is a solution of (1)-(2) with n=0, we have that (u0,0,0) is a solution of (19). However, (u0,0,0) 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, ∫Ωnαdx=α1>0 and ∫Ωcαdx=α2>0. Thus, by considering the change of variable n^α=nα-α1Ω-1 and c^α=cα-α2Ω-1, we can rewrite (19) as follows:(20)Givenα=α2,α2∈0,1×0,1finduα,n^α,c^α∈V×H~1Ω×H~1Ω:(21)Sca0uα,v+b0uα,uα,v=γScn^αg,v+F,v,(22)an^α,ϕ+buα,n^α,ϕ=χn^α+α1Ωrc^α+α2Ω∇c^α,∇ϕ+fn,ϕ,(23)δac^α,φ+buα,c^α,φ=-βrc^α+α2Ωn^α+α1Ω,φ+δ∫∂ΩU∇c^α·νφdS+fc,φ,(24)∀v,ϕ,φ∈V×H~1Ω×H~1Ω.


### 2.2. Some A Priori Estimates foruα,n^α, and c^α

Proposition 2.
Consider that the assumptions for the existence result of Theorem1 are satisfied. If we assume that (uα,n^α,c^α) is a solution of (20)–(24), then n^αH~1(Ω)≤Γ0 with Γ0 defined on (13). Furthermore, the following estimates are valid:(25)uαV≤CpoiγgΓ0+FL2Ω,c^αH~1Ω≤Θ2CpoiδβCpoirL∞RΓ0+fcL2Ω.Proof.
In order to prove the estimates, we select the test functions(v,ϕ,φ)=(uα,n^α,c^α) in (21)–(23). From (21) and (10), we deduce that(26)uαV≤γgCpoi2n^αH~1Ω+Sc-1CpoiFL2Ω.Now, by the trace inequality and integration by parts, we have that (27)∫∂Ω∇n^α·νn^αdS≤Ctrn^α∇n^α·νW1,1Ω≤CtrCpoin^αH~1Ω2+Ctr∫∂Ω∇n^α·νn^αdS,which implies that(28)∫∂Ω∇n^α·νn^αdS≤CtrCpoi1-Ctrn^αH~1Ω2.Here, we have used the fact that 1-Ctr>0, as a consequence of the assumption (16). Then, by integration by parts we get the bound(29)n^αrc^α+α2Ω∇c^α,∇n^α=∇∫0c^αrm+α2Ωdm,∇n^α22=-∫0c^αrm+α2Ωdm,Δn^α2+∫∂Ω∫0c^αrm+α2Ωdm∇n^α22·νdS≤2rL1R∫∂Ωn^α∇n^α·νdS≤2rL1RCtrCpoi1-Ctrn^αH~1Ω2.From (22), using the properties (10) and the inequality (29), we have that (30)n^αH~1Ω2=χn^αrc^α+α2Ω∇c^α,∇n^α+χα1Ωrc^α+α2Ω∇c^α,∇n^α+fn,ϕ≤2χrL1RCtrCpoi1-Ctrn^αH~1Ω2+χα1ΩrL∞Rc^αH~1Ωn^αH~1Ω+CpoifnL2Ωn^αH~1Ω,or equivalently, we get the following estimate for n^α:(31)n^αH~1Ω≤Θ1χα1ΩrL∞Rc^αH~1Ω+CpoifnL2Ω,with Θ1 being defined in (12). Similarly, from (23) and (28) with c^α instead of n^α, we deduce that(32)c^αH~1Ω≤Θ2CpoiδβCpoirL∞Rn^αH~1Ω+fcL2Ω,where Θ2 is given in (12). Now, replacing the estimate (32) in (31) and applying (16), we deduce the existence of Γ0 defined in (13) such that n^αH~1Ω≤Γ0. We notice that the second and third relation in (16) imply that Θi>1, i=1,2, and Ω>χβα1rL∞R2Cpoi2Θ1Θ2, respectively, that is, Γ>0 under (16). Moreover, from (26) and (31), we deduce the estimates given in (25) for uαV and c^αH~1Ω, concluding the proof of the Proposition.


