A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem
Francisco Guillén-González, Exequiel Mallea-Zepeda, María Ángeles Rodríguez-Bellido
aa r X i v : . [ m a t h . O C ] A ug A regularity criterion for a 3D chemo-repulsion system and itsapplication to a bilinear optimal control problem
F. Guillén-González , E. Mallea-Zepeda , M.A. Rodríguez-Bellido , Dpto. Ecuaciones Diferenciales y Análisis Numérico and IMUS Universidad de Sevilla,Facultad de Matemáticas, C/ Tarfia, S/N, 41012, Spain Departamento de Matemática, Universidad de Tarapacá, Arica, Chile
Abstract
In this paper we study a bilinear optimal control problem associated to a 3D chemo-repulsionmodel with linear production. We prove the existence of weak solutions and we establish aregularity criterion to get global in time strong solutions. As a consequence, we deduce theexistence of a global optimal solution with bilinear control and, using a Lagrange multiplierstheorem, we derive first-order optimality conditions for local optimal solutions.
Keywords:
Chemo-repulsion and production model, weak solutions, strong solutions, bilinearoptimal control, optimality conditions.
The chemotaxis phenomenon is understood as the directed movement of live organisms in response tochemical gradients. Keller and Segel [18] proposed a mathematical model that describes chemotacticaggregation of cellular slime molds which move preferentially towards relatively high concentrationsof a chemical substance secreted by the amoebae themselves, which is called chemo-attraction withproduction. When the regions of high chemical concentration generate a repulsive effect on theorganisms, the phenomenon is called chemo-repulsion . E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
1n this work we study an optimal control problem subject to a chemo-repulsion with linearproduction system in which a bilinear control acts injecting or extracting chemical substance on asubdomain of control Ω c ⊂ Ω . Specifically, we consider Ω ⊂ R be a simply connected boundeddomain with boundary ∂ Ω of class C and (0 , T ) a time interval, with < T < + ∞ . Then westudy a control problem related to the following system in the time-space domain Q := (0 , T ) × Ω , ∂ t u − ∆ u = ∇ · ( u ∇ v ) ,∂ t v − ∆ v + v = u + f v χ Ω c , (1)with initial conditions u (0 , · ) = u ≥ , v (0 , · ) = v ≥ in Ω , (2)and non-flux boundary conditions ∂u∂ n = 0 , ∂v∂ n = 0 on (0 , T ) × ∂ Ω , (3)where n denotes the outward unit normal vector to ∂ Ω . In (1), the unknowns are the cell density u ( t, x ) ≥ and chemical concentration v ( t, x ) ≥ . The function f = f ( t, x ) denotes a bilinearcontrol acting in the chemical equation. We observe that in the subdomains of Ω where f ≥ thechemical substance is injected, and conversely where f ≤ the chemical substance is extracted.System (1)-(3) without control (i.e. f ≡ ) has been studied in [10], [32]. In [10], the authorsproved the global existence and uniqueness of smooth classical solutions in 2D domains, and globalexistence of weak solutions in dimension 3 and 4. In [32], on a bounded convex domain Ω ⊂ R n ( n ≥ ), 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 intime classical solution. This result is not applicable in our case, because g ( u ) = u does not satisfiesthe hypothesis imposed in [32].There is an extensive literature devoted to the study of control problems with PDEs, see forinstance [2, 6, 7, 17, 19, 21, 24, 25, 31, 35] and references therein. In all previous works, the control isof distributed or boundary type. As far as know, the literature related to optimal control problemswith 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 is2lso scarce. In [13, 29] a 1D problem is studied. In [13] the authors analyzed two problems for achemoattractant model. The bilinear control acts on the whole Ω in the cells equation. The existenceof optimal control is proved and an optimality system is derived. Also, a numerical scheme for theoptimality system is designed and some numerical simulations are presented. In [29] a boundarycontrol problem for a chemotaxis reaction-diffusion system is studied. The control acts on theboundary for the chemical substance, and the existence of optimal solution is proved. A distributedoptimal 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-Bellidoet al. [27] study a distributive optimal control problem related to a D stationary chemotaxismodel coupled with the Navier-Stokes equations ( chemotaxis-fluid system ). The authors provethe 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 anextreme problem for a chemoattractant D model, in which the control variable is distributed inthe chemical equation. They prove the existence of optimal solutions, and derive an optimalitysystem, using the fact that the state is differentiable with respect to the control. Other studiesrelated to controllability for the nonstationary Keller-Segel model and nonstationary chemotaxis-fluid system can be consulted in [8] and [9], respectively.In [16], an optimal bilinear control problem related to strong solutions of system (1)-(3) in2D domains was studied, proving the existence and uniqueness of global strong solutions, and theexistence of global optimal control. Moreover, using a Lagrange multiplier theorem, first-orderoptimality 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 toderive optimality conditions, which will be more complicated because the PDE system is consideredin 3D domains. In this case, we distinguish two different types of solutions: weak and strong . Theexistence 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 regularityof weak solutions. In order to overcome this problem, we introduce a regularity criterion that allowsto obtain a (unique) strong solution of (1)-(3) (see Theorem 3). As far as we know, there are noresults of global in time regularity of weak solutions of system (1)-(3) in D domains. This is similarto what happens with the Navier-Stokes equations (see [33]).3n this work, we deal with strong solutions of (1)-(3) which allows us to analyze the controlproblem. However, we are going to prove the existence of an optimal control associated to strongsolutions, assuming the existence of controls such that the associated strong solution exists. Fol-lowing the ideas of [6, 7], we consider a regularity criterion in the objective functional such that anyweak 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 functionalspaces to be used and we state a regularity result for linear parabolic-Neumann problems that willbe 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 theAppendix) and passing to the limit, prove the existence of weak solutions of system (1)-(3). InSection 4 we give the definition of strong solutions of (1)-(3), and we establish a regularity criterionunder which weak solutions of (1)-(3) are also strong solutions. Section 5 is dedicated to the study ofa bilinear control problem related to strong solutions of system (1)-(3), proving the existence of anoptimal solution and deriving the first-order optimality conditions based on a Lagrange multipliersargument in Banach spaces. Finally, we obtain a regularity result for these Lagrange multipliers. We will introduce some notations. We will use the Lebesgue space L p (Ω) , ≤ p ≤ + ∞ , withnorm denoted by k · k L p . In particular, the L -norm and its inner product will denoted by k · k and ( · , · ) , respectively. We consider the usual Sobolev spaces W m,p (Ω) = { u ∈ L p (Ω) : k ∂ α u k L p < + ∞ , ∀| α | ≤ m } , with norm denoted by k · k W m,q . When p = 2 , we write H m (Ω) := W m, (Ω) andwe denote the respective norm by k · k H m . Also, we use the space W m,p n (Ω) = { u ∈ W m,p (Ω) : ∂u∂ n =0 on ∂ Ω } ( m ≥ ) and its norm denoted by k · k 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 Bochnersense, and its norm will be denoted by k · k L p ( X ) . For simplicity we denote L p ( Q ) := L p (0 , T ; L p ) and its norm by k · k 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 k · k C ( X ) . The topological dual space of aBanach space X will be denoted by X ′ , and the duality for a pair X and X ′ by h· , ·i X ′ or simply by h· , ·i unless this leads to ambiguity. Moreover, the letters C , K , C , K , C , K ,..., denote positiveconstants, independent of state ( u, v ) and control f , but its value may change from line to line.4n order to study the existence of solution of system (1)-(3), we define the space c W − /p,p (Ω) := W − /p,p (Ω) if p < ,W − /p,p n (Ω) if p > , and we will often use the following regularity result for the heat equation (see [12, p. 344]). Lemma 1.
Let < p < + ∞ , u ∈ c W − /p,p (Ω) and g ∈ L p ( Q ) . Then the problem ∂ t u − ∆ u = g in Q,u (0 , · ) = u in Ω ,∂u∂ n = 0 on (0 , T ) × ∂ Ω , admits a unique solution u such that u ∈ C ([0 , T ]; c W − /p,p ) ∩ L p ( W ,p ) , ∂ t u ∈ L p ( Q ) . Moreover, there exists a positive constant C := C ( p, Ω , T ) such that k u k C ( c W − /p,p ) + k ∂ t u k L p ( Q ) + k u k L p ( W ,p ) ≤ C ( k g k L p ( Q ) + k u k c W − /p,p ) . For simplicity, in what follows we will use the following notation X p := { u ∈ C ([0 , T ]; c W − /p,p ) ∩ L p ( W ,p ) : ∂ t u ∈ L p ( Q ) } , and its norm will be denoted by k · k X p . In fact, u ∈ X p iff u ∈ W , p (Ω) := { u ∈ L p ( W ,p ) : ∂ t u ∈ L p ( Q ) } and u ∈ C ([0 , T ]; c W − /p,p ) .Throughtout this paper, we will use the following equivalent norms in H (Ω) and H (Ω) , re-spectively (see [26] for details): k u k H ≃ k∇ u k + (cid:18)Z Ω u (cid:19) , ∀ u ∈ H (Ω) , (4) k u k H ≃ k ∆ u k + (cid:18)Z Ω u (cid:19) , ∀ u ∈ H n (Ω) , (5)5nd the classical interpolation inequality in D domains k u k L ≤ C k u k / k u k / H , ∀ u ∈ H (Ω) . (6) Remark 1.
