AN OPTIMAL CONTROL PROBLEM RELATED TO A 3D-CHEMOTAXIS-NAVIER-STOKES MODEL

. In this paper, we study an optimal control problem associated to a 3D-chemotaxis-Navier-Stokes model. First we prove the existence of global weak solutions of the state equations with a linear reaction term on the chemical concentration equation, and an external source on the velocity equation, both acting as controls on the system. Second, we establish a regularity criterion to get global-in-time strong solutions. Finally, we prove the existence of an optimal solution, and we establish a ﬁrst-order optimality condition.


Introduction
The chemotaxis phenomenon describes the movement of cells directed by the concentration gradient of a chemical substance in their environment. One of the most interesting phenomena in chemotaxis corresponds to the movement of cells toward the increasing concentration of a chemical signal which is consumed by cells themselves. Also, has been observed that interactions between cells and the chemical signal with liquid environments play an important role in various biological processes; indeed, it was observed that when bacteria of the species Bacillus subtilis are suspended in water, some spatial patterns may spontaneously emerge from initially almost homogeneous distributions of bacteria [15,48]. A mathematical model to describe this processes was proposed in [46]; it is given by the following system of Partial Differential Equations: n t + u · ∇n = D n ∆n − χ∇ · (n∇c), c t + u · ∇c = D c ∆c − γcn, ρ (u t + (u · ∇)u) = η∆u − ∇π + n∇Φ, ∇ · u = 0, (1.1) where n = n(x, t) ≥ 0, c = c(x, t) ≥ 0, π(x, t) and u(x, t) denote respectively the cell density, the concentration of an attractive chemical signal, the hydrostatic pressure, and the velocity of the fluid at position x ∈ Ω ⊆ R 3 and time t ∈ (0, T ], T > 0. This model describes the interaction between a type of cells (e.g., bacteria), and a chemical signal which is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. The equation for the velocity field u is described by the incompressible Navier-Stokes system with forcing term given by n∇Φ, which represents the effects due to density variations caused by cell aggregation. The parameters χ, D n , D c , ρ, γ and η are positive constants that represent the chemotactic coefficient, the cell diffusion coefficient, the chemical diffusion coefficient, the fluid density, the chemical consumed rate and the viscosity of fluid, respectively.
In this paper we are interested in the mathematical formulation and theoretical analysis of an optimal control problem for a chemotaxis process occurring inside a viscous and incompressible fluid. The control of the velocity, the proliferation of organisms and the concentration of chemicals in diverse environments have significant applications in science and biological processes. In fact, in several applications, the respective biological setting requires to control the proliferation and death of cells, for example, bacterial pattern formation [47,54] or endothelial cell movement and growth in response to a chemical substance known as tumor angiogenesis factor (TAF), which have a significant role in the process of cancer cell invasion of neighboring tissue [7,8,34]. In this paper we propose the analysis of an optimal control problem given by the minimization of a general cost functional subject to constraints, where the state equations are given by the model (1.1). The controls act on the concentration and the velocity equations as follows        n t + u · ∇n = D n ∆n − χ∇ · (n∇c), c t + u · ∇c = D c ∆c − γcn + gc, u t + (u · ∇)u = η∆u − ∇π + n∇Φ + f, ∇ · u = 0, (1.2) in Ω T = Ω × (0, T ), where g and f are given controls acting on the chemical concentration and the fluid velocity, respectively. Notice that in the region of Ω where g ≥ 0 the control acts as a proliferation source of the chemical substance, and inversely, in the region of Ω where g ≤ 0 the control acts as a degradation source of the chemical substance. This kind of control is commonly known as bilinear control. It is worthwhile to remark that, with this kind of bilinear control we are able to guarantee the non-negativity of the unknowns n, c of (1.2); if we consider a distributed control, that is, if we take in the second equation (1.2) g in place of cg, we have to assume that g ≥ 0 in order to guarantee the non-negativity of c. System (1.2) is completed with the following initial and non-flux boundary data where ν denotes the outward unit normal vector to ∂Ω. Thus, the aim of this paper is to analyze an optimal control problem in such a way that any admissible state is a strong solution of (1. where A × B denotes a suitable set of admissible controls and the functions n, c, u are subject to satisfy the state system (1.2)-(1.3). The functionsn,c,ū are given and denote the desired states for density, concentration and velocity, and the parameters α 1 , α 2 > 0 stand for the cost coefficients of the control. The exact mathematical formulation will be given in Section 3. Without loss of generality, from now on we will assume D n = D c = η = 1. We will prove the solvability of the optimal control problem and state the first-order optimality conditions.