### 2.3. Proof of Theorem1

To prove the existence, we can apply the Gossez theorem [9, 14]. Let us first define the mapping G:V×H~1(Ω)×H~1(Ω)→V×H~1Ω×H~1Ω′ by the relation (33)Gu,n,c,v,ϕ,φ=λ1Sca0u,v+b0u,u,v-γScng,v-F,v+λ2an,ϕ+bu,n,ϕ-χn+α1Ωrc+α2Ω∇c,∇ϕ-fn,ϕ+λ3δac,φ+bu,c,φ+βrc+α2Ωn+α1Ω,φ-δ∫∂ΩU∇c·νφdS-fc,φ,∀u,n,c,v,ϕ,φ∈V×H~1Ω×H~1Ω, with ·,· denoting the duality pairing between V×H~1(Ω)×H~1(Ω) and V×H~1Ω×H~1Ω′ and λ1, λ2, and λ3 are positive fixed constant. From (10), (12), and (29), we then have that (34)Gu,n,c,u,n,c≥λ1ScuV2-λ1γScgCpoi2nH~1ΩuV+λ23Θ1nH~1Ω2+λ23Θ1nH~1Ω2-λ2χα1ΩrL∞RcH~1ΩnH~1Ω+λ3δ2Θ2cH~1Ω2+λ3δ2Θ2cH~1Ω2-λ3βCpoi2rL∞RcH~1ΩnH~1Ω+λ23Θ1nH~1Ω2-Cpoiλ1FL2ΩuV+λ2fnL2ΩnH~1Ω+λ3fcL2ΩcH~1Ω≔Υ1+Υ2-Υ3.Now, selecting λ1, λ2, λ3 and r such that (35)λ1<4λ23Θ1γ2g2ScCpoi4,λ2<4δΩ2λ36Θ1Θ2χα1rL∞R2,λ3<4δλ26Θ1Θ2βCpoi2rL∞R2r<Υ1+Υ2Cpoiλ1FL2Ω+λ2fnL2Ω+λ3fcL2Ω, we can prove that G(u,n,c),(u,n,c) is positive for all (u,n,c)∈V×H~1(Ω)×H~1(Ω) such that (u,n,c)V×H~1(Ω)×H~1(Ω)=r. Moreover, we notice that it is straightforward to deduce that G is continuous between the weak topologies of V×H~1(Ω)×H~1(Ω) and V×H~1Ω×H~1Ω′. Thus, there is (u,n,c)∈B¯r(0)⊂V×H~1(Ω)×H~1(Ω) such that G(u,n,c),(u,n,c)=0, concluding the proof of existence.To prove the uniqueness we consider that there are two solutions(ui,ni,ci), i=1,2, satisfying (21)–(23). Then, subtracting, selecting the test functions (v,ϕ,φ)=(u1-u2,n1-n2,c1-c2), using (10), (16), (17), and applying Proposition 2, we get(36)u1-u2V≤Γ1n1-n2H~1Ω,(37)n1-n2H~1Ω≤Γ2C1u1-u2Vn1H~1Ω+rL∞Rc1-c2H~1Ω,(38)c1-c2H~1Ω≤C1Γ3u1-u2Vc2H~1Ω+C12rLipRn1H~1Ωc1-c2H~1Ω,with Γi being defined in (13)–(15). From (38), Proposition 2, and the first inequality in (18), we have that(39)c1-c2H~1Ω≤C1Γ3Θ2Cpoiδ1-C12rLipRΓ0βCpoirL∞RΓ0+fcL2Ωu1-u2V.Then, replacing (39) in (37), using Proposition 2 to estimate n1H~1Ω, we obtain the bound n1-n2H~1Ω≤ΠΓ1-1u1-u2V with Π being defined in (18). Now, using this estimate in (36), we get that u1-u2V≤Πu1-u2V. Thus using the fact that Π≤1 we deduce that u1=u2 on V, which also implies that n1=n2 and c1=c2 on H~1(Ω), concluding the uniqueness proof.