The problem (1)-(3) is conservative in u , because the total mass R Ω u ( t ) remains con-stant in time. In fact, integrating (1) in Ω we have ddt (cid:18)Z Ω u (cid:19) = 0 , i.e. Z Ω u ( t ) = Z Ω u := m , ∀ t > . Also, integrating (1) in Ω we deduce that R Ω v satisfies ddt (cid:18)Z Ω v (cid:19) + Z Ω v = m + Z Ω f v χ Ω c , ∀ t > . Definition 1. (Weak solution) Let f ∈ L ( Q c ) := L (0 , T ; L (Ω c )) , u ∈ L (Ω) , v ∈ H (Ω) with u ≥ and v ≥ in Ω , a pair ( u, v ) is called weak solution of problem (1)-(3) in (0 , T ) , if u ≥ , v ≥ , u ∈ L / ( Q ) ∩ L / ( W , / ) , ∂ t u ∈ [ L ( W , )] ′ , (7) v ∈ L ∞ ( H ) ∩ L ( H ) , ∂ t v ∈ L / ( Q ) , (8) the following variational formulation holds for the u -equation − Z T h u, ∂ t u i + Z T ( ∇ u, ∇ u ) + Z T ( u ∇ v, ∇ u ) = ( u , u (0)) , ∀ u ∈ X u , (9) the v -equation (1) holds pointwisely a.e. ( t, x ) ∈ Q , and the initial and boundary conditions for v (2) -(3) are satisfied. The space X u given in (9) is defined as follow X u = { u ∈ L ( W , ) : ∂ t u ∈ L / ( Q ) and u ( T ) = 0 in Ω } . emark 2. This definition of weak solution implies, in particular, that u ∈ L ∞ ( L ) and Z Ω u ( t ) = Z Ω u = m . Also, each term of (9) has sense. In particular, from (7) - (8) one has that u ∇ v ∈ L / ( Q ) . Theorem 1. (Existence of weak solutions of (1)-(3)) There exists a weak solution ( u, v ) of system(1)-(3) in the sense of Definition 1. The proof of this theorem follows from the two next subsections.
In order to prove Theorem 1, we will study the following family of regularized problems related tosystem (1)-(3), for any ε ∈ (0 , . Given an adequate regularization ( u ε , v ε ) of initial data ( u , v ) ,we define ( u ε , z ε ) as the solution of ∂ t u ε − ∆ u ε = ∇ · ( u ε ∇ v ( z ε )) in Q,∂ t z ε − ∆ z ε + z ε = u ε + f v ( z ε ) + χ Ω c in Q,u ε (0) = u ε , z ε (0) = v ε − ε ∆ v ε in Ω ∂u ε ∂ n = 0 , ∂z ε ∂ n = 0 on (0 , T ) × ∂ Ω , (10)where v ε := v ( z ε ) is the unique solution of the problem v ε − ε ∆ v ε = z ε in Ω ,∂v ε ∂ n = 0 on ∂ Ω , (11)and v + := max { v, } ≥ .We choose the initial conditions u ε and v ε , with u ε ≥ in Ω , such that ( u ε , v ε − ε ∆ v ε ) ∈ W / , / (Ω) × W / , / n (Ω) and ( u ε , v ε − ε ∆ v ε ) → ( u , v ) in L (Ω) × H (Ω) , as ε → . (12)In the remaining of this section, we will denote v ( z ε ) only by v ε .7 efinition 2. Let u ε ∈ W / , / (Ω) , v ε − ε ∆ v ε ∈ W / , / n (Ω) with u ε ≥ in Ω , and f ∈ L ( Q c ) .We say that a pair ( u ε , z ε ) is a (strong) solution of problem (10) in (0 , T ) , if u ε ≥ in Q , ( u ε , z ε ) ∈ X / × X / , the equations (10) -(10) holds pointwisely a.e. ( t, x ) ∈ Q , and the initial and boundary conditions(10) -(10) are satisfied. Remark 3.
Integrating (10) in Ω we have Z Ω u ε ( t ) = Z Ω u ε := m ε ∀ t > . (13) In fact, k u ε ( t ) k L = k u ε k L := m ε . Moreover, integrating (10) in Ω we deduce ddt (cid:18)Z Ω z ε (cid:19) + Z Ω z ε = m ε + Z Ω f v ε + χ Ω c , which implies ddt (cid:18)Z Ω z ε (cid:19) + (cid:18)Z Ω z ε (cid:19) ≤ (cid:18) m ε + Z Ω f v ε + χ Ω c (cid:19) . Theorem 2.
There exists a strong solution ( u ε , z ε ) ∈ X / × X / of system (10) in (0 , T ) in thesense of Definition 2. The proof of Theorem 2 is carried out in the Appendix. ε → . From the energy inequality (116) (see the proof of Lemma 10 in the Appendix) and the conservativityproperty (13) we deduce the following estimates (uniformly with respect to ε ) {∇√ u ε + 1 } ε> is bounded in L ( Q ) , {√ u ε + 1 } ε> is bounded in L ∞ ( L ) ∩ L ( L ) ֒ → L / ( Q ) ∩ L ( L / ) , { v ε } ε> is bounded in L ∞ ( H ) ∩ L ( H ) , {√ ε ∆ v ε } ε> is bounded in L ∞ ( L ) ∩ L ( H ) , (14)8hich implies { u ε } ε> is bounded in L / ( Q ) ∩ L ( L / ) , { z ε } ε> is bounded in L ∞ ( L ) ∩ L ( H ) , { ∂ t u ε } ε> is bounded in [ L ( W , )] ′ , { ∂ t z ε } ε> is bounded in [ L ( H )] ′ . (15)On the other hand, taking into account that ∇ u ε = 2 √ u ε + 1 ∇√ u ε + 1 , from (14) and (14) wededuce that { u ε } ε> is bounded in L / ( W , / ) . (16)Also, from (14) we have that {∇ v ε } ε> is bounded in L ∞ ( L ) ∩ L ( H ) ֒ → L / ( Q ) , which jointlyto (15) implies that { u ε ∇ v ε } ε> is bounded in L / ( Q ) . (17)Notice that from (11) and (14) we obtain that z ε − v ε = − ε ∆ v ε → as ε → , in the L ∞ ( L ) ∩ L ( H ) -norm. (18)Therefore, from (14), (15), (16) and (18), we deduce that there exists limit functions ( u, v ) suchthat u ∈ L / ( Q ) ∩ L / ( W , / ) ,v ∈ L ∞ ( H ) ∩ L ( H ) , and for some subsequence of { ( u ε , v ε , z ε ) } ε> , still denoted by { ( u ε , v ε , z ε ) } ε> , the following con-vergences holds, as ε → , u ε → u weakly in L / ( Q ) ∩ L / ( W , / ) ,v ε → v weakly in L ( H ) and weakly* in L ∞ ( H ) ,z ε → v weakly in L ( H ) and weakly* in L ∞ ( L ) ,∂ t u ε → ∂ t u weakly* in [ L ( W , )] ′ ,∂ t z ε → ∂ t v weakly* in [ L ( H )] ′ . (19)9e will verify that ( u, v ) is a weak solution of (1)-(3). From (15) , (16) and the Aubin-Lions lemma(see [22, Théorème 5.1, p. 58]) we deduce that { u ε } ε> is relatively compact in L / ( L ) ( and also in L p ( Q ) , ∀ p < / . (20)Thus, from (19) , (20) and taking into account (17) we have u ε ∇ v ε → u ∇ v weakly in L / ( Q ) . (21)On the other hand, from (19) , (19) , [22, Théorème 5.1, p. 58] and [30, Corollary 4] we obtain z ε → v strongly in L ( Q ) ∩ C ([0 , T ]; ( H ) ′ ) . (22)Thus, from (18), (19) and (22) we deduce that v ε converges to v strongly in L ( Q ) , which implies v ε + → v + strongly in L ( Q ) . Then, using that { v ε } ε> is bounded in L ∞ ( H ) ∩ L ( H ) ֒ → L ( Q ) and f ∈ L ( Q c ) , we deduce f v ε + χ Ω c → f v + χ Ω c weakly in L / ( Q ) . (23)Also from (22), z ε (0) converges to v (0) in H (Ω) ′ , then from (12) and the uniqueness of the limitwe have v (0) = v , which is the initial condition given in (2) .Therefore, taking to the limit in the regularized problem (10), as ε → , and taking into account(12), (19), (21) and (23) we conclude that ( u, v ) satisfies the weak formulation − Z T h u, ∂ t u i + Z T ( ∇ u, ∇ u ) + Z T ( u ∇ v, ∇ u ) = ( u , u (0)) ∀ u ∈ X u , (24) Z T h ∂ t v, z i + Z T ( ∇ v, ∇ z ) + Z T ( v, ¯ z ) = Z T ( u, z ) + Z T ( f v + χ Ω c , z ) ∀ z ∈ L ( H ) . (25)Integrating by parts in (25), and using that u ∈ L / ( Q ) and v ∈ L ( H ) , we deduce that v is the10nique solution of the problem ∂ t v − ∆ v + v = u + f v + χ Ω c in L / ( Q ) ,v (0) = v in Ω ,∂v∂ n = 0 on (0 , T ) × ∂ Ω . (26)Finally, we will check the positivity of ( u, v ) . Indeed, the positivity of u follow from (20) and thefact that u ε ≥ a.e. ( t, x ) ∈ Q (see Lemma 10 in the Appendix). In order to check that v ≥ , wetest (26) by v − := min { v, } ≤ , taking into account that u ≥ , and using that v − = 0 if v ≥ , ∇ v − = ∇ v if v ≤ and ∇ v − = 0 if v > , we obtain ddt k v − k + k∇ v − k + k v − k = ( u, v − ) + ( f v + χ Ω c , v − ) ≤ , which implies that v − ≡ , then v ≥ a.e. ( t, x ) ∈ Q . Thus, since v + ≡ v then v ≥ is also asolution of the v -equation (1) . In this section we will give a regularity criterion of system (1)-(3).
