This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.
Citation: Yunfei Yuan, Changchun Liu. Optimal control for the coupled chemotaxis-fluid models in two space dimensions[J]. Electronic Research Archive, 2021, 29(6): 4269-4296. doi: 10.3934/era.2021085
[1] | Yunfei Yuan, Changchun Liu . Optimal control for the coupled chemotaxis-fluid models in two space dimensions. Electronic Research Archive, 2021, 29(6): 4269-4296. doi: 10.3934/era.2021085 |
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This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.
In this paper, we study the coupled chemotaxis-fluid models with the initial-bounary conditions
{nt+u⋅∇n=Δn−∇⋅(n∇c)+γn−μn2,in Q≡(0,T)×Ω,ct+u⋅∇c=Δc−c+n+f,in Q,ut+u⋅∇u=Δu−∇π+n∇φ,in Q,∇⋅u=0,in Q,∂n∂ν=∂c∂ν=0,u=0,on (0,T)×∂Ω,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),in Ω, | (1.1) |
where
In order to understand the development of system (1.1), let us mention some previous contributions in this direction. Jin [11] dealed with the time periodic problem of (1.1) in spatial dimension
Espejo and Suzuki [6] discussed the chemotaxis-fluid model
nt+u⋅∇n=Δn−∇⋅(n∇c)+n(γ−μn), | (1.2) |
ct+u⋅∇c=Δc−c+n, | (1.3) |
ut=Δu−∇π+n∇φ, | (1.4) |
∇⋅u=0, | (1.5) |
∂n∂ν=∂c∂ν=0,u=0. | (1.6) |
They proved the global existence of weak solution. Tao and Winkler [17] proved the existence of global classical solution and the uniform boundedness. Tao and Winkler [18] also obtained the global classical solution and uniform boundedness under the condition of
The optimal control problems governed by the coupled partial differential equations is important. Colli et al. [4] studied the distributed control problem for a phase-field system of conserved type with a possibly singular potential. Liu and Zhang [14] considered the optimal control of a new mechanochemical model with state constraint. Chen et al. [3] studied the distributed optimal control problem for the coupled Allen-Cahn/Cahn-Hilliard equations. Recently, Guillén-González et al. [9] studied a bilinear optimal control problem for the chemo-repulsion model with the linear production term. The existence, uniqueness and regularity of strong solutions of this model are deduced. They also derived the first-order optimality conditions by using a Lagrange multipliers theorem. Frigeri et al. [8] studied an optimal control problem for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential. Some other results can be found in [2,5,13,15,19].
In this paper, we discuss the optimal control problem for (1.1). We adjust the external source
In this section, we will construct the existence and some priori estimates of the linearized problem for the chemotaxis-Navier-Stokes system in a bounded domain
In the following lemmas we will state the Gagliardo-Nirenberg interpolation inequality [7].
Lemma 2.1. Let
1p−lN=a(1q−kN)+(1−a)1r. | (2.1) |
Then, for any
‖Dlu‖Lp⩽c1‖Dku‖aLq‖u‖1−aLr+c2‖u‖Lr |
with the following exception: If
The following log-interpolation inequality has been proved by [1].
Lemma 2.2. Let
‖u‖3L3(Ω)≤δ‖u‖2H1(Ω)‖(u+1)log(u+1)‖L1(Ω)+p(δ−1)‖u‖L1(Ω), |
where
We first consider the existence of solutions to the linear problem of system (1.1). Assume functions
{ut−Δu+ˆu⋅∇u=−∇π+ˆn∇φ,in Q,∇⋅u=0,in Q,u=0,on ∂Ω,u(x,0)=u0(x),in Ω. | (2.2) |
By using fixed point method, the existence of solutions can be easily obtained. Therefore, we ignore the process of proof and just give the regularity estimate.