## 2.1. Variational Formulation

By standard arguments, the variational formulation of (1)–(6) is given by(19)Findu,n,c∈V×H1Ω×H1ΩsuchthatSca0u,v+b0u,u,v=γScng,v+F,v,∀v∈V,an,ϕ+bu,n,ϕ=χnrc∇c,∇ϕ+fn,ϕ,∀ϕ∈H1Ω,δac,φ+bu,c,φ=-βrcn,φ+δ∫∂ΩU∇c·νφdS+fc,φ,∀φ∈H1Ω.We notice that if fc=fn=0 and u0 is a solution of (1)-(2) with n=0, we have that (u0,0,0) is a solution of (19). However, (u0,0,0) 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, ∫Ωnαdx=α1>0 and ∫Ωcαdx=α2>0. Thus, by considering the change of variable n^α=nα-α1Ω-1 and c^α=cα-α2Ω-1, we can rewrite (19) as follows:(20)Givenα=α2,α2∈0,1×0,1finduα,n^α,c^α∈V×H~1Ω×H~1Ω:(21)Sca0uα,v+b0uα,uα,v=γScn^αg,v+F,v,(22)an^α,ϕ+buα,n^α,ϕ=χn^α+α1Ωrc^α+α2Ω∇c^α,∇ϕ+fn,ϕ,(23)δac^α,φ+buα,c^α,φ=-βrc^α+α2Ωn^α+α1Ω,φ+δ∫∂ΩU∇c^α·νφdS+fc,φ,(24)∀v,ϕ,φ∈V×H~1Ω×H~1Ω.


## 2.2. Some A Priori Estimates foruα,n^α, and c^α

Proposition 2.
Consider that the assumptions for the existence result of Theorem1 are satisfied. If we assume that (uα,n^α,c^α) is a solution of (20)–(24), then n^αH~1(Ω)≤Γ0 with Γ0 defined on (13). Furthermore, the following estimates are valid:(25)uαV≤CpoiγgΓ0+FL2Ω,c^αH~1Ω≤Θ2CpoiδβCpoirL∞RΓ0+fcL2Ω.Proof.
In order to prove the estimates, we select the test functions(v,ϕ,φ)=(uα,n^α,c^α) in (21)–(23). From (21) and (10), we deduce that(26)uαV≤γgCpoi2n^αH~1Ω+Sc-1CpoiFL2Ω.Now, by the trace inequality and integration by parts, we have that (27)∫∂Ω∇n^α·νn^αdS≤Ctrn^α∇n^α·νW1,1Ω≤CtrCpoin^αH~1Ω2+Ctr∫∂Ω∇n^α·νn^αdS,which implies that(28)∫∂Ω∇n^α·νn^αdS≤CtrCpoi1-Ctrn^αH~1Ω2.Here, we have used the fact that 1-Ctr>0, as a consequence of the assumption (16). Then, by integration by parts we get the bound(29)n^αrc^α+α2Ω∇c^α,∇n^α=∇∫0c^αrm+α2Ωdm,∇n^α22=-∫0c^αrm+α2Ωdm,Δn^α2+∫∂Ω∫0c^αrm+α2Ωdm∇n^α22·νdS≤2rL1R∫∂Ωn^α∇n^α·νdS≤2rL1RCtrCpoi1-Ctrn^αH~1Ω2.From (22), using the properties (10) and the inequality (29), we have that (30)n^αH~1Ω2=χn^αrc^α+α2Ω∇c^α,∇n^α+χα1Ωrc^α+α2Ω∇c^α,∇n^α+fn,ϕ≤2χrL1RCtrCpoi1-Ctrn^αH~1Ω2+χα1ΩrL∞Rc^αH~1Ωn^αH~1Ω+CpoifnL2Ωn^αH~1Ω,or equivalently, we get the following estimate for n^α:(31)n^αH~1Ω≤Θ1χα1ΩrL∞Rc^αH~1Ω+CpoifnL2Ω,with Θ1 being defined in (12). Similarly, from (23) and (28) with c^α instead of n^α, we deduce that(32)c^αH~1Ω≤Θ2CpoiδβCpoirL∞Rn^αH~1Ω+fcL2Ω,where Θ2 is given in (12). Now, replacing the estimate (32) in (31) and applying (16), we deduce the existence of Γ0 defined in (13) such that n^αH~1Ω≤Γ0. We notice that the second and third relation in (16) imply that Θi>1, i=1,2, and Ω>χβα1rL∞R2Cpoi2Θ1Θ2, respectively, that is, Γ>0 under (16). Moreover, from (26) and (31), we deduce the estimates given in (25) for uαV and c^αH~1Ω, concluding the proof of the Proposition.