Definition 3. (Strong solution of problem (1)-(3)) Let f ∈ L ( Q c ) , u ∈ H (Ω) , v ∈ W / , n (Ω) with u ≥ and v ≥ in Ω . A pair ( u, v ) is called strong solution of problem (1)-(3) in (0,T), if u ≥ , v ≥ in Q , ( u, v ) ∈ X × X , (27) the system (1) holds pointwisely a.e. ( t, x ) ∈ Q , and the initial and boundary conditions (2) and (3)are satisfied. Remark 4.
Using the interpolation inequality (6), Gronwall lemma and proceeding as for theNavier-Stokes equations (see [33]), we can deduce the uniqueness of strong solutions of (1)-(3).
Theorem 3. (Regularity Criterion) Let ( u, v ) be a weak solution of (1)-(3). If, in addition, u ∈ (Ω) , v ∈ W / , n (Ω) and the following regularity criterion holds u ∈ L / ( Q ) , (28) then ( u, v ) is a strong solution of (1)-(3) in sense of Definition 3. Moreover, there exists a positiveconstant K = K ( k u k H , k v k W / , n , k f k L ( Q ) ) such that k u, v k X × X ≤ K. (29)The proof of this theorem follows from the two next subsections. 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 Theorem3, different interpolation results will be used together with some embeddings results that will bestated below.As a consequence of the interpolation inequality k u k L p ≤ k u k − θL p k u k θL p , with p = 1 − θp + θp and θ ∈ [0 , we have the following result Lemma 2.
Let p , p , q , q , p, q ≥ such that q = 1 − θq + θq and p = 1 − θp + θp , with θ ∈ [0 , . Then, L p ( L q ) ∩ L p ( L q ) ֒ → L p ( L q ) . (30)Using the Sobolev embedding W r,p (Ω) ֒ → L q (Ω) , with q = 1 p − rN , N is the space-dimension and the Gagliardo-Nirenberg inequality (see [14, Theorem 10.1]) W s,p (Ω) ∩ L p (Ω) ֒ → L p (Ω) , with p = θ (cid:18) p − sN (cid:19) + 1 − θp and θ ∈ [0 , we deduce the following result Lemma 3.
Let p , q , p , p, q ≥ such that q = 1 − θq + θ (cid:18) p − rN (cid:19) and p = θp with θ ∈ [0 , and r > . Then, L ∞ ( L q ) ∩ L p ( W r,p ) ֒ → L p ( L q ) . Lemma 4. ([1, Theorem 7.58, p.218]) Let < p < , and r, s > such that s = N (cid:18) − p (cid:19) + r. Then, W r,p (Ω) ֒ → H s (Ω) . Lemma 5. ([23, Théorème 9.6, p. 49]) Let p , p , p ≥ and s , s , s > such that s = (1 − θ ) s + θs and p = 1 − θp + θp , with θ ∈ [0 , . Then, L p ( H s ) ∩ L p ( H s ) ֒ → L p ( H s ) . Proof.
The proof is carried out into four steps:Step 1: v ∈ X / From Theorem 1, we know that there exists a weak solution ( u, v ) of system (1)-(3) in the senseof Definition 1. Thus, in particular v ∈ L ( Q ) and then f vχ Ω c ∈ L / ( Q ) , which implies, usinghypothesis (28), that u + f vχ Ω c ∈ L / ( Q ) . Then, applying Lemma 1 (for p = 20 / ) to equation131) , we have v ∈ X / . In particular, using Sobolev embeddings we have v ∈ L ∞ ( Q ) , (31) ∇ v ∈ L ∞ ( L ) ∩ L / ( W , / ) ֒ → L ∞ ( L ) ∩ L / ( L ) . (32)Embedding (30) for p = ∞ , q = 4 , p = 20 / and q = 60 (see Lemma 2) implies p = q = 20 / hence ∇ v ∈ L ∞ ( L ) ∩ L / ( L ) ֒ → L / ( Q ) . (33)Step 2: u ∈ L ∞ ( L ) ∩ L ( H ) .Starting from u ∈ L / ( Q ) ∩ L / ( W , / ) and v ∈ X / , we improve the regularity of u by abootstrapping argument in eigth sub-steps: i) u ∈ X / : Using that ( u, ∆ v ) ∈ L / ( Q ) × L / ( Q ) (hence u ∆ v ∈ L / ( Q ) ), and ( ∇ u, ∇ v ) ∈ L / ( Q ) × L / ( Q ) (hence ∇ u · ∇ v ∈ L / ( Q ) ) we have ∇ · ( u ∇ v ) = u ∆ v + ∇ u · ∇ v ∈ L / ( Q ) . Thus, applying Lemma 1 (for p = 20 / ) to equation (1) we obtain that u ∈ X / . ii) u ∈ X / : Since u ∈ X / , then by Sobolev embeddings ∇ u ∈ L / ( W , / ) ֒ → L / ( L / ) . (34)Moreover, using (30) in (32) (for p = ∞ , q = 4 , p = 20 / , q = 60 and p = 20 , hence q = 60 / ),we obtain ∇ v ∈ L ∞ ( L ) ∩ L ( L / ) . (35)Thus, from (34) and (35) we have ∇ u · ∇ v ∈ L / ( L / ) ∩ L ( L / ) . Then, owing to (30) appliedto ( p , q ) = (20 / , / and ( p , q ) = (1 , / implies that p = q = 10 / , hence ∇ u · ∇ v ∈ L / ( Q ) . u ∆ v ∈ L / ( Q ) , we have ∇ · ( u ∇ v ) ∈ L / ( Q ) . Then, applying Lemma 1 (for p = 10 / ) to(1) we deduce that u ∈ X / . iii) u ∈ X / : Since u ∈ X / , then ∇ u ∈ L / ( W , / ) ֒ → L / ( L / ) . (36)Now, using (30) in (32) (for p = ∞ , q = 4 , p = 20 / , q = 60 and p = 10 , hence q = 60 / ), weobtain ∇ v ∈ L ∞ ( L ) ∩ L ( L / ) , which jointly to (36) implies ∇ u · ∇ v ∈ L / ( L / ) ∩ L ( L / ) . Then using (30) with ( p , q ) =(10 / , / , ( p , q ) = (1 , / implies that p = q = 20 / , hence ∇ u · ∇ v ∈ L / ( Q ) . Since u ∆ v ∈ L / ( Q ) , we have ∇ · ( u ∇ v ) ∈ L / ( Q ) . Then, applying Lemma 1 (for p = 20 / )to (1) we deduce that u ∈ X / . iv) u ∈ X / : Since u ∈ X / then ∇ u ∈ L / ( W , / ) ֒ → L / ( L / ) , and, from (33), ∇ v ∈ L ∞ ( L ) ∩ L / ( Q ) , then ∇ u · ∇ v ∈ L / ( L / ) ∩ L ( L / ) , which thanksto (30) applied to ( p , q ) = (20 / , / , ( p , q ) = (1 , / implies p = q = 5 / hence ∇ u · ∇ v ∈ L / ( Q ) . Since u ∆ v ∈ L / ( Q ) , we obtain that ∇ · ( u ∇ v ) ∈ L / ( Q ) and, applying Lemma 1 (for p = 5 / )to equation (1) we deduce u ∈ X / . v) u ∈ X / : Using that u ∈ X / , then ∇ u ∈ L / ( W , / ) ֒ → L / ( L / ) . (37)15sing (30) in (32) (for p = ∞ , q = 4 , p = 20 / , q = 60 and p = 5 , hence q = 60 / ), we obtain ∇ v ∈ L ∞ ( L ) ∩ L ( L / ); then from the latter regularity and (37) we have ∇ u · ∇ v ∈ L / ( L / ) ∩ L ( L / ) , which thanksto (30) applied to ( p , q ) = (5 / , / , ( p , q ) = (1 , implies p = q = 4 / , hence ∇ u · ∇ v ∈ L / ( Q ) . Since u ∆ v ∈ L / ( Q ) , we obtain ∇ · ( u ∇ v ) ∈ L / ( Q ) . Then, applying Lemma 1 to equation (1) we have u ∈ X / . vi) u ∈ X / : Since u ∈ X / , then ∇ u ∈ L / ( W , / ) ֒ → L / ( L / ) , again using (30) in (32) (for p = ∞ , q = 4 , p = 20 / , q = 60 and p = 4 , hence q = 12 ), weobtain ∇ v ∈ L ∞ ( L ) ∩ L ( L ) and ∇ u · ∇ v ∈ L / ( L / ) ∩ L ( L ) , which thanks to (30) applied to ( p , q ) = (4 / , / , ( p , q ) =(1 , implies p = q = 10 / , hence ∇ u · ∇ v ∈ L / ( Q ) . Since u ∆ v ∈ L / ( Q ) , we obtain ∇ · ( u ∇ v ) ∈ L / ( Q ) , and applying Lemma 1 (for p = 10 / ) toequation (1) we have u ∈ X / . vii) u ∈ X / : Since u ∈ X / , then u ∈ L ∞ ( W / , / ) ∩ L / ( W , / ) ֒ → L ∞ ( L ) ∩ L / ( L ) ֒ → L / ( Q ) , ∇ u ∈ L / ( W , / ) ֒ → L / ( L / ) . (38)This time, we use (30) in (32) (for p = ∞ , q = 4 , p = 20 / , q = 60 and p = 10 / , hence q = 20 ),16e obtain ∇ v ∈ L ∞ ( L ) ∩ L / ( L ) , the latter regularity, (38) and the fact that ∆ v ∈ L / ( Q ) implies u ∆ v ∈ L / ( Q ) and ∇ u · ∇ v ∈ L / ( L / ) ∩ L ( L / ) . From (30) applied to ( p , q ) = (10 / , / , ( p , q ) = (1 , / one has p = q = 20 / hence ∇ u · ∇ v ∈ L / ( Q ) . Then, applying Lemma 1 (for p = 20 / ) to equation (1) we have u ∈ X / . viii) u ∈ L ∞ ( L ) ∩ L ( H ) : From Lemma 4, we know that W / , / (Ω) ֒ → H / (Ω) and W , / (Ω) ֒ → H / (Ω) . Therefore, from u ∈ X / we can deduce u ∈ L ∞ ( H / ) ∩ L / ( H / ) . Moreover, from Lemma 5 for ( p , s ) = ( ∞ , / , ( p , s ) = (20 / , / we have that u ∈ L ( H / ) ֒ → L ( H ) . Therefore, from the latter regularity and (38) we deduce u ∈ L ∞ ( L ) ∩ L ( H ) ֒ → L / ( Q ) . (39)Step 3: ( u, v ) ∈ X / × X / , u ∈ L ( Q ) and ∇ u ∈ L / ( Q ) .From (31), (39) and the fact that f ∈ L ( Q ) we obtain u + f v ∈ L / ( Q ) . Then applying Lemma1 (for p = 10 / ) to equation (1) we have that v ∈ X / . In particular, from Lemma 3 (for p = p = 10 / , q = 6 , r = 1 and p = q = 10 ) we obtain ∇ v ∈ L ∞ ( L ) ∩ L / ( W , / ) ֒ → L ( Q ) . Then, using that ( u, ∆ v ) ∈ L / ( Q ) × L / ( Q ) , ∇ v ∈ L ( Q ) and taking into account that ∇ u ∈ L ( Q ) we have ∇ · ( u ∇ v ) = u ∆ v + ∇ u · ∇ v ∈ L / ( Q ) . Thus, applying Lemma 1 (for p = 5 / ) to equation (1) we obtain that u ∈ X / . Moreover, from17obolev embeddings and again Lemma 3 (for p = p = 5 / , q = 3 , r = 2 and p = q = 5 ) we have u ∈ L ∞ ( L ) ∩ L / ( W , / ) ֒ → L ( Q ) . (40)From Lemma 4 we have the embeddings W / , / (Ω) ֒ → H / (Ω) and W , / (Ω) ֒ → H / (Ω) .Thus, since u ∈ X / , one has u ∈ L ∞ ( H / ) ∩ L / ( H / ) . Moreover, from Lemma 5 (for ( p , s ) = ( ∞ , / and ( p , s ) = (5 / , / ), we have u ∈ L / ( H / ) , and in particular ∇ u ∈ L / ( H / ) ֒ → L / ( Q ) .Step 4: ( u, v ) ∈ X × X .From (31), (40), and using that f ∈ L ( Q c ) , we have u + f vχ Ω c ∈ L ( Q ) . Then, applying Lemma 1(for p = 4 ) to equation (1) we deduce that v ∈ X and satisfies the estimate k v k X ≤ C ( k u + f v k L ( Q ) + k v k W / , n ) ≤ C ( k u k L ( Q ) + k f k L ( Q ) k v k L ∞ ( Q ) + k v k W / , n ) ≤ C ( k u k W / , / , k v k W / , n , k f k L ( Q ) ) . (41)In particular, by Sobolev embeddings and Lemma 3 (for p = p = 4 , q = 12 , r = 1 hence p = q = 20 ) we have ∇ v ∈ L ∞ ( L ) ∩ L ( W , ) ֒ → L ( Q ) .Now, using that ( u, ∆ v ) ∈ L ( Q ) × L ( Q ) and ( ∇ u, ∇ v ) ∈ L / ( Q ) × L ( Q ) we obtain ∇ · ( u ∇ v ) = u ∆ v + ∇ u · ∇ v ∈ L ( Q ) . Therefore, applying Lemma 1 (for p = 2 ) to equation (1) we deduce that u ∈ X and k u k X ≤ C ( k u k L ( Q ) k ∆ v k L ( Q ) + k∇ u k L / ( Q ) k∇ v k L ( Q ) + k u k H ) ≤ C ( k u k H , k v k W / , n , k f k L ( Q ) ) . (42)Finally, we observe that estimate (29) follows from (41) and (42).18 The Optimal Control Problem
In this section we establish the statement of the bilinear control problem. Following [6, 7], weformulate the control problem in such a way that any admissible state is a strong solution of (1)-(3). Since there is no existence result of global in time strong solutions of (1)-(3), we have to choosea suitable objective functional.We suppose that
F ⊂ L ( Q c ) := L (0 , T ; L (Ω c )) is a nonempty and convex set, (43)where Ω c ⊂ Ω is the control domain. We consider data u ∈ H (Ω) , v ∈ W / , n (Ω) with u ≥ and v ≥ in Ω , and the function f ∈ F describing the bilinear control acting on the v -equation.Now, we define the following constrained minimization problem related to system (1)-(3): Find ( u, v, f ) ∈ X × X × F such that the functional J ( u, v, f ) := 7 α u Z T k u ( t ) − u d ( t ) k / L / (Ω) dt + α v Z T k v ( t ) − v d ( t ) k L (Ω) dt + α f Z T k f ( t ) k L (Ω c ) dt is minimized, subject to ( u, v, f ) satisfies the PDE system (1)-(3). (44)Here ( u d , v d ) ∈ L / ( Q ) × L ( Q ) represent the desires states (see the beginning of the proof ofTheorem 7 below to justify the regularity required for u d ∈ L / ( Q ) ) and the real numbers α u , α v and α f measure the cost of the states and control, respectively. These numbers satisfy α u > and α v , α f ≥ . The admissible set for the optimal control problem (44) is defined by S ad = { s = ( u, v, f ) ∈ X × X × F : s is a strong solution of (1)-(3) in (0 , T ) } . The functional J defined in (44) describes the deviation of the cell density u and the chemicalconcentration v from a desired cell density u d and chemical concentration v d respectively, plus thecost of the control measured in the L -norm. We also observe that if ( u, v ) is a weak solution of191)-(3) in (0 , T ) such that J ( u, v, f ) < + ∞ , then by Theorem 3, ( u, v ) is a strong solution of (1)-(3)in (0 , T ) . In what follows, we will assume the hypothesis S ad = ∅ . (45) Remark 5.
The reason for choosing the first term of the objective functional in the L / -normis that any weak solution of (1)-(3) satisfying J ( u, v, f ) < + ∞ satisfies that u ∈ L / ( Q ) andtherefore, in virtue of Theorem 3, let us to state that ( u, v ) is the unique solution of (1)-(3) in thesense of Definition 3. Thus, we reduce the admissible states of problem (44) to the strong solutionsof (1)-(3). With this formulation we are going to prove the existence of a global optimal solutionand derive the optimality conditions associated to any local optimal solution. Definition 4.
An element (˜ u, ˜ v, ˜ f ) ∈ S ad will be called a global optimal solution of problem (44) if J (˜ u, ˜ v, ˜ f ) = min ( u,v,f ) ∈S ad J ( u, v, f ) . (46) Theorem 4.
Let u ∈ H (Ω) and v ∈ W / , n (Ω) with u ≥ and v ≥ in Ω . We assume thateither α f > or F is bounded in L ( Q c ) and hypothesis (45), then the bilinear optimal controlproblem (44) has at least one global optimal solution (˜ u, ˜ v, ˜ f ) ∈ S ad .Proof. From hypothesis (45) S ad = ∅ . Let { s m } m ∈ N := { ( u m , v m , f m ) } m ∈ N ⊂ S ad be a minimizingsequence of J , that is, lim m → + ∞ J ( s m ) = inf s ∈S ad J ( s ) . Then, by definition of S ad , for each m ∈ N , s m satisfies system (1) a.e. ( t, x ) ∈ Q .From the definition of J and the assumption α f > or F is bounded in L ( Q c ) , it follows that { f m } m ∈ N is bounded in L ( Q c ) (47)and { u m } m ∈ N is bounded in L / ( Q ) . C , independent of m , such that k u m , v m k X × X ≤ C. (48)Therefore, from (47), (48), and taking into account that F is a closed convex subset of L ( Q c ) (hence is weakly closed in L ( Q c ) ), we deduce that there exists ˜ s = (˜ u, ˜ v, ˜ f ) ∈ X × X × F suchthat, for some subsequence of { s m } m ∈ N , still denoted by { s m } m ∈ N , the following convergences hold,as m → + ∞ : u m → ˜ u weakly in L ( H ) and weakly* in L ∞ ( H ) , (49) v m → ˜ v weakly in L ( W , ) and weakly* in L ∞ ( W / , n ) , (50) ∂ t u m → ∂ t ˜ u weakly in L ( Q ) , (51) ∂ t v m → ∂ t ˜ v weakly in L ( Q ) , (52) f m → ˜ f weakly in L ( Q c ) , and ˜ f ∈ F . (53)From (49)-(52), the Aubin-Lions lemma (see [22, Théorème 5.1, p. 58] and [30, Corollary 4]) andusing Sobolev embedding, we have u m → ˜ u strongly in C ([0 , T ]; L p ) ∩ L ( W ,p ) ∀ p < , (54) v m → ˜ v strongly in C ([0 , T ]; L q ) ∩ L ( W ,q ) ∀ q < + ∞ . (55)In particular, we can control the limit of the nonlinear terms of (1) as follows ∇ · ( u m ∇ v m ) → ∇ · (˜ u ∇ ˜ v ) weakly in L / ( Q ) , (56) f m v m χ Ω c → ˜ f ˜ v χ Ω c weakly in L ( Q ) . (57)Moreover, from (54) and (55) we have that ( u m (0) , v m (0)) converges to (˜ u (0) , ˜ v (0)) in L p (Ω) × L q (Ω) ,and since u m (0) = u , v m (0) = v , we deduce that ˜ u (0) = u and ˜ v (0) = v . Thus ˜ s satisfies theinitial conditions given in (2). Therefore, considering the convergences (49)-(57), we can pass to thelimit in (1) satisfied by ( u m , v m , f m ) , as m goes to + ∞ , and we conclude that ˜ s = (˜ u, ˜ v, ˜ f ) is also a21olution of the system (1) pointwisely, that is, ˜ s ∈ S ad . Therefore, lim m → + ∞ J ( s m ) = inf s ∈S ad J ( s ) ≤ J (˜ s ) . (58)On the other hand, since J is lower semicontinuous on S ad , we have J (˜ s ) ≤ lim inf m → + ∞ J ( s m ) , whichjointly to (58), implies (46). We will derive the first-order necessary optimality conditions for a local optimal solution (˜ u, ˜ v, ˜ f ) of problem (44), applying a Lagrange multipliers theorem. We will base on a generic result givenby Zowe et al [36] on the existence of Lagrange multipliers in Banach spaces. In order to introducethe concepts and results given in [36] we consider the following optimization problem min x ∈ M J ( x ) subject to G ( x ) ∈ N , (59)where J : X → R is a functional, G : X → Y is an operator, X and Y are Banach spaces, M is anonempty closed convex subset of X and N is a nonempty closed convex cone in Y with vertex atthe origin. The admissible set for problem (59) is defined by S = { x ∈ M : G ( x ) ∈ N } . For a subset A of X (or Y ), A + denotes its polar cone, that is A + = { ρ ∈ X ′ : h ρ, a i X ′ ≥ , ∀ a ∈ A } . Definition 5. (Lagrange multiplier) Let ˜ x ∈ S be a local optimal solution for problem (59). Supposethat J and G are Fréchet differentiable in ˜ x , with derivatives J ′ (˜ x ) and G ′ (˜ x ) , respectively. Then,any ξ ∈ Y ′ is called a Lagrange multiplier for (59) at the point ˜ x if ξ ∈ N + , h ξ, G (˜ x ) i Y ′ = 0 ,J ′ (˜ x ) − ξ ◦ G ′ (˜ x ) ∈ C (˜ x ) + , (60)22 here C (˜ x ) = { θ ( x − ˜ x ) : x ∈ M , θ ≥ } is the conical hull of ˜ x in M . Definition 6.