Lemma 2.3. Let
Proof. Multiplying the first equation of (2.2) by
12ddt∫Ωu2dx+∫Ω|∇u|2dx+∫Ωu2dx=∫Ωˆn∇φ⋅udx+∫Ωu2dx≤‖ˆn‖L2‖u‖L2+‖u‖2L2≤C(‖ˆn‖2L2+‖u‖2L2). |
By Gronwall's inequality, we have
‖u‖2L2+∫T0‖u‖2H1dτ≤C(∫T0‖ˆn‖2L2dτ+‖u0‖2L2). |
Operating the Helmholtz projection operator
ut+Au+P(ˆu⋅∇u)=P(ˆn∇φ), |
where
12ddt∫Ω|∇u|2dx+∫Ω|Δu|2dx+∫Ω|∇u|2dx=∫ΩP(ˆu∇u)Δudx−∫ΩP(ˆn∇φ)Δudx+∫Ω|∇u|2dx. |
For the terms on the right, we have
∫ΩP(ˆu∇u)Δudx−∫ΩP(ˆn∇φ)Δudx+∫Ω|∇u|2dx≤‖ˆu‖L4‖∇u‖L4‖Δu‖L2+‖ˆn‖L2‖Δu‖L2+‖∇u‖2L2≤‖ˆu‖L4‖∇u‖1/2L2‖Δu‖3/2L2+‖ˆu‖L4‖∇u‖L2‖Δu‖L2+‖ˆn‖L2‖Δu‖L2+‖∇u‖2L2≤12‖Δu‖2L2+C(‖ˆu‖4L4+‖ˆu‖2L4+1)‖∇u‖2L2+‖ˆn‖2L2. |
Therefore, we get
ddt‖∇u‖2L2+‖∇u‖2H1≤C(‖ˆu‖4L4+‖ˆu‖2L4+1)‖∇u‖2L2+C‖ˆn‖2L2+C. |
By Gronwall's inequality, we derive
‖∇u‖2L2+∫T0‖∇u‖2H1dτ≤C. |
Multiplying the first equation of (2.2) by
∫T0∫Ω|ut|2dxdt≤C. |
Summing up, we complete the proof.
For the above solution
{ct−Δc+u⋅∇c+c=ˆn++f,in Q,∂c∂ν=0,on (0,T)×∂Ω,c(x,0)=c0(x),in Ω. | (2.3) |
Along with fixed point method, the existence of solutions can be easily obtained. Thus we omit the proof and only give the regularity estimate.
Lemma 2.4. Let
Proof. Multiplying the first equation of (2.3) by
12ddt∫Ωc2dx+∫Ω|∇c|2dx+∫Ωc2dx≤‖ˆn‖L2‖c‖L2+‖f‖L2‖c‖L2. |
Therefore, we have
‖c‖2L2+‖c‖2H1≤C(‖c0‖2L2+∫t0(‖ˆn‖2L2+‖f‖2L2)dτ). |
Multiplying the first equation of (2.3) by
12ddt∫Ω|∇c|2dx+∫Ω|Δc|2dx+∫Ω|∇c|2dx=∫Ωu∇cΔcdx−∫ΩΔcˆndx−∫ΩΔcfdx. |
Using the Young inequality and the Hölder inequality, we obtain
∫Ωu∇cΔcdx−∫ΩΔcˆndx−∫ΩΔcfdx≤‖u‖L4‖∇c‖L4‖Δc‖L2+‖ˆn‖L2‖Δc‖L2+‖f‖L2‖Δc‖L2≤C‖u‖H1(‖∇c‖12L2‖Δc‖12L2+‖∇c‖L2)‖Δc‖L2+‖ˆn‖L2‖Δc‖L2+‖f‖L2‖Δc‖L2=C‖u‖H1‖∇c‖12L2‖Δc‖32L2+C‖∇c‖L2‖Δc‖L2+‖ˆn‖L2‖Δc‖L2+‖f‖L2‖Δc‖L2≤12‖Δc‖2L2+C‖u‖4H1‖∇c‖2L2+C(‖ˆn‖2L2+‖f‖2L2). |
Combining this and above inequalities, we conclude
ddt‖∇c‖2L2+‖∇c‖2H1≤C‖u‖4H1‖∇c‖2L2+C(‖ˆn‖2L2+‖f‖2L2). |
We therefore verify that
‖∇c‖2L2+∫t0‖∇c‖2H1≤C(∫t0‖ˆn‖2L2dτ+∫t0‖f‖2L2dτ). |
Applying
12ddt∫Ω|Δc|2dx+∫Ω|∇Δc|2dx+∫Ω|Δc|2dx=∫Ω∇(u∇c)∇Δcdx−∫Ω∇ˆn+∇Δcdx−∫Ω∇f∇Δcdx. |
For the terms on the right, we obtain
∫Ω∇(u∇c)∇Δcdx−∫Ω∇ˆn+∇Δcdx−∫Ω∇f∇Δcdx≤‖∇Δc‖L2(‖u‖L4‖Δc‖L4+‖∇u‖L4‖∇c‖L4)+‖∇ˆn‖L2‖∇Δc‖L2+‖∇f‖L2‖∇Δc‖L2≤‖∇Δc‖L2(‖u‖L4‖Δc‖12L2‖∇Δc‖12L2+‖u‖L4‖Δc‖L2+‖∇u‖12L2‖Δu‖12L2‖∇c‖12L2‖Δc‖12L2+‖∇u‖L2‖∇c‖12L2‖Δc‖12L2+‖∇u‖12L2‖Δu‖12L2‖∇c‖L2+‖∇u‖L2‖∇c‖L2)+‖∇ˆn‖L2‖∇Δc‖L2+‖∇f‖L2‖∇Δc‖L2≤12‖∇Δc‖2L2+C(1+‖Δc‖2L2+‖Δu‖2L2+‖∇ˆn‖2L2+‖∇f‖2L2). |
Straightforward calculations yield
‖Δc‖2L2+∫t0‖Δc‖2H1dτ≤C(1+∫t0‖ˆn‖2H1dτ+∫t0‖f‖2H1dτ). |
Multiplying the first equation of (2.3) by
∫T0∫Ω|ct|2dxdt≤C, |
and thereby precisely arrive at the conclusion.