In order to study the optimal control problem described above, firstly we need to clarify the existence and uniqueness of a solution for (1.2)-(1.3). The mathematical understanding of the existence and uniqueness of a solution for (1.2)-(1.3) (with f, g = 0) is quite challenging, due to the coupling between the Navier-Stokes equations and the chemotaxis system. Indeed, as is well known, the 3D Navier-Stokes system has not a satisfactory existence theory; it is known the existence of global weak solutions u ∈ L 2 (0, T ; H 1 0 (Ω))∩C w ([0, T ]; L 2 (Ω)), for initial data in L 2 (Ω). However, in contrast to the 2D case, the uniqueness is still an open problem. On the other hand, for the chemotaxis subsystem of (1.2), obtained upon neglecting the fluid interactions, contrary to the 2D case, only certain weak solutions are known to exist globally (cf. [43]). The full chemotaxis-fluid model (1.2)-(1.3) (with f, g = 0) has been analyzed in [4,16,27,44,49,48,50,52,57]. In [48], the existence of global classical solutions was proved in two-dimensional bounded convex domains. The results of [48] were extended to nonconvex domains in [28]. Results of convergence of classical solutions to the corresponding stationary model were analyzed in [51,56]. In [48] was also proved the existence of global weak solutions in bounded three-dimensional convex domains, considering the Stokes system in place of Navier-Stokes equation, that is, neglecting the nonlinear term (u · ∇)u in the fluid equation. The existence of global weak solutions for the full three-dimensional bounded convex domains Chemotaxis Navier-Stokes model was obtained in [50]. There, the author also proved that any eventual energy solution becomes smooth after some waiting time. The existence of weak solutions for (1.2)-(1.3) given in [50] (with f, g = 0) was obtained as the limit of smooth solutions to suitably regularized problems, where appropriate compactness properties are derived on the basis of a priori estimates gained from an energy-type inequality, which combines the standard L 2 dissipation property of the fluid evolution with a quasi-dissipative structure associated with the chemotaxis subsystem. The existence of classical solutions for the regularized problems are obtained by applying a fixed point procedure to ensure local existence and uniqueness of smooth solutions, and deriving a suitable extension criterion for such solutions, by using a priori estimates. A further step toward a qualitative understanding of the chemotaxis-fluid interaction described by (1.2), in terms of the possible effects of the chemotaxis-driven forcing on the motion, and the latter on the distribution of cells was given in [52]. The author shows that any mutual influence will in fact dissappear asymptotically, in that the large-time behavior of solutions is essentially governed by the decoupled chemotaxis only and Navier-Stokes subsystems on neglecting the components u and [n, c], respectively. Variants of (1.2)-(1.3) including nonlinear cell diffusion, that is, replacing ∆n by the porous medium-type diffusion term ∆ m n for m > 1, and assuming that χ is a chemotactic sensitivity tensor, that is, replacing χ∇ · (n∇c) by ∇ · (nS(n, c, x)∇c), where S(n, c, x) = (s i,j ) N ×N is a matrix-valued function, have been analyzed in [4,16,27,44,49,53,57] and some references therein. In these cases, convenient boundary conditions are assumed. System (1.2)-(1.3) with logistic source (f (n) = ςn − µn 2 with µ > 0), has been analyzed in [33]. Considering bounded domains of R 3 , in [33] the author analyzed the existence of weak solutions and proved that the weak solutions become smooth after some waiting