Let ˜ x ∈ S be a local optimal solution for problem (59). We say that ˜ x is a regularpoint if G ′ (˜ x )[ C (˜ x )] − N ( G (˜ x )) = Y , where N ( G (˜ x )) = { ( θ ( n − G (˜ x )) : n ∈ N , θ ≥ } is the conical hull of G (˜ x ) in N . Theorem 5. ([36, Theorem 3.1]) Let ˜ x ∈ S be a local optimal solution for problem (59). If ˜ x is aregular point, then the set of Lagrange multipliers for (59) at ˜ x is nonempty. Now, we will reformulate the optimal control problem (44) in the abstract setting (59). Weconsider the following Banach spaces X := W u × W v × L ( Q c ) , Y := L ( Q ) × L ( Q ) × H (Ω) × W / , n (Ω) , where W u := (cid:26) u ∈ X : ∂u∂ n = 0 on (0 , T ) × ∂ Ω (cid:27) , (61) W v := (cid:26) v ∈ X : ∂v∂ n = 0 on (0 , T ) × ∂ Ω (cid:27) , (62)and the operator G = ( G , G , G , G ) : X → Y , where G : X → L ( Q ) , G : X → L ( Q ) , G : X → H (Ω) , G : X → W / , n (Ω) are defined at each point s = ( u, v, f ) ∈ X by G ( s ) = ∂ t u − ∆ u − ∇ · ( u ∇ v ) ,G ( s ) = ∂ t v − ∆ v + v − u − f v χ Ω c ,G ( s ) = u (0) − u ,G ( s ) = v (0) − v . min s ∈ M J ( s ) subject to G ( s ) = , (63)where M := W u × W v × F . and F is defined in (43).We observe that M is a closed convex subset of X , N = { } and the set of admissible solutionsis rewritten as S ad = { s = ( u, v, f ) ∈ M : G ( s ) = } . (64)Concerning to the differentiability of the constraint operator G and the functional J we have thefollowing results. Lemma 6.
The functional J : X → R is Fréchet differentiable and the derivative of J in ˜ s =(˜ u, ˜ v, ˜ f ) ∈ X in the direction r = ( U, V, F ) ∈ X is J ′ (˜ s )[ r ] = α u Z T Z Ω sgn(˜ u − u d ) | ˜ u − u d | / U + α v Z T Z Ω (˜ v − v d ) V + α f Z T Z Ω c ( ˜ f ) F. (65) Lemma 7.
The operator G : X → Y is Fréchet differentiable and the derivative of G in ˜ s =(˜ u, ˜ v, ˜ f ) ∈ X in the direction r = ( U, V, F ) ∈ X is the linear operator G ′ (˜ s )[ r ] = ( G ′ (˜ s )[ r ] , G ′ (˜ s )[ r ] , G ′ (˜ s )[ r ] , G ′ (˜ s )[ r ]) defined by G ′ (˜ s )[ r ] = ∂ t U − ∆ U − ∇ · ( U ∇ ˜ v ) − ∇ · (˜ u ∇ V ) ,G ′ (˜ s )[ r ] = ∂ t V − ∆ V + V − U − ˜ f V χ Ω c − F ˜ v,G ′ (˜ s )[ r ] = U (0) ,G ′ (˜ s )[ r ] = V (0) . (66)We wish to prove the existence of Lagrange multipliers, which is guaranteed if a local optimalsolution of problem (63) is a regular point of operator G (in virtue of Theorem 5). Remark 6.
Since for problem (63) N = { } , then N ( G (˜ s )) = { } . Thus, from Definition 6we conclude that ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad is a regular point if for any ( g u , g v , U , V ) ∈ Y there exists = ( U, V, F ) ∈ W u × W v × C ( ˜ f ) such that G ′ (˜ s )[ r ] = ( g u , g v , U , V ) , where C ( ˜ f ) := { θ ( f − ˜ f ) : θ ≥ , f ∈ F } is the conical hull of ˜ f in F . Lemma 8.
Let ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad ( S ad defined in (64)), then ˜ s is a regular point.Proof. Let ( g u , g v , U , V ) ∈ Y . Since ∈ C ( ˜ f ) = { θ ( f − ˜ f ) : θ ≥ , f ∈ F } , it is sufficient to showthe existence of ( U, V ) ∈ W u × W v solving the linear problem ∂ t U − ∆ U − ∇ · ( U ∇ ˜ v ) − ∇ · (˜ u ∇ V ) = g u in Q,∂ t V − ∆ V + V − U − ˜ f V χ Ω c = g v in Q,U (0) = U , V (0) = V in Ω ,∂U∂ n = 0 , ∂V∂ n = 0 on (0 , T ) × ∂ Ω . (67)Since (67) is a linear system we argue in a formal manner, proving that any regular enough solutionis bounded in W u × W v . A detailed proof can be made by using, for instance, a Galerkin method.Testing (67) by U and (67) by − ∆ V , we have ddt ( k U k + k∇ V k ) + k∇ U k + k∇ V k + k ∆ V k ≤ | ( U ∇ ˜ v, ∇ U ) | + | (˜ u ∇ V, ∇ U ) | + | ( g u , U ) | + | ( U, ∆ V ) | + | ( ˜ f V χ Ω c , ∆ V ) | + | ( g v , ∆ V ) | . (68)Using the Hölder and Young inequalities on the terms on the right side of (68) and taking into25ccount (6) we obtain | ( U ∇ ˜ v, ∇ U ) | ≤ k U k L k∇ ˜ v k L k∇ U k ≤ C k U k / k∇ ˜ v k L k U k / H ≤ δ k U k H + C δ k∇ ˜ v k L k U k , (69) | (˜ u ∇ V, ∇ U ) | ≤ k ˜ u k L k∇ V k L k∇ U k ≤ δ k∇ U k + C δ k ˜ u k L k∇ V k / k∇ V k / H ≤ δ ( k∇ U k + k∇ V k H ) + C δ k ˜ u k L k∇ V k , (70) | ( g u , U ) | ≤ δ k U k + C δ k g u k , (71) | ( U, ∆ V ) | ≤ δ k ∆ V k + C δ k U k , (72) | ( ˜ f V χ Ω c , ∆ V ) | ≤ k ˜ f k L k V k L k ∆ V k ≤ δ k ∆ V k + C δ k ˜ f k L k V k H , (73) | ( g v , ∆ V ) | ≤ δ k ∆ v k + C δ k g v k . (74)On the other hand, testing by V in (67) we obtain ddt k V k + k∇ V k + k V k ≤ | ( U, V ) | + | ( ˜ f V χ Ω c , V ) | + | ( g v , V ) |≤ δ k V k H + C δ k U k + C δ k ˜ f k L k V k + C δ k g v k . (75)Summing the inequalities (68) and (75), and then adding k U k to both sides of the inequalityobtained, and taking into account (69)-(74), for δ small enough, we have ddt ( k U k + k V k H ) + C k U k H + C k V k H ≤ C (1 + k∇ ˜ v k L ) k U k + C ( k g u k + k g v k )+ C k ˜ u k L k∇ V k + C k ˜ f k L k V k H . (76)From (76) and Gronwall lemma we deduce that there exists a positive constant C that depends on T , k U k , k V k H , k ˜ u k L ( L ) , k∇ ˜ v k L ( L ) , k ˜ f k L ( L ) , k g u k L ( Q ) and k g v k L ( Q ) such that k U, V k L ∞ ( L × H ) ∩ L ( H × H ) ≤ C. (77)In particular, from (77) we obtain that ( U, V ) ∈ L / ( Q ) × L ( Q ) , and since ˜ f ∈ L ( Q c ) we have ˜ f V χ Ω c ∈ L / ( Q ) . Then, applying Lemma 1 (for p = 20 / ) to (67) , we deduce that V ∈ X / .