With above solutions
{nt−Δn+u⋅∇n+n=−∇⋅(n∇c)+(1+γ)ˆn+−μˆn+n,in Q,∂n∂ν|∂Ω=0,n(x,0)=n0(x),in Ω. | (2.4) |
By a similar argument as the above two problems, the existence of solutions can be easily obtained. Therefore, we only give the regularity estimate.
Lemma 2.5. Suppose
Proof. Firstly, we verify the nonnegativity of
ddt∫A(t)ndx−∫∂A(t)∂n∂νds+∫A(t)ndx=(1+γ)∫A(t)ˆn+dx−μ∫A(t)ˆn+ndx. |
Since
∫A(t)ndxdτ+∫t0∫A(t)ndxdτ=0. |
Then, we get
Next, multiplying the first equation of (2.4) by
12ddt∫Ωn2dx+∫Ω(n2+|∇n|2)dx+μ∫Ωˆn+n2dx=∫Ωn∇c∇ndx+(1+γ)∫Ωnˆn+dx≤‖n‖L4‖∇c‖L4‖∇n‖L2+(1+γ)‖ˆn‖L2‖n‖L2≤C(‖n‖12L2‖∇n‖12L2+‖n‖L2)‖c‖H2‖∇n‖L2+(1+γ)‖ˆn‖L2‖n‖L2≤C(‖n‖2L2‖c‖4H2+‖n‖2L2‖c‖2H2+‖ˆn‖L2)+12‖n‖2H1. |
So, we derive that
‖n‖2L2+∫T0‖n‖2H1dt≤C(1+∫T0‖ˆn‖2L2dt). |
Multiplying the first equation of (2.4) by
12ddt∫Ω|∇n|2dx+∫Ω|Δn|2dx+∫Ω|∇n|2dx=∫Ωu∇nΔndx+∫Ω(∇⋅(n∇c)Δn−(1+γ)ˆn+Δn+μˆn+nΔn)dx≤‖u‖L4‖∇n‖L4‖Δn‖L2+‖n‖L4‖Δc‖L4‖Δn‖L2+‖∇n‖L4‖∇c‖L4‖Δn‖L2+(1+γ)‖ˆn‖L2‖Δn‖L2+μ‖n‖L4‖ˆn‖L4‖Δn‖L2≤C‖u‖H1(‖∇n‖12L2‖Δn‖12L2+‖∇n‖L2)‖Δn‖L2+‖n‖L4(‖Δc‖12L2‖∇Δc‖12L2+‖Δc‖L2)‖Δn‖L2+μ‖n‖L4‖ˆn‖L4‖Δn‖L2+(‖∇n‖12L2‖Δn‖12L2+‖∇n‖L2)‖∇c‖H1‖Δn‖L2+(1+γ)‖ˆn‖L2‖Δn‖L2≤12‖Δn‖2L2+C(‖∇n‖2L2+‖n‖4L4+‖Δc‖4L2+‖∇Δc‖2L2+‖ˆn‖2L2+‖ˆn‖4L4)≤12‖Δn‖2L2+C(1+‖∇n‖2L2+‖n‖4L2+‖n‖2L2‖∇n‖2L2+‖∇Δc‖2L2+‖ˆn‖2L2+‖ˆn‖4L4). |
Straightforward calculations yield
‖∇n‖2L2+∫T0∫Ω(|Δn|2+|∇n|2+ˆn+|∇n|2)dxdt≤C. |
Multiplying the first equation of (2.4) by
∫T0∫Ω|nt|2dxdt≤C. |
The proof is complete.