time; moreover, convergence to the steady state (ς/µ, 0, 0) was also established. In [3], it was considered an exchange of oxygen between the fluid and its environment, which leads to a different boundary conditions to (1.2); then, by requiring sufficiently smooth initial data, it was proved the existence of a unique global classical solution for N = 2, as well as the existence of a global weak solution for N = 3. Concerning smooth bounded domains and initial data in L p -spaces, results about local and global-in-time existence of solutions were obtained in [17]. On the other hand, in the whole space R N , N = 2, 3, the local and global-in-time existence of solutions for (1.2) has been studied in [11,12,13,29,55], and some references therein. In particular, a class of small global solutions in weak-L p spaces including double attraction nonlinear terms in the density equation was obtained in [29]. In [55] the local well-posedness for initial data in the nonhomogeneous Besov spaces class B s p,r × B s+1 p,r × B s+1 p,r where 1 < p < ∞, 1 ≤ r ≤ ∞ and s > 3/p + 1 was analyzed. In [13] the local-in-time existence of solutions for large initial data, as well as global-in-time existence for small initial data and some smallness condition on the gravitational term, were obtained in critical homogeneous Besov spaces. Later, an extension criterion for local-in-time solutions was proved in [11]. In [12] the global-in-time existence of solutions for small initial data in the critical triple of homogeneous Besov spacesḂ −2+3/p p,1 ×Ḃ 3/p p,1 ×Ḃ −1+3/p p,1 with 1 ≤ p < 3 was proved. More recently, in [18,20] results on existence and asymptotic behavior of small global solutions in Besov-Morrey spaces were obtained.
However, the derivation of the energy-like estimates when f, g = 0 in (1.2) do not follow directly due principally to the term gc in the derivation of energy-type estimates and uniform estimates for the chemical concentration, by considering a nonregular control g ∈ L p (L p ) for some p > 1. (see details in Section 3). Therefore, the first aim of this paper is to prove the existence of weak solutions for (1.2). In comparison with the results of [50], our definition of weak solution establishes that the equation for the chemoattractant is satisfied a.e. in Ω × (0, T ), that is, our weak solution is less weak than the one in [50]. Moreover, following [28,33], we do not assume the convexity condition on Ω as required in [48,50]. In order to prove the existence of weak solutions of the control system we consider a family of regular solutions to a suitable regularized problem, which is a little different from that considered in [50]. Indeed, we introduce a decoupling through an auxiliary elliptic problem which allows to gain regularity for the chemical equation as well as to obtain an energy-type inequality after testing and combining conveniently the density equation and the concentration equation (see details in Section 3.1).
In connection to the existence of weak solutions, it is worthwhile to remark that the uniqueness of weak solutions in 3D is an open problem; consequently, due to the lack of uniqueness of weak solutions nor the existence of strong solutions, establishing optimality conditions when analyzing control problems becomes nontrivial. As a consequence, most of the studies devoted to the control problems in fluid mechanics and related models assume that Ω ⊂ R 2 (cf. [2,5]). In order to overcome this difficulty, the second aim of this paper is to introduce a regularity criterion which allow us to get a unique strong solution (cf. Section 4). This regularity criterion is motivated by the corresponding one for the Navier-Stokes system [2,5].
