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A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem

In this paper we study a bilinear optimal control problem associated to a 3D chemo-repulsion model with linear production. We prove the existence of weak solutions and we establish a regularity criterion to get global in time strong solutions. As a consequence, we deduce the existence of a global optimal solution with bilinear control and, using a Lagrange multipliers theorem, we derive first-order optimality conditions for local optimal solutions.


Introduction
The chemotaxis phenomenon is understood as the directed movement of live organisms in response to chemical gradients. Keller and Segel [18] proposed a mathematical model that describes chemotactic aggregation of cellular slime molds which move preferentially towards relatively high concentrations of a chemical substance secreted by the amoebae themselves, which is called chemo-attraction with production. When the regions of high chemical concentration generate a repulsive effect on the organisms, the phenomenon is called chemo-repulsion.
In this work we study an optimal control problem subject to a chemo-repulsion with linear production system in which a bilinear control acts injecting or extracting chemical substance on a subdomain of control Ω c ⊂ Ω. Specifically, we consider Ω ⊂ R 3 be a simply connected bounded domain with boundary ∂Ω of class C 2 and (0, T ) a time interval, with 0 < T < +∞. Then we study a control problem related to the following system in the time-space domain Q := (0, T ) × Ω, with initial conditions and non-flux boundary conditions where n denotes the outward unit normal vector to ∂Ω. In (1), the unknowns are the cell density u(t, x) ≥ 0 and chemical concentration v(t, x) ≥ 0. The function f = f (t, x) denotes a bilinear control acting in the chemical equation. We observe that in the subdomains of Ω where f ≥ 0 the chemical substance is injected, and conversely where f ≤ 0 the chemical substance is extracted.
System (1)-(3) without control (i.e. f ≡ 0) has been studied in [10], [32]. In [10], the authors proved the global existence and uniqueness of smooth classical solutions in 2D domains, and global existence of weak solutions in dimension 3 and 4. In [32], on a bounded convex domain Ω ⊂ R n (n ≥ 3), it is proved that a modified system of (1)-(3), changing the chemotactic term ∇ · (u∇v) by ∇ · (g(u)∇v) with an adequate density-dependent chemotactic function g(u), has a unique global in time classical solution. This result is not applicable in our case, because g(u) = u does not satisfies the hypothesis imposed in [32].
There is an extensive literature devoted to the study of control problems with PDEs, see for instance [2,6,7,17,19,21,24,25,31,35] and references therein. In all previous works, the control is of distributed or boundary type. As far as know, the literature related to optimal control problems with PDEs and bilinear control is scarce, see [4,13,16,20,34].
In the context of optimal control problems associated to chemotaxis models, the literature is also scarce. In [13,29] a 1D problem is studied. In [13] the authors analyzed two problems for a chemoattractant model. The bilinear control acts on the whole Ω in the cells equation. The existence of optimal control is proved and an optimality system is derived. Also, a numerical scheme for the optimality system is designed and some numerical simulations are presented. In [29] a boundary control problem for a chemotaxis reaction-diffusion system is studied. The control acts on the boundary for the chemical substance, and the existence of optimal solution is proved. A distributed optimal control problem for a two-dimensional model of cancer invasion has been studied in [11], proving the existence of optimal solution and deriving an optimality system. Rodríguez-Bellido et al. [27] study a distributive optimal control problem related to a 3D stationary chemotaxis model coupled with the Navier-Stokes equations (chemotaxis-fluid system). The authors prove the existence of an optimal solution and derive an optimality system using a penalty method, taking into account that the relation control-state is multivalued. Ryu and Yagi [28] study an extreme problem for a chemoattractant 2D model, in which the control variable is distributed in the chemical equation. They prove the existence of optimal solutions, and derive an optimality system, using the fact that the state is differentiable with respect to the control. Other studies related to controllability for the nonstationary Keller-Segel model and nonstationary chemotaxisfluid system can be consulted in [8] and [9], respectively.
In [16], an optimal bilinear control problem related to strong solutions of system (1)-(3) in 2D domains was studied, proving the existence and uniqueness of global strong solutions, and the existence of global optimal control. Moreover, using a Lagrange multiplier theorem, first-order optimality conditions are derived. Now, this paper can be seen as a 3D version of [16]. In fact, similarly to [16], the main objective now is to prove the existence of global optimal solutions and to derive optimality conditions, which will be more complicated because the PDE system is considered in 3D domains. In this case, we distinguish two different types of solutions: weak and strong. The existence of weak solutions can be obtained under minimal assumptions (see Theorem 1). However, such result is not sufficient to carry out the study of the control problem, due to the lack of regularity of weak solutions. In order to overcome this problem, we introduce a regularity criterion that allows to obtain a (unique) strong solution of (1)-(3) (see Theorem 3). As far as we know, there are no results of global in time regularity of weak solutions of system (1)-(3) in 3D domains. This is similar to what happens with the Navier-Stokes equations (see [33]).
In this work, we deal with strong solutions of (1)-(3) which allows us to analyze the control problem. However, we are going to prove the existence of an optimal control associated to strong solutions, assuming the existence of controls such that the associated strong solution exists. Following the ideas of [6,7], we consider a regularity criterion in the objective functional such that any weak solution of (1)-(3) with this regularity is also a strong solution.
The paper is organized as follow: In Section 2, we fix the notation, introduce the functional spaces to be used and we state a regularity result for linear parabolic-Neumann problems that will be used throughout this work. In Section 3 we give the definition of weak solutions of (1)-(3) and, by introducing a family of regularized problems related to (1)-(3) (its existence is deduced in the Appendix) and passing to the limit, prove the existence of weak solutions of system (1)-(3). In Section 4 we give the definition of strong solutions of (1)-(3), and we establish a regularity criterion under which weak solutions of (1)-(3) are also strong solutions. Section 5 is dedicated to the study of a bilinear control problem related to strong solutions of system (1)-(3), proving the existence of an optimal solution and deriving the first-order optimality conditions based on a Lagrange multipliers argument in Banach spaces. Finally, we obtain a regularity result for these Lagrange multipliers.