26y Sobolev embeddings V ∈ L ∞ ( Q ) , so that ˜ f V χ Ω c ∈ L ( Q ) . Thus, using that U ∈ L / ( Q ) ,again by Lemma 1 (for p = 10 / ) we obtain that V ∈ X / . (78)Now, testing (67) by − ∆ U we have ddt k∇ U k + k ∆ U k ≤ | ( U ∆˜ v, ∆ U ) | + | ( ∇ U · ∇ ˜ v, ∆ U ) | + | (˜ u ∆ V, ∆ U ) | + | ( ∇ ˜ u · ∇ V, ∆ U ) | + | ( g u , ∆ U ) | . (79)Applying the Hölder and Young inequalities to the terms on the right side of (79), and using (6),we have | ( U ∆˜ v, ∆ U ) | ≤ k U k L k ∆˜ v k L k ∆ U k ≤ C k U k H k ∆˜ v k L k ∆ U k≤ δ k U k H + C δ k U k H k ∆˜ v k L , (80) | ( ∇ U · ∇ ˜ v, ∆ U ) | ≤ k∇ U k L k∇ ˜ v k L k ∆ U k ≤ C k∇ U k / k∇ ˜ v k L k U k / H ≤ δ k U k H + C δ k∇ U k k∇ ˜ v k L , (81) | (˜ u ∆ V, ∆ U ) | ≤ k ˜ u k L k ∆ V k L k ∆ U k ≤ C k ˜ u k H k ∆ V k L k ∆ U k≤ δ k U k H + C δ k ˜ u k H k ∆ V k L , (82) | ( ∇ ˜ u · ∇ V, ∆ U ) | ≤ k∇ ˜ u k L k∇ V k L k ∆ U k ≤ C k∇ ˜ u k L k∇ V k H k ∆ U k≤ δ k U k H + C δ k∇ ˜ u k L k V k W / , / , (83) | ( g u , ∆ U ) | ≤ δ k ∆ U k + C δ k g u k . (84)Now, we observe that ddt (cid:18)Z Ω U (cid:19) = Z Ω g u , which implies ddt (cid:18)Z Ω U (cid:19) = (cid:18)Z Ω g u (cid:19) (cid:18)Z Ω U (cid:19) ≤ C δ (cid:18)Z Ω g u (cid:19) + δ (cid:18)Z Ω U (cid:19) (85)and (cid:12)(cid:12)(cid:12)(cid:12)Z Ω U ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z Ω U + Z t Z Ω g u (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. (86)Summing inequalities (79), (85) and (86), and taking into account (80)-(84), for δ small enough, we27btain ddt k U k H + C k U k H ≤ C k U k H k ∆˜ v k L + C k∇ U k k∇ ˜ v k L + C k ˜ u k H k ∆ V k L + C k∇ ˜ u k L k V k W / , / + C k g u k + C. (87)We observe that from (78) we have V ∈ L ∞ ( W / , / ) ∩ L / ( W , / ) , and we know that ˜ u ∈ X , ˜ v ∈ X . Then, from (87) and Gronwall lemma we deduce U ∈ L ∞ ( H ) ∩ L ( H ) ֒ → L ( Q ) . Now, since U ∈ L ( Q ) and ˜ f V χ Ω c ∈ L ( Q ) , we have U + ˜ f V χ Ω c ∈ L ( Q ) . Then, from (67) andLemma 1 (for p = 4 ) we conclude that V ∈ X .Finally, using that (˜ u, U ) ∈ L ( Q ) , (∆˜ v, ∆ V ) ∈ L ( Q ) , ( ∇ ˜ u, ∇ U ) ∈ L / ( Q ) , and ( ∇ ˜ v, ∇ V ) ∈ L ( Q ) we deduce ∇ · ( U ∇ ˜ v ) + ∇ · (˜ u ∇ V ) ∈ L / ֒ → L ( Q ) . (88)Therefore, thanks to (88), applying Lemma 1 (for p = 2 ) to (67) , we conclude that U ∈ X . Thus,the proof is finished. Remark 7.
Using a classical comparison argument, inequality (6) and Gronwall lemma, the unique-ness of solutions of system (67) is deduced.
Now we show the existence of Lagrange multiplier for problem (44) associated to any localoptimal solution ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad . Theorem 6.
Let ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad be a local optimal solution for the control problem (44). Then,there exist a Lagrange multiplier ξ = ( λ, η, ϕ , ϕ ) ∈ L ( Q ) × L / ( Q ) × ( H (Ω)) ′ × ( W / , n (Ω)) ′ such that for all ( U, V, F ) ∈ W u × W v × C ( ˜ f ) α u Z T Z Ω sgn(˜ u − u d ) | ˜ u − u d | / U + α v Z T Z Ω (˜ v − v d ) V + α f Z T Z Ω c ( ˜ f ) F − Z T Z Ω (cid:18) ∂ t U − ∆ U − ∇ · ( U ∇ ˜ v ) − ∇ · (˜ u ∇ V ) (cid:19) λ − Z T Z Ω (cid:18) ∂ t V − ∆ V + V − U − ˜ f V χ Ω c (cid:19) η − Z Ω U (0) ϕ − Z Ω V (0) ϕ + Z T Z Ω c F ˜ vη ≥ . (89)28 roof. From Lemma 8, ˜ s ∈ S ad is a regular point, then from Theorem 5 there exists a Lagrangemultiplier ξ = ( λ, η, ϕ , ϕ ) ∈ L ( Q ) × L / ( Q ) × ( H (Ω)) ′ × ( W / , n (Ω)) ′ such that by (60) onemust satisfy J ′ (˜ s )[ r ] − h R ′ (˜ s )[ r ] , λ i − h R ′ (˜ s )[ r ] , η i − h R ′ (˜ s )[ r ] , ϕ i − h R ′ (˜ s )[ r ] , ϕ i ≥ , (90)for all r = ( U, V, F ) ∈ W u × W v × C ( ˜ f ) . Thus, the proof follows from (65), (66) and (90).From Theorem 6, we derive an optimality system for problem (44), by considering the spaces W u = { u ∈ W u : u (0) = 0 } , W v = { v ∈ W v : v (0) = 0 } . Corollary 1.
Let ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad be a local optimal solution for the control problem (44). Thenthe Lagrange multiplier ( λ, η ) ∈ L ( Q ) × L / ( Q ) , provided by Theorem 6, satisfies the system Z T Z Ω (cid:18) ∂ t U − ∆ U − ∇ · ( U ∇ ˜ v ) (cid:19) λ − Z T Z Ω U η = α u Z T Z Ω sgn(˜ u − u d ) | ˜ u − u d | / U, ∀ U ∈ W u , (91) Z T Z Ω (cid:18) ∂ t V − ∆ V + V (cid:19) η − Z T Z Ω c ˜ f V η − Z T Z Ω ∇ · (˜ u ∇ V ) λ = α v Z T Z Ω (˜ v − v d ) V, ∀ V ∈ W v , (92) and the optimality condition Z T Z Ω c ( α f ( ˜ f ) + ˜ vη )( f − ˜ f ) ≥ ∀ f ∈ F . (93) Proof.
From (89), taking ( V, F ) = (0 , , and using that W u is a vectorial space, we have (91).Similarly, taking ( U, F ) = (0 , in (89), and taking into account that W v is a vectorial space, wededuce (92). Finally, taking ( U, V ) = (0 , in (89) we have α f Z T Z Ω c ( ˜ f ) F + Z T Z Ω c ˜ vηF ≥ ∀ F ∈ C ( ˜ f ) . Thus, choosing F = θ ( f − ˜ f ) ∈ C ( ˜ f ) for all f ∈ F and θ ≥ in the last inequality, we have (93).29 emark 8. A pair ( λ, η ) ∈ L ( Q ) × L / ( Q ) satisfying (91)-(92) corresponds to the concept of veryweak solution of the linear system ∂ t λ + ∆ λ − ∇ λ · ∇ ˜ v + η = − α u sgn(˜ u − u d ) | ˜ u − u d | / in Q,∂ t η + ∆ η + ∇ · (˜ u ∇ λ ) − η + ˜ f η χ Ω c = − α v (˜ v − v d ) in Q,λ ( T ) = 0 , η ( T ) = 0 in Ω ,∂λ∂ n = 0 , ∂η∂ n = 0 on (0 , T ) × ∂ Ω . (94) Theorem 7.
Let ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad be a local optimal solution for the problem (44) and u d ∈ L / ( Q ) . Then the system (94) has a unique solution ( λ, η ) such that λ ∈ X , (95) η ∈ X / . (96) Proof.