Introduce the spaces
Xu=L4(0,T;L4(Ω)),Xn=L4(0,T;L4(Ω))∩L2(0,T;H1(Ω)),Yu=L∞(0,T;H1(Ω))∩L2(0,T;H2(Ω)),Yn=L∞(0,T;H1(Ω))∩L2(0,T;H2(Ω)). |
Define a map
F:Xu×Xn→Xu×Xn,F(ˆu,ˆn)=(u,n), |
where the
{nt−Δn+u⋅∇n+n=−∇⋅(n∇c)+(1+γ)ˆn+−μˆn+n,in (0,T)×Ω≡Q,ct−Δc+u⋅∇c+c=ˆn++f,in (0,T)×Ω≡Q,ut−Δu+ˆu⋅∇u=−∇π+ˆn∇φ,in (0,T)×Ω≡Q,∇⋅u=0,in (0,T)×Ω≡Q,∂n∂ν=∂c∂ν=0,u=0,on (0,T)×∂Ω,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),in Ω. |
Next, we use fixed point method to prove the local existence of solutions of the problem (1.1).
Lemma 2.6. The map
Proof. Let
From Lemma 2.6,
{nt−Δn+u⋅∇n+n=−∇⋅(n∇c)+α(1+γ)n−μn2,in Q,ct−Δc+u⋅∇c+c=n+αf,in Q,ut−Δu+u⋅∇u=−∇π+αn∇φ,in Q,∇⋅u=0,in Q,∂n∂ν=∂c∂ν=0,u=0,on (0,T)×∂Ω,n(x,0)=n0(x),c(x,0)=c0(x),u(x,0)=u0(x),in Ω. | (3.1) |
In order to prove the existence of solution, we first give some a priori estimates.
Lemma 3.1. Let
‖n‖L1+∫t0(‖n‖L1+‖n‖L2)dτ≤C, | (3.2) |
‖∇u‖2L2+∫t0‖∇u‖2H1dτ≤C, | (3.3) |
‖∇c‖2L2+∫t0‖∇c‖2H1dτ≤C. | (3.4) |
Proof. With Lemma 2.5 in hand, we get
ddt∫Ωndx+∫Ωndx+μ∫Ωn2dx=α(1+γ)∫Ωndx≤μ2∫Ωn2dx+C. |
Solving this differential inequality, we obtain that
‖n‖L1+∫t0(‖n‖L1+‖n‖L2)dτ≤C. |
Multiplying the third equation of (3.1) by
12ddt∫Ωu2dx+∫Ω|∇u|2dx+∫Ωu2dx=α∫Ωn∇φ⋅udx+∫Ωu2dx≤‖n‖L2‖u‖L2+‖u‖2L2≤C(‖n‖2L2+‖u‖2L2). |
Therefore, we see that
‖u‖2L2+∫t0‖u‖H1dτ≤C. |
By the Gagliardo-Nirenberg interpolation inequality, we deduce that
∫t0‖u‖4L4dτ≤C∫t0(‖u‖2L2‖∇u‖2L2d+‖u‖2L2)τ≤‖u‖2L2∫t0‖∇u‖2L2dτ+∫t0‖u‖2L2dτ≤C. |
Multiplying the third equation of (3.1) by
ddt‖∇u‖2L2+‖∇u‖2H1≤C(‖u‖4L4+‖u‖2L4+1)‖∇u‖2L2+C‖n‖2L2+C. |
Thus, we know
‖∇u‖2L2+∫t0‖∇u‖2H1dτ≤C. |
Multiplying the second equation of (3.1) by
12ddt∫Ωc2dx+∫Ω|∇c|2dx+∫Ωc2dx≤‖n‖L2‖c‖L2+α‖f‖L2‖c‖L2. |
Then, we have
Multiplying the second equation of (3.1) by
Further, we have
The proof is complete.
Lemma 3.2. Let
(3.5) |
Proof. We rewrite the first equation of (3.1) as
Multiplying the above equation by
For
For the term
For the rest term
Combining
(3.6) |
Multiplying the second equation of (3.1) by
Straightforward calculations yield
(3.7) |
Combing (3.6) and (3.7), it follows that
Taking
The proof is complete.
Lemma 3.3. Assume
(3.8) |
Proof. Taking the
Here, we note that
From Lemma 2.2 and (3.2), it follows that
As an immediate consequence
(3.9) |
Applying
For
For the term
Along with
(3.10) |
Combining (3.9) and (3.10), it follows that
By choosing
The proof is complete.
Lemma 3.4. Assume
(3.11) |
Proof. Taking the
For the term
For the term
For the term
Combine the estimates about
By taking
Therefore, this proof is complete.