Preliminaries
We will introduce some notations. We will use the Lebesgue space L p (Ω), 1 ≤ p ≤ +∞, with norm denoted by · L p . In particular, the L 2 -norm and its inner product will denoted by · and (·, ·), respectively. We consider the usual Sobolev spaces W m,p (Ω) = {u ∈ L p (Ω) : ∂ α u L p < +∞, ∀|α| ≤ m}, with norm denoted by · W m,q . When p = 2, we write H m (Ω) := W m,2 (Ω) and we denote the respective norm by · H m . Also, we use the space W m,p n (Ω) = {u ∈ W m,p (Ω) : ∂u ∂n = 0 on ∂Ω} (m ≥ 2) and its norm denoted by · W m,p n . If X is a Banach space, we denote by L p (X) the space of valued functions in X defined on the interval [0, T ] that are integrable in the Bochner sense, and its norm will be denoted by · L p (X) . For simplicity we denote L p (Q) := L p (0, T ; L p ) and its norm by · L p (Q) . We also denote by C([0, T ]; X) the space of continuous functions from [0, T ] into a Banach space X, whose norm is given by · C(X) . The topological dual space of a Banach space X will be denoted by X ′ , and the duality for a pair X and X ′ by ·, · X ′ or simply by ·, · unless this leads to ambiguity. Moreover, the letters C, K, C 0 , K 0 , C 1 , K 1 ,..., denote positive constants, independent of state (u, v) and control f , but its value may change from line to line.
In order to study the existence of solution of system (1)-(3), we define the space and we will often use the following regularity result for the heat equation (see [12, p. 344]). Lemma 1. Let 1 < p < +∞, u 0 ∈ W 2−2/p,p (Ω) and g ∈ L p (Q). Then the problem admits a unique solution u such that Moreover, there exists a positive constant C := C(p, Ω, T ) such that For simplicity, in what follows we will use the following notation and its norm will be denoted by · Xp . In fact, u ∈ X p iff u ∈ W 2,1 Throughtout this paper, we will use the following equivalent norms in H 1 (Ω) and H 2 (Ω), respectively (see [26] for details): and the classical interpolation inequality in 3D domains Remark 1. The problem (1)-(3) is conservative in u, because the total mass Ω u(t) remains constant in time. In fact, integrating (1) 1 in Ω we have 3 Existence of Weak Solutions of Problem (1)-(3) the following variational formulation holds for the u-equation the v-equation (1) 2 holds pointwisely a.e. (t, x) ∈ Q, and the initial and boundary conditions for v (2) 2 -(3) 2 are satisfied. The space X u given in (9) is defined as follow X u = {u ∈ L 10 (W 1,10 ) : ∂ t u ∈ L 5/2 (Q) and u(T ) = 0 in Ω}.