Since the desired state u d ∈ L / ( Q ) , we have that h (˜ u ) := sgn(˜ u − u d ) | ˜ u − u d | / ∈ L ( Q ) .In fact, ˜ u is more regular because assuming ˜ u ∈ L / ( Q ) , it can be proved that ˜ u ∈ L ∞ ( H ) ∩ L ( H ) ֒ → L ( Q ) (see the proof of the Theorem 3 for more details).Let s = T − t , with t ∈ (0 , T ) and ˜ λ ( s ) = λ ( t ) , ˜ η ( s ) = η ( t ) . Then, system (94) is equivalent to ∂ s ˜ λ − ∆˜ λ + ∇ ˜ λ · ∇ ˜ v − ˜ η = α u h (˜ u ) in Q,∂ s ˜ η − ∆˜ η − ∇ · (˜ u ∇ ˜ λ ) + ˜ η − ˜ f ˜ η χ Ω c = α v (˜ v − v d ) in Q, ˜ λ (0) = 0 , ˜ η (0) = 0 in Ω ,∂ ˜ λ∂ n = 0 , ∂ ˜ η∂ n = 0 on (0 , T ) × ∂ Ω . (97)Testing (97) by − ∆˜ λ and (97) by ˜ η , and using Hölder and Young inequalities, we can obtain dds ( k∇ ˜ λ k + k ˜ η k ) + k ∆˜ λ k + k ˜ η k H ≤ δ ( k∇ ˜ λ k H + k ∆˜ λ k + k∇ ˜ η k ) + C δ (1 + k ˜ f k / L ) k ˜ η k + C δ ( k ˜ u k L + k∇ ˜ v k L ) k∇ ˜ λ k + C δ ( k h (˜ u ) k + k ˜ v − v d k ) . (98)Now, since ∂ ˜ λ∂ n = 0 on ∂ Ω , then by [3, Corollary 3.5] we have k∇ ˜ λ k H ≃ k∇ ˜ λ k + k ∆˜ λ k . (99)30hus, taking δ small enough, from (98) and (99) we deduce the following energy inequality dds ( k∇ ˜ λ k + k ˜ η k ) + C ( k∇ ˜ λ k H + k ˜ η k H ) ≤ C ( k ˜ u k L + k∇ ˜ v k L + 1) k∇ ˜ λ k + C (1 + k ˜ f k / L ) k ˜ η k + C ( k h (˜ u ) k + k ˜ v − v d k ) , which, jointly with Gronwall lemma, implies ( ∇ ˜ λ, ˜ η ) ∈ L ∞ ( L ) ∩ L ( H ) ֒ → L / ( Q ) . In particular, using that ( ∇ ˜ λ, ∇ ˜ v ) ∈ L / ( Q ) × L ( Q ) , we have ∇ ˜ λ · ∇ ˜ v ∈ L / ( Q ) ֒ → L ( Q ) .Thus, applying Lemma 1 (for p = 2 ) to (97) , we deduce (95).On the other hand, since ˜ f ∈ L ( Q c ) , ˜ η ∈ L / ( Q ) , we have ˜ f ˜ η χ Ω c ∈ L / ( Q ) . (100)Now, taking into account that ˜ u ∈ L ∞ ( H ) ∩ L ( H ) ֒ → L ( Q ) , ∆˜ λ ∈ L ( Q ) , and ∇ ˜ u, ∇ ˜ λ ∈ L / ( Q ) , we deduce ∇ · (˜ u ∇ ˜ λ ) = ˜ u ∆˜ λ + ∇ ˜ u · ∇ ˜ λ ∈ L / ( Q ) . (101)Therefore, from (97) , (100), (101) and Lemma 1 (for p = 5 / ) we obtain (96).In the following result, we obtain more regularity for the Lagrange multiplier ( λ, η ) than providedby Theorem 6. Theorem 8.
Let ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad be a local optimal solution for the control problem (44). Thenthe Lagrange multiplier, provided by Theorem 6, satisfies ( λ, η ) ∈ X × X / .Proof. Let ( λ, η ) be the Lagrange multiplier given in Theorem 6, which is a very weak solution ofproblem (94). In particular, ( λ, η ) satisfies (91)-(92).On the other hand, from Theorem 7, system (94) has a unique solution ( λ, η ) ∈ X × X / .Then, it suffices to identify ( λ, η ) with ( λ, η ) . With this objective, we consider the unique solution ( U, V ) ∈ W u ×W v of linear system (67) for g u := λ − λ ∈ L ( Q ) and g v := sgn( η − η ) | η − η | / ∈ L ( Q ) (see Lemma 8 and Remark 7). Then, written (94) for ( λ, η ) (instead of ( λ, η ) ), testing the first31quation by U , and the second one by V , and integrating by parts in Ω , we obtain Z T Z Ω (cid:18) ∂ t U − ∆ U − ∇ · ( U ∇ ˜ v ) (cid:19) λ − Z T Z Ω U η = α u Z T Z Ω sgn(˜ u − u d ) | ˜ u − u d | / U, (102) Z T Z Ω (cid:18) ∂ t V − ∆ V + V − ˜ f V χ Ω c (cid:19) η − Z T Z Ω ∇ · (˜ u ∇ V ) λ = α v Z T Z Ω (˜ v − v d ) V. (103)Making the difference between (91) for ( λ, η ) and (102) for ( λ, η ) , and between (92) and (103), andthen adding the respective equations, since the right-hand side terms vanish, we have Z T Z Ω (cid:18) ∂ t U − ∆ U − ∇ · ( U ∇ ˜ v ) − ∇ · (˜ u ∇ V ) (cid:19) ( λ − λ )+ Z T Z Ω (cid:18) ∂ t V − ∆ V + V − U − ˜ f V χ Ω c (cid:19) ( η − η ) = 0 . (104)Therefore, taking into account that ( U, V ) is the unique solution of (67) for g u = λ − λ and g v = sgn( η − η ) | η − η | / , from (104) we deduce k λ − λ k L ( Q ) + k η − η k / L / ( Q ) = 0 , which implies that ( λ, η ) = ( λ, η ) in L ( Q ) × L / ( Q ) . As a consequence of the regularity of ( λ, η ) we deduce that ( λ, η ) ∈ X × X / . Corollary 2. (Optimality System) Let ˜ s = (˜ u, ˜ v, ˜ f ) ∈ S ad be a local optimal solution for the controlproblem (44). Then, the Lagrange multiplier ( λ, η ) ∈ X × X / satisfies the optimality system ∂ t λ + ∆ λ − ∇ λ · ∇ ˜ v + η = − α u sgn(˜ u − u d ) | ˜ u − u d | / a.e. ( t, x ) ∈ Q,∂ t η + ∆ η + ∇ · (˜ u ∇ λ ) − η + ˜ f η χ Ω c = − α v (˜ v − v d ) a.e. ( t, x ) ∈ Q,λ ( T ) = 0 , η ( T ) = 0 in Ω ,∂λ∂ n = 0 , ∂η∂ n = 0 on (0 , T ) × ∂ Ω , Z T Z Ω c ( α f ( ˜ f ) + ˜ v η )( f − ˜ f ) ≥ ∀ f ∈ F . (105) Remark 9.
If there is no convexity constraint on the control, that is,
F ≡ L ( Q c ) , then (105) ecomes α f ( ˜ f ) χ Ω c + ˜ v η χ Ω c = 0 . Thus, the control ˜ f is given by ˜ f = (cid:18) − α f ˜ v η (cid:19) / χ Ω c . Appendix: Existence of Strong Solutions of Problem (10)
In this appendix we will prove Theorem 2.Let us introduce the weak space X := L ∞ ( L ) ∩ L ( H ) . We define the operator R : X × X → X / × X / ֒ → X × X by R ( u ε , z ε ) = ( u ε , z ε ) the solutionof the decoupled linear problem ∂ t u ε − ∆ u ε = ∇ · ( u ε + ∇ v ( z ε )) in Q,∂ t z ε − ∆ z ε = u ε + f v ( z ε ) + χ Ω c − z ε in Q,u ε (0) = u ε , z ε (0) = v ε − ε ∆ v ε in Ω ,∂u ε ∂ n = 0 , ∂z ε ∂ n = 0 on (0 , T ) × ∂ Ω , (106)where v ε := v ( z ε ) is the unique solution of problem (11). In this Appendix, we will denote v ( z ε ) only by v ε . Then, a solution of system (10) is a fixed point of R . Therefore, in order to prove theexistence of solution to system (10) we will use the Leray-Schauder fixed point theorem. In thefollowing lemmas, we will prove the hypotheses of such fixed point theorem. Lemma 9.
The operator R : X × X → X × X is well defined and compact.Proof.
Let ( u ε , z ε ) ∈ X × X . Then, from the H and H -regularity of problem (11) (see [15,Theorem 2.4.2.7 and Theorem 2.5.11] respectively) we have v ε ∈ L ∞ ( H ) ∩ L ( H ) . Thus, weconclude that ∇ v ε ∈ L ∞ ( H ) ∩ L ( H ) ֒ → L ( Q ) , and taking into account that ( u ε , z ε ) ∈ X × X ,we have ∇ · ( u ε + ∇ v ε ) = u ε + ∆ v ε + ∇ u ε + · ∇ v ε ∈ L / ( Q ) . Then, by Lemma 1 (for p = 5 / ), there33xists a unique solution u ε ∈ X / of (106) such that k u ε k X / ≤ C ( k u ε k W / , / , k u ε k X , k z ε k X ) . (107)Now, since X ֒ → L / ( Q ) and v ε ∈ L ∞ ( Q ) , we have u ε + f v ε + χ Ω c − z ε ∈ L / ( Q ) . Then, byLemma 1 (for p = 10 / ), there exists a unique solution z ε of (106) belonging to X / such that k z ε k X / ≤ C ( k z ε k W / , / n , k u ε k X , k z ε k X , k f k L ( Q ) ) . (108)Therefore, R is well defined. The compactness of R is consequence of estimates (107) and(108), and the compact embedding X / × X / ֒ → X × X . Indeed, it suffices to prove only thecompact embedding X / ֒ → X , because X / ֒ → X / . Let u ∈ X / , then from Lemma 4 we have W / , / (Ω) ֒ → H / (Ω) and W , / (Ω) ֒ → H / (Ω) ; thus u ∈ X / ֒ → L ∞ ( H / ) ∩ L / ( H / ) . (109)Then, from (109) and Lemma 5 (for ( p , s ) = ( ∞ , / and ( p , s ) = (5 / , / ) we deduce that u ∈ L ∞ ( H / ) ∩ L / ( H / ) ֒ → L ( H / ) . (110)Therefore, since the embedding H / (Ω) ֒ → H (Ω) is compact and ∂ t u ∈ L / ( Q ) , from [22,Théorème 5.1, p. 58] and (110) we obtain that X / is compactly embedded in X . Lemma 10.