## 2.3. Proof of Theorem1

To prove the existence, we can apply the Gossez theorem [9, 14]. Let us first define the mapping G:V×H~1(Ω)×H~1(Ω)→V×H~1Ω×H~1Ω′ by the relation (33)Gu,n,c,v,ϕ,φ=λ1Sca0u,v+b0u,u,v-γScng,v-F,v+λ2an,ϕ+bu,n,ϕ-χn+α1Ωrc+α2Ω∇c,∇ϕ-fn,ϕ+λ3δac,φ+bu,c,φ+βrc+α2Ωn+α1Ω,φ-δ∫∂ΩU∇c·νφdS-fc,φ,∀u,n,c,v,ϕ,φ∈V×H~1Ω×H~1Ω, with ·,· denoting the duality pairing between V×H~1(Ω)×H~1(Ω) and V×H~1Ω×H~1Ω′ and λ1, λ2, and λ3 are positive fixed constant. From (10), (12), and (29), we then have that (34)Gu,n,c,u,n,c≥λ1ScuV2-λ1γScgCpoi2nH~1ΩuV+λ23Θ1nH~1Ω2+λ23Θ1nH~1Ω2-λ2χα1ΩrL∞RcH~1ΩnH~1Ω+λ3δ2Θ2cH~1Ω2+λ3δ2Θ2cH~1Ω2-λ3βCpoi2rL∞RcH~1ΩnH~1Ω+λ23Θ1nH~1Ω2-Cpoiλ1FL2ΩuV+λ2fnL2ΩnH~1Ω+λ3fcL2ΩcH~1Ω≔Υ1+Υ2-Υ3.Now, selecting λ1, λ2, λ3 and r such that (35)λ1<4λ23Θ1γ2g2ScCpoi4,λ2<4δΩ2λ36Θ1Θ2χα1rL∞R2,λ3<4δλ26Θ1Θ2βCpoi2rL∞R2r<Υ1+Υ2Cpoiλ1FL2Ω+λ2fnL2Ω+λ3fcL2Ω, we can prove that G(u,n,c),(u,n,c) is positive for all (u,n,c)∈V×H~1(Ω)×H~1(Ω) such that (u,n,c)V×H~1(Ω)×H~1(Ω)=r. Moreover, we notice that it is straightforward to deduce that G is continuous between the weak topologies of V×H~1(Ω)×H~1(Ω) and V×H~1Ω×H~1Ω′. Thus, there is (u,n,c)∈B¯r(0)⊂V×H~1(Ω)×H~1(Ω) such that G(u,n,c),(u,n,c)=0, concluding the proof of existence.To prove the uniqueness we consider that there are two solutions(ui,ni,ci), i=1,2, satisfying (21)–(23). Then, subtracting, selecting the test functions (v,ϕ,φ)=(u1-u2,n1-n2,c1-c2), using (10), (16), (17), and applying Proposition 2, we get(36)u1-u2V≤Γ1n1-n2H~1Ω,(37)n1-n2H~1Ω≤Γ2C1u1-u2Vn1H~1Ω+rL∞Rc1-c2H~1Ω,(38)c1-c2H~1Ω≤C1Γ3u1-u2Vc2H~1Ω+C12rLipRn1H~1Ωc1-c2H~1Ω,with Γi being defined in (13)–(15). From (38), Proposition 2, and the first inequality in (18), we have that(39)c1-c2H~1Ω≤C1Γ3Θ2Cpoiδ1-C12rLipRΓ0βCpoirL∞RΓ0+fcL2Ωu1-u2V.Then, replacing (39) in (37), using Proposition 2 to estimate n1H~1Ω, we obtain the bound n1-n2H~1Ω≤ΠΓ1-1u1-u2V with Π being defined in (18). Now, using this estimate in (36), we get that u1-u2V≤Πu1-u2V. Thus using the fact that Π≤1 we deduce that u1=u2 on V, which also implies that n1=n2 and c1=c2 on H~1(Ω), concluding the uniqueness proof.



---

*Source: 4051812-2018-04-30.xml*