Remark 2.
This definition of weak solution implies, in particular, that Also, each term of (9) has sense. In particular, from (7)-(8) one has that u∇v ∈ L 10/9 (Q). The proof of this theorem follows from the two next subsections.

Regularized Problem
In order to prove Theorem 1, we will study the following family of regularized problems related to system (1)-(3), for any ε ∈ (0, 1). Given an adequate regularization where v ε := v(z ε ) is the unique solution of the problem and v + := max{v, 0} ≥ 0.
We choose the initial conditions u ε 0 and v ε 0 , with u ε In the remaining of this section, we will denote v(z ε ) only by v ε .
The proof of Theorem 2 is carried out in the Appendix.
Therefore, taking to the limit in the regularized problem (10), as ε → 0, and taking into account (12), (19), (21) and (23) we conclude that (u, v) satisfies the weak formulation Integrating by parts in (25), and using that u ∈ L 5/3 (Q) and v ∈ L 2 (H 2 ), we deduce that v is the unique solution of the problem Finally, we will check the positivity of (u, v). Indeed, the positivity of u follow from (20) and the fact that u ε ≥ 0 a.e. (t, x) ∈ Q (see Lemma 10 in the Appendix). In order to check that v ≥ 0, we

Regularity Criterion
In this section we will give a regularity criterion of system (1)-(3).
(Ω) and the following regularity criterion holds then (u, v) is a strong solution of (1)-(3) in sense of Definition 3. Moreover, there exists a positive The proof of this theorem follows from the two next subsections.

Interpolation and embedding results
In order to proof Theorem 3, starting from the regularity of u and v, we will get the regularity for ∇ · (u∇v) which improves the regularity for u. With this new regularity for u, the regularity for ∇ · (u∇v) is improved several times using a bootstraping argument. Along the proof of Theorem 3, different interpolation results will be used together with some embeddings results that will be stated below.
As a consequence of the interpolation inequality we have the following result Then, Using the Sobolev embedding where N is the space-dimension and the Gagliardo-Nirenberg inequality (see [14,Theorem 10.1]) we deduce the following result Then, Then, Then,

The Optimal Control Problem
In this section we establish the statement of the bilinear control problem. Following [6,7], we formulate the control problem in such a way that any admissible state is a strong solution of (1)- where Ω c ⊂ Ω is the control domain. We consider data u 0 ∈ H 1 (Ω), v 0 ∈ W

3/2,4 n
(Ω) with u 0 ≥ 0 and v 0 ≥ 0 in Ω, and the function f ∈ F describing the bilinear control acting on the v-equation.
Here (u d , v d ) ∈ L 26/7 (Q) × L 2 (Q) represent the desires states (see the beginning of the proof of Theorem 7 below to justify the regularity required for u d ∈ L 26/7 (Q)) and the real numbers α u , α v and α f measure the cost of the states and control, respectively. These numbers satisfy The admissible set for the optimal control problem (44) is defined by  From the definition of J and the assumption α f > 0 or F is bounded in L 4 (Q c ), it follows that

Existence of Global Optimal Solution
and {u m } m∈N is bounded in L 20/7 (Q).
From (29) there exists a positive constant C, independent of m, such that Therefore, from (47), (48), and taking into account that F is a closed convex subset of L 4 (Q c ) (hence is weakly closed in L 4 (Q c )), we deduce that there existss = (ũ,ṽ,f ) ∈ X 2 × X 4 × F such that, for some subsequence of {s m } m∈N , still denoted by {s m } m∈N , the following convergences hold, as m → +∞: v m →ṽ weakly in L 4 (W 2,4 ) and weakly* in f m →f weakly in L 4 (Q c ), andf ∈ F.