On the other hand, having decided what kind of solutions of the Chemotaxis-Navier-Stokes equations (1.2)-(1.3) are going to be considered, the third aim of this paper is to analyze an optimal control problem. Here we recall that in past years, significant progress has been made in mathematical analysis and numerics of optimal control problems for viscous flows described by the Navier-Stokes equations and related models (see for instance, [2,5,22,35] and references therein). However, from the optimal control point of view, the literature related is scarce, and most of the results are devoted to the control theory governed by chemotaxis models without fluids (cf. [14,21,23,24,25,39]). In these references, the authors proved the existence of optimal controls and derived an optimality systems. However, as far as we know, optimal control problems for evolutive attractive chemotaxis-Navier-Stokes models have not been studied previously. In [38] the authors analyze an optimal distributed control problem where the state equations are given by a stationary chemotaxis model coupled with the Navier-Stokes equations. The control was given through a distributed force and a coefficient of chemotactic sensitivity, leading the chemical concentration, the cell density, and the velocity field towards a given target concentration, density and velocity, respectively. In spite of, we observe that in [9,10] the authors obtained some results related to the controllability for the nonstationary Keller-Segel system and the nonstationary chemotaxis-fluid model with consumption of chemoattractant substance associated to a chemotaxis system, based on Carleman estimates for the solutions of the adjoint system. We formulate the control problem in such a way that any optimal state is a strong solution, and prove the existence of an optimal solution assuming controls on the concentration and velocity, such that the associated strong solution exists.
The rest of the paper is organized as follows: in Section 2 we introduce some basic notations and preliminary results. In Section 3, we give the definition of weak solutions for system (1.2)-(1.3) and prove that they exist indeed. In Section 4, we establish a regularity criterion under which weak solutions of (1.2)-(1.3) are also strong solutions. In Section 5, we analyze an optimal control problem related to the strong solutions of (1.2)-(1.3); we prove the existence of an optimal solution and derive first-order optimality conditions.
Along this work, we are going to fix a time 0 < T < ∞ but arbitrary. In this sense, we introduce the following notation. For Y being a Banach space, L p (Y ) := L p (0, T ; Y ), 1 ≤ p ≤ ∞, denotes the space of Bochner integrable functions defined on the interval [0, T ] with values in Y , endowed with the usual norm · L p (Y ) . We also consider the space The topological dual of a Banach space Y will be denoted by Y , and the duality product by ·, · Y or simply ·, · when there is no confusion. For simplicity in the notation, we use u for both scalar and vector valued functions. Also, the letter C will denote a general positive constant which may change from line to line or even within the same line.
We will denote by X p the space

Existence of weak solutions
In this section we prove the existence of global weak solutions for the control system (1.2)-(1.3). We obtain the existence of weak solutions as the limit of smooth solutions to suitable regularized problems, where appropriate compactness properties are derived from a priori estimates gained from an energy-type inequality. We start by establishing the notion of weak solutions for (1.2)-(1.3).
In order to obtain the existence of weak solutions, we consider the following assumption on the data: We prove the following theorem of global existence of weak solutions.   3) is obtained through as the limit of a sequence of strong solutions of the following approximated problems: given > 0, find [n , w , u , π ] such that: in Ω × (0, T ), where Y = (1 + A) −1 is the Yosida approximation [42], and c is the unique solution of the elliptic problem c − ∆c = w in Ω, In (3.2), n + = max{n , 0} ≥ 0, w + = max{w , 0} ≥ 0, and for all ∈ (0, 1), n 0, , c 0, , u 0, are initial data satisfying: Remark 3.2. Approximation (3.2) differs from that considered in [50], in the following aspects. First, we decouple the cross-diffusion term by introducing the elliptic problem (3.3). This allows enhancing regularity for the new variable c , which, in particular, enables the use of the Leray-Schauder fixed point theorem. This procedure is not required in [50] due to the use of the semigroup theory to get existence of local strong solutions. However, in our case, we were able to consider a different class of initial data providing a new class of global weak solutions, weaker than those in [50, Definition 2.1]. Second, our approximation includes some non-negative terms at the right hand side of (3.2), in order to guarantee the non negativeness of n , w . Approximation in the w -equation allows to cancel the cross-diffusion in the nequation with the consumption term in the w -equation. Finally, the Yosida approximation for the u -equation was the regularization for the velocity used in [50].