Lemma 3.5. The operator
Proof. Let
Let
Theorem 3.1. Let
(3.12) |
Proof. From Lemmas 3.1, 3.3 and 3.4, it is easy to verify the existence of solution and (3.11). Therefore, we will prove the uniqueness of the solution in the following part. For convenience, we set
(3.13) |
(3.14) |
(3.15) |
(3.16) |
(3.17) |
(3.18) |
Taking the
For the term
For the term
For the term
With the use of estimates
(3.19) |
Taking the
Then, we get
(3.20) |
Taking the
Straightforward calculations yield
(3.21) |
Then, a combination of (3.19), (3.20) and (3.21) yields
By choosing
Applying Gronwall's lemma to the resulting differential inequality, we finally obtain the uniqueness of the solution.
In this section, we will prove the existence of the optimal solution of control problem. The method we use for treating this problem was inspired by some ideas of Guillén-González et al [9]. Assume
Minimize the cost functional
(4.1) |
subject to the system (1.1). Moreover, the nonnegative constants
The set of admissible solutions of optimal control problem (4.1) is defined by
The function space
where
Now, we prove the existence of a global optimal control for problem (1.1).
Theorem 4.1. Suppose
Proof. Along with Theorem 3.1, we conduct that
(4.2) |
According to the definition of
(4.3) |
Observing that
According to the Aubin-Lions lemma [16] and the compact embedding theorems, we obtain
Since
Recalling that
Therefore, we get that
On the other hand, we deduce from the weak lower semicontinuity of the cost functional
Therefore, this implies that
In order to derive the first-order necessary optimality conditions for a local optimal solution of problem (4.1). To this end, we will use a result on existence of Lagrange multipliers in Banach spaces ([20]). First, we discuss the following problem
(5.1) |
where
We consider the following Banach spaces
where
and the operator
which are defined at each point
(5.2) |
The function spaces are given as follows
We see that
(5.3) |
Taking the differentiability of
Lemma 5.1. The functional
(5.4) |
Lemma 5.2. The operator
defined by
Lemma 5.3. Let
Proof. For any fixed
(5.5) |
Next, we use Leray-Schauder's fixed point method to prove the existence of solutions of the problem (5.5), the operator
(5.6) |
The system (5.6) is complemented by the corresponding Neumann boundary and initial conditions. Similar to the proof of Lemmas 2.3, 2.4, 2.5 and 2.6, we conduct that operator
Similar to the proof of Theorem 3.1,
(5.7) |
complemented by the corresponding Neumann boundary and initial conditions.
Taking the
By the Poincaré inequality and Young's inequality, we have
(5.8) |
Taking the
With the Poincaré inequality and Young's inequality in hand, we see that
(5.9) |
Taking the
For the term
For the term
For the term
Therefore, combining
(5.10) |
Taking the
For the term
For the term
For the term
For the term
Therefore, by choosing
(5.11) |
By choosing
Applying Gronwall's lemma to the resulting differential inequality, we obatin
(5.12) |
Taking the
With the use of the Gagliardo-Nirenberg interpolation inequality, we derive
and
For the term
By choosing
(5.13) |
Applying
For the first term
Similarly, for the term
For the rest term
By choosing
(5.14) |
From (5.13) and (5.14), along with
Applying Gronwall's lemma to the resulting differential inequality, we know
Taking the
With the Gagliardo-Nirenberg interpolation inequality in hand, we can estimate
Similar to above estimates, we see
Similarly, we derive
and
For the rest terms, we know
Therefore, Taking
Applying Gronwall's lemma to the resulting differential inequality, we know
Therefore, from Leray-Schauder theorem, we derive the existence of solution for (5.5). Along with the regularity of
Theorem 5.1. Assume that
(5.15) |
where
Proof. With the Lemma 5.3 in hand, we get that
for all
Corollary 5.1. Assume that
(5.16) |
(5.17) |
(5.18) |
which corresponds to the linear system
(5.19) |
subject to the following boundary and final conditions
and the following identities hold
(5.20) |
Proof. By taking
By choosing
Theorem 5.2. Under the assumptions of Theorem 5.1, system (5.19) has a unique weak solution such that
Proof. For convenience, we set
(5.21) |
subject to the following boundary and final conditions
Following an analogous reasoning as in the proof of Lemma 5.3, we omit the process and just give a number of a priori estimates as follows.
Taking the
Then, we have
(5.22) |
Taking the
Thus, we get
(5.23) |
Taking the
As an immediate consequence, we obtain
(5.24) |
Taking the
Therefore, we see that
(5.25) |
Combining (5.22)-(5.25) and taking
Applying Gronwall's lemma to the resulting differential inequality, we know
The proof is complete.
The authors would like to express their deep thanks to the referee's valuable suggestions for the revision and improvement of the manuscript.
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