Optimality System Related to Local Optimal Solutions
We will derive the first-order necessary optimality conditions for a local optimal solution (ũ,ṽ,f ) of problem (44), applying a Lagrange multipliers theorem. We will base on a generic result given by Zowe et al [36] on the existence of Lagrange multipliers in Banach spaces. In order to introduce the concepts and results given in [36] we consider the following optimization problem where J : X → R is a functional, G : X → Y is an operator, X and Y are Banach spaces, M is a nonempty closed convex subset of X and N is a nonempty closed convex cone in Y with vertex at the origin. The admissible set for problem (59) is defined by For a subset A of X (or Y), A + denotes its polar cone, that is Definition 5. (Lagrange multiplier) Letx ∈ S be a local optimal solution for problem (59). Suppose that J and G are Fréchet differentiable inx, with derivatives J ′ (x) and G ′ (x), respectively. Then, any ξ ∈ Y ′ is called a Lagrange multiplier for (59) at the pointx if where C(x) = {θ(x −x) : x ∈ M, θ ≥ 0} is the conical hull ofx in M.
Definition 6. Letx ∈ S be a local optimal solution for problem (59). We say thatx is a regular point if Now, we will reformulate the optimal control problem (44) in the abstract setting (59). We consider the following Banach spaces and the operator G = (G 1 , G 2 , G 3 , G 4 ) : X → Y, where Thus, the optimal control problem (44) is reformulated as follows where and F is defined in (43).
We observe that M is a closed convex subset of X, N = {0} and the set of admissible solutions is rewritten as Concerning to the differentiability of the constraint operator G and the functional J we have the following results.
Lemma 7. The operator G : X → Y is Fréchet differentiable and the derivative of G ins = (ũ,ṽ,f ) ∈ X in the direction r = (U, V, F ) ∈ X is the linear operator We wish to prove the existence of Lagrange multipliers, which is guaranteed if a local optimal solution of problem (63) is a regular point of operator G (in virtue of Theorem 5). we conclude thats = (ũ,ṽ,f ) ∈ S ad is a regular point if for any (g u , g v , U 0 , V 0 ) ∈ Y there exists Lemma 8. Lets = (ũ,ṽ,f ) ∈ S ad (S ad defined in (64)), thens is a regular point.
Since (67) is a linear system we argue in a formal manner, proving that any regular enough solution is bounded in W u × W v . A detailed proof can be made by using, for instance, a Galerkin method.
Testing Using the Hölder and Young inequalities on the terms on the right side of (68) and taking into account (6) we obtain On the other hand, testing by V in (67) 2 we obtain Summing the inequalities (68) and (75), and then adding U 2 to both sides of the inequality obtained, and taking into account (69)-(74), for δ small enough, we have From (76) and Gronwall lemma we deduce that there exists a positive constant C that depends on In particular, from (77) we obtain that (U, V ) ∈ L 10/3 (Q) × L 10 (Q), and sincef ∈ L 4 (Q c ) we havẽ f V χ Ωc ∈ L 20/7 (Q). Then, applying Lemma 1 (for p = 20/7) to (67) 1 , we deduce that V ∈ X 20/7 .
Remark 7. Using a classical comparison argument, inequality (6) and Gronwall lemma, the uniqueness of solutions of system (67) is deduced.
Proof. Since the desired state In fact,ũ is more regular because assumingũ ∈ L 20/7 (Q), it can be proved thatũ ∈ L ∞ (H 1 ) ∩ L 2 (H 2 ) ֒→ L 10 (Q) (see the proof of the Theorem 3 for more details).
In the following result, we obtain more regularity for the Lagrange multiplier (λ, η) than provided by Theorem 6.
Thus, the controlf is given byf Appendix: Existence of Strong Solutions of Problem (10) In this appendix we will prove Theorem 2.
Lemma 9. The operator R : X × X → X × X is well defined and compact.
Integrating (121) in (0, T ) we have Thus, from (122) and (123) we deduce that u ε is bounded in X . Consequently, the fixed points of αR are bounded in X × X , independently of α > 0. For α = 0 the result is trivial.