The existence of strong solutions for (3.2)-(3.4) is given in the next proposition.
Proof. The proof will be carried out through the Leray-Schauder fixed point theorem. For that let us define the Banach space X = L ∞ (L 2 ) ∩ L 2 (H 1 ) and the operator Γ : X × X → X × X by Γ([n ,w ]) = [n , w ], where n , w are the two first components of the solution [n , w , u , π ] for the following system: withc being the unique solution of (3.3) and right hand sidew (note that the first and second equations are decoupled between them in system (3.5)). We divide the proof in four steps.
Using the last estimate we are going to bound n in X. Testing (3.8) 1 by n and integrating in Ω we get Since c is the solution of (3.3) and w ∈ X, we know that In particular Then, by applying the Gronwall inequality in (3.10) we get n X ≤ C( ).
Step four. Γ : X × X → X × X is continuous.
In particular [n m ,w m ] m∈N is bounded in X × X, and by (3.6), (3.7), [n m , w m ] = Γ[n m ,w m ] is bounded in X 5/3 × X 5/3 . Then there exists a subsequence, still denoted by [n m , w m ] m∈N , and a pair of functions [n ,ŵ ] ∈ X 5/3 × X 5/3 such that (3.12) and the corresponding elliptic problem is satisfied As we did in step 1, sincen m ∈ X, from (3.12) 3 we have u m ∈ X 2 and Also, testing (3.12) 3 by Au m , we get Then, from last two inequalities we get From (3.14), (3.15), there exists u ∈ X 2 such that u m → u , weakly in X 2 and strongly in X.
Therefore, from (3.11) and (3.16) we can pass to the limit, as m → ∞, in system (3.12). Strong estimates for n m and w m in X 5/3 allow us to pass to the limit in (3.12) 1,2 . In order to pass to the limit in the velocity equation it is enough to see that wherec is the unique solution of the elliptic problem (3.3) with right hand sidew . Namely, [n ,ŵ ] = Γ[n ,w ]. We have proved that, for an arbitrary sequence [n m , This implies the continuity of Γ.
From the previous steps the operator Γ satisfies the hypothesis of the Leray-Schauder fixed point theorem. Thus, we conclude that the map Γ has a fixed point [n , w ], that is, Γ[n , w ] = [n , w ], which provides a solution of system (3.2).
Step five. Uniqueness of the solution. We proceed as follows. First, observe that where we have used the classical Sobolev embedding and Lemma 7.
3.2. Uniform estimates. The aim of this section is to find proper estimates, independent of , allowing the pass to the limit in (3.2)-(3.3).
Next, we are going to derive a quasi-energy estimate for system (3.2), which is the key for deriving strong convergences allowing the pass to the limit as → 0. For that we test system (3.2) with suitable functions in such a way that the regularized chemotaxis term is absorbed by the regularized consumption signal. First, multiplying (3.2) 1 by ln n , integrating on Ω, and using the mass conservation property for n (we already proved n + = n ) gives , for any > 0 on (0, T ). By using (3.2) 2 , and following Lemma 2.8 in [33] one has By Lemma 2.7(vi) in [33], there exist k 1 > 0, k 2 > 0 such that for any > 0: Having into account identity ∆w = w ∆ ln w + |∇w | 2 w , we have Moreover, following the proof of Lemma 2.8 in [33], there exists k 3 > 0 such that for any > 0: 2 Then, putting together (3.30)-(3.33), we obtain where we have used the uniform bound to w provided by Lemma 3.4 (observe that, for q > 3, w 0, ∈ W 1,q (Ω) → L ∞ (Ω)). Note that 2γ Ω ∇(∆c ) · ∇n 1 + n = 2γ 1/2 Ω ∇(∆c ) · ∇n Now, for the velocity u we have the following estimate: there exist δ 3 , C > 0 such that for every > 0 Indeed, multiplying (3.2) 3 by u , and integrating in Ω Moreover, since n ∈ L 1 (Ω), by the Gagliardo-Niremberg inequality Finally, by using Lemma 3.4, we get and accordingly. Recalling that ∇w L 2 (L 2 ) ≤ C uniformly in (see (3.28)), then, by (3.41) we have ∇c L 2 (L 2 ) ≤ C uniformly in . Moreover, √ ∆c is bounded in L 2 (L 2 ) uniformly in . Thus, integrating in time (3.39), in particular, we obtain that and since Ω |∇w | 2 = Ω |∇w | 2 w w and w is uniformly bounded (Lemma 3.4), we get Moreover, taking ∆ in (3.3), and testing by ∆c , give us 1 2 Therefore, by (3.43) {c } is bounded in L 2 (H 2 ), { √ ∆c } is bounded in L 2 (H 1 ).