Let ( u ε , v ε − ε ∆ v ε ) ∈ W / , / (Ω) × W / , / n (Ω) with u ε ≥ in Ω and f ∈ L ( Q c ) .Then, the fixed points of αR are bounded in X × X , independently of α ∈ [0 , , with u ε ≥ .Proof. We assume α ∈ (0 , . Notice that if ( u ε , z ε ) is a fixed point of αR ( u ε , z ε ) , then ( u ε , z ε ) satisfies ∂ t u ε − ∆ u ε = α ∇ · ( u ε + ∇ v ε ) in Q,∂ t z ε − ∆ z ε = α u ε + α f v ε + χ Ω c − α z ε in Q,u ε (0) = u ε , z ε (0) = v ε − ε ∆ v ε in Ω ,∂u ε ∂ n = 0 , ∂z ε ∂ n = 0 on (0 , T ) × ∂ Ω . (111)The proof is carried out in three steps: 34tep 1: u ε ≥ and Z Ω u ( t ) = m ε . Let ( u ε , v ε ) be a solution of (111), then ∂ t u ε , ∆ v ε and ∇ · ( u ε + ∇ v ε ) belong to L / ( Q ) . Testing(111) by u ε − ∈ X ֒ → L / ( Q ) ֒ → L / ( Q ) , where u ε − := min { u ε , } ≤ , and taking into accountthat u ε − = 0 if u ε ≥ ; ∇ u ε − = ∇ u ε if u ε ≤ , and ∇ u ε − = 0 if u ε > , we have ddt k u ε − k + k∇ u ε − k = − α ( u ε + ∇ v ε , ∇ u ε − ) = 0 , which implies that u ε − ≡ and, consequently, u ε ≥ and, therefore, u ε + = u ε . Finally, integrating(111) in Ω and using (13) we obtain Z Ω u ε ( t ) = m ε .Step 2: z ε is bounded in X .We observe that u ε + 1 ≥ and u ε + 1 ∈ L ∞ ( L ) . Then, in particular, u ε + 1 ∈ L ( Q ) and
25 ln( u ε + 1) = ln( u ε + 1) / ≤ ( u ε + 1) / ∈ L / ( Q ) , hence ln( u ε + 1) ∈ L / ( Q ) .Now, testing (111) by ln( u ε + 1) ∈ L / ( Q ) and (111) by − ∆ v ε ∈ L / ( W , / ) (rewrittenin terms of v ε ) we have ddt (cid:18)Z Ω ( u ε + 1) ln( u ε + 1) + 12 k∇ v ε k + ε k ∆ v ε k (cid:19) + 4 k∇√ u ε + 1 k + k ∆ v ε k + α k∇ v ε k + αε k ∆ v ε k + ε k∇ (∆ v ε ) k = − α Z Ω u ε u ε + 1 ∇ v ε · ∇ u ε + α Z Ω ∇ u ε · ∇ v ε − α Z Ω f v ε + χ Ω c ∆ v ε = α Z Ω u ε + 1 ∇ u ε · ∇ v ε − α Z Ω f v ε + χ Ω c ∆ v ε . (112)Applying Hölder and Young inequalities, we have α Z Ω u ε + 1 ∇ u ε · ∇ v ε ≤ α Z Ω |∇ u ε | u ε + 1 + α Z Ω |∇ v ε | u ε + 1 ≤ α k∇√ u ε + 1 k + α k∇ v ε k , (113) − α Z Ω f v ε + χ Ω c ∆ v ε ≤ α k f k L k v ε k L k ∆ v ε k ≤ δ k v ε k H + α C δ k f k L k v ε k H . (114)Moreover, integrating (111) in Ω , using (13), and taking into account that v ε is the unique35olution of the problem (11), we have ddt (cid:18)Z Ω v ε (cid:19) + Z Ω v ε = α m ε + α Z Ω f v ε + χ Ω c . Multiplying this equation by Z Ω v ε and using the Hölder and Young inequalities we obtain ddt (cid:18)Z Ω v ε (cid:19) + (cid:18)Z Ω v ε (cid:19) = α m ε (cid:18)Z Ω v ε (cid:19) + α (cid:18)Z Ω f v ε + χ Ω c (cid:19) (cid:18)Z Ω v ε (cid:19) ≤ (cid:18)Z Ω v ε (cid:19) + α ( m ε ) C + α C k f k k v ε k . (115)Adding (115) to (112), then replacing (113) and (114) in the resulting inequality, and taking intoaccount that α ≤ , we obtain ddt (cid:18)Z Ω ( u ε + 1) ln( u ε + 1) + 12 k v ε k H + ε k ∆ v ε k (cid:19) + 2 k∇√ u ε + 1 k + C k v ε k H + ε k∇ (∆ v ε ) k ≤ C (( m ε ) + k f k L k v ε k H ) . (116)From (116) and Gronwall lemma we deduce that k v ε k L ∞ (0 ,T ; H (Ω)) ≤ ε exp( A ( T )) (cid:0) k u ε k + k v ε k H + C ( m ε ) T (cid:1) := K ε ( m ε , T, k u ε k , k v ε k H , A ( T )) , (117)where A ( T ) := C Z T k f ( s ) k L ds = C k f k L ( L ) . Now, integrating (116) in (0,T) and using (117) we obtain Z T k v ε ( s ) k H ds ≤ ε C k u ε k + k v ε k H + ( m ε ) T + ( sup ≤ s ≤ T k v ε ( s ) k H ) A ( T ) ! := K ε ( m ε , T, k u ε k , k v ε k H , A ( T )) . (118)Therefore, from (117) and (118) we conclude that v ε is bounded in L ∞ (0 , T ; H (Ω)) ∩ L (0 , T ; H (Ω)) (independently of α ∈ (0 , ), which implies that z ε is bounded in X . Step 3: u ε is bounded in X . 36esting (111) by u ε we have ddt k u ε k + k∇ u ε k = − α ( u ε ∇ v ε , ∇ u ε ) . (119)Applying Hölder and Young inequalities, and using (6), we obtain − α ( u ε ∇ v ε , ∇ u ε ) ≤ α k u ε k L k∇ v ε k L k∇ u ε k ≤ C k u ε k / k∇ v ε k L k u ε k / H ≤ k u ε k H + C k∇ v ε k L k u ε k . (120)Replacing (120) in (119), and taking into account that ( m ε ) = (cid:18)Z Ω u ε ( t ) (cid:19) , we have ddt k u ε k + k u ε k H ≤ C k∇ v ε k L k u ε k + 2( m ε ) . (121)In particular, using (6), (117), we obtain k∇ v ε k L ≤ C ( K ε ) . Then, we can apply the Gronwall lemma in (121), obtaining k u ε k L ∞ (0 ,T ; L (Ω)) ≤ exp( C ( K ε ) )( k u ε k + 2( m ε ) T ) := K ε ( m ε , T, k u ε k , k v ε k H , A ( T )) . (122)Integrating (121) in (0 , T ) we have Z T k u ε ( s ) k H ds ≤ k u ε k + 2( m ε ) T + C ( K ε ) Z T k u ε ( s ) k ds ≤ k u ε k + 2( m ε ) T + C ( K ε ) K ε T := K ε ( m ε , T, k u ε k , k v ε k H , A ( T )) . (123)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 > . For α = 0 the result is trivial. Lemma 11.
The operator R : X × X → X × X , defined in (106), is continuous. roof. Let { ( u εm , z εm ) } m ∈ N ⊂ X × X be a sequence such that ( u εm , z εm ) → ( u ε , z ε ) in X × X . (124)In particular, { ( u εm , z εm ) } m ∈ N is bounded in X × X , thus, from (107) and (108) we deduce thatsequence { ( u εm , z εm ) := R ( u εm , z εm ) } m ∈ N is bounded in X / × X / . Then, there exists a subsequenceof { R ( u εm , z εm ) } m ∈ N , still denoted by { R ( u εm , z εm ) } m ∈ N , and an element ( b u ε , b z ε ) ∈ X / × X / suchthat R ( u εm , z εm ) → ( b u ε , b z ε ) weakly in X / × X / and strongly in X × X . (125)Now, we consider system (106) written for ( u ε , z ε ) = R ( u εm , z εm ) and ( u ε , z ε ) = ( u εm , z εm ) . From(124) and (125), taking the limit in the system depending on m , as m goes to + ∞ , we deduce that ( b u ε , b z ε ) = R (lim m → + ∞ ( u εm , z εm )) . Then, by uniqueness of limit the whole sequence { R ( u εm , z εm ) } m ∈ N converges to ( b u ε , b z ε ) strongly in X × X . Thus, operator R : X × X → X × X is continuous.Consequently, from Lemmas 9, 10 and 11, it follows that the operator R satisfy the hypothesesof the Leray-Schauder fixed point theorem. Thus, we conclude that the map R has a fixed point ( u ε , z ε ) , that is R ( u ε , z ε ) = ( u ε , z ε ) , which is a solution of system (10). Acknowledgments
F. Guillén-González and M.A. Rodríguez-Bellido have been supported by MINECO grant MTM2015-69875-P (Ministerio de Economía y Competitividad, Spain) with the participation of FEDER. E.Mallea-Zepeda has been supported by Proyecto UTA-Mayor 4740-18 (Universidad de Tarapacá,Chile). Also, E. Mallea-Zepeda expresses his gratitude to Instituto de Matemáticas Universidad deSevilla and Dpto. de Ecuaciones Diferenciales y Análisis Numérico of Universidad de Sevilla fortheir hospitality during his research stay in both centers.
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