Regularity criterion
In this section we are going to prove that under the extra assumption [n, u] ∈ L 20/7 (L 20/7 ) × L 8 (L 4 ), weak solutions have better regularity properties. This is possible by a bootstrapping argument. It is worth noting that this extra regularity assumption is going to be obtained from the definition of the functional in the control problem, providing the relation between the solutions of the PDE, the cost functional and the existence of optimal solutions.
Proof. The idea is to use a bootstrapping argument following the ideas of [24]. From Theorem 3.2, there exists a weak solution [n, c, u] of system (1.2) in the sense of Definition 3.1. The regularity is carried out into several steps: Step 1: u ∈ X 2 . Since u ∈ L 8 (L 4 ), from Theorem 4.2.1 in [5], we have u ∈ X 2 → L 10 (L 10 ). The last embedding is consequence of Lemma 7.3 below, applied to q 1 = 6, p 1 = p 2 = 2, r = 2. Moreover, Step 2: c ∈ X 20/7 . As a general fact in the following steps, n ∈ L 20/7 (L 20/7 ) and u ∈ L 10 (L 10 ). Moreover, from the weak regularity we have c ∈ X 2 → L 10 (L 10 ).
Notice that we cannot go further in the regularity of c because cn cannot be improved beyond L 20/7 (L 20/7 ). That is why we work now to improve the regularity of n, and go back later again on the regularity of c.
Notice that the regularity of c cannot be improved, because g ∈ L 4 (L 4 ) by hypothesis.
Remark 4.1. Notice that, following the proof above, if we assume g ∈ L 10/3 (L 10/3 ), we can get a strong solution in the class X 5/3 × X 10/3 × X 2 .
Before formulating the optimal control problem, let us state the following result concerning a "linearized" version of System (1.2).
, and [n, c, u] ∈ X 4 × X 4 × X 2 with div u = 0 in Ω T . Then, there exists a unique element [n,c,ū] ∈ X 4 × X 4 × X 2 and some π ∈ L 2 (H 1 ) solution of the following problem: for some π ∈ L 2 (H 1 ), which is unique up to the addition of a function of L 2 (0, T ).
Proof. The idea is to use the Leray-Schauder fixed-point theorem. Indeed, let X = L ∞ (L 2 ) ∩ L 2 (H 1 ) and Γ be the operator where [n,c] is the solution of the coupled system (4.14) Following [6], Theorem 4.2.1, since v ∈ L 2 (L 2 ) and n d ∈ X → L 10/3 (L 10/3 ), there exists a unique solutionū ∈ X 2 → L 10 (L 10 ) of (4.14) 3 . This impliesū · ∇c ∈ L 20/3 (L 20/3 ) and we can follow the ideas in the proof of Proposition 3.3 to prove the existence of a unique solution [n,c] ∈ X × X.
Step 5:n ∈ X 4 . The proof follows Step 7 in the proof of Theorem 4.2.
Thus, following the ideas in [6], we conclude that (4.15) is satisfied for some π ∈ L 2 (H 1 ). In this sense, from Lemma 4.3,

∂G ∂[n, c, u]
[n, c, u, g, f ] : As a consequence, the set of controls [g, f ] ∈ L 4 (L 4 ) × L 2 (L 2 ) for which there exists a strong solution [n, c, u] is open. This set will be denoted by A × B.

The control problem
In this section, we define the bilinear optimal control problem. We derive the optimality system and prove the existence of one global optimal solution. As is explained in [5,6,24], allowing the weak solutions of (1.2) to be admissible states imply the optimality system can not be obtained. Indeed, the lack of uniqueness of solutions for the state system imply, eventually, that the Gateaux derivative of the functional is not uniquely determined. That is why the control problem is formulated in such a way that any optimal state is a strong solution of (1.2).
For the rest of this work we fix a finite time T > 0. We recall the following notation L p (Y ) := L p (0, T ; Y ). Let  The choose of the L 20/7 (L 20/7 ) and L 8 (L 4 ) norms for n and u, respectively, in the cost functional, is made in line with Theorem 4.2. That is, if [n, c, u] is a weak solution of (1.2) in (0, T ) such that J([g, f ]) < ∞, then [n, u] ∈ L 20/7 (L 20/7 ) × L 8 (L 4 ) and, by Theorem (4.2), [n, c, u] ∈ X 4 × X 4 × X 2 is a strong solution of (1.2) in (0, T ). In what follows, we will make the following assumption: Remark 5.1. The hypothesis (5.2) holds. Indeed, following [41], given > 0, there exist f ∈ L 2 (L 2 ) with f L 1 (L 1 ) < , and a unique solution u ∈ X 2 of Then, taking u ∈ X 2 the solution of (5.3), we solve the equation c t − ∆c = −u · ∇c, with Neumann boundary condition ∂c ∂ν = 0, and initial data c(0) = c 0 . In fact, it is clear the existence of a unique weak solution c ∈ L ∞ (L 2 ) ∩ L 2 (H 1 ) of this initial and boundary value problem.
By the regularity of [n d , c d , u d ], and as a consequence of Corollary 4.4, we state now the following differentiability property for the functional J.
In order to derive the dual system, let us multiply last system, respectively, by test functions [p, q, r] such that lim |x|→∞ [p, q, r] = 0. After integrating in space and time, recalling that (·, ·) denotes the product in L 2 we get      Observe that the regularity assumed on the objective [n d , c d , u d ] and the regularity [n, c, u] ∈ X 4 × X 4 × X 2 imply |n − n d | 6/7 (n − n d ), u − u d 4 L 4 |u − u d | 2 (u − u d ) ∈ L 2 (L 2 ). The next theorem establishes the existence of at least one solution of the optimal control problem (P), as well as the first order optimality conditions satisfied by any local minimum of the optimization problem.

Conclusions
In this paper, we study an optimal control problem associated to a 3D-chemotaxis-Navier-Stokes model. First we prove the existence of global weak solutions of the state equations with a linear reaction term on the chemical concentration equation, and an external source on the velocity equation, both acting as controls on the system. Second, we establish a regularity criterion to get global-in-time strong solutions. Finally, we prove the existence of an optimal solution, and we establish a first-order optimality condition. Some open questions and future work in this direction may be addressed; for example, the derivation of second order necessary and sufficient optimality conditions, the proposal of a numerical scheme to approximate the control problem in three dimensions providing some error estimates for the difference between locally optimal controls and their discrete approximations.

Appendix
In order to get some regularity results, frequently we use the following embedding results Lemma 7.1. Let p, q, p 1 , p 2 , q 1 , q 2 ≥ 1 verifying Then, L p 1 (L q 1 ) ∩ L p 2 (L q 2 ) → L p (L q ). Then, L p 1 (H s 1 ) ∩ L p 2 (H s 2 ) → L p (H s ).