Weak solutions and optimal control of hemivariational evolutionary Navier-Stokes equations under Rauch condition

In this paper we consider evolutionary Navier-Stokes equations subject to the nonslip boundary condition together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity. Under Rauch condition, we use the Galerkin approximation method and a weak precompactness criteria to ensure the convergence to a desired solution. Moreover a control problem associated with such system of equations is studied with the help of a stability result with respect to the external forces. In the end of this paper, a more general condition due to Z. Naniewicz, namely the directional growth condition, is considered and all the results are reexamined.


Introduction
In many engineering situations, one deals with fluids flow problems in tubes or channels, or for semipermeable walls and membranes. In practice, hydraulic control devices are used as a mechanism allowing the adjustment of orifices dimensions so that the normal velocity on the boundary of the tube is regulated to reduce the dynamic pressure. The model that describes usually this situation is repesented by Navier-Stokes equations for incompressible viscous fluids with the nonslip boundary conditions together with a Clarke subdifferential relation between the dynamic pressure and the normal component of the velocity. The resulting multivalued subdifferential boundary condition leads, after a standard variational transformation, to the so-called hemivariational inequality.
The theory of hemivariational inequalities was introduced for the first time by Panagiotopoulos [29,33,34,35,36] for the sake of generalization of the classical convex variational theory to a nonconvex one. The main tool in this effort is the generalized gradient of Clarke and Rockafellar [12,11,39]. From this perspective the litterature have seen a fast emergence of applications in a mathematical and mechanical point of view, see [34,35,27,28,30,31,32] for more details. Among the main applications of this theory, we mention Newtonian and non-Newtonian Navier-Stokes equations and their variants (Oseen model, heat-conducting fluids, miscible liquids,. . . ) with nonstandard boundary conditions ensuing from multivalued nonmonotone friction law with leak, slip or nonslip conditions..
Over the last two decades intensive research has been conducted on hemivatiational inequalities for stationary and non-stationary Navier-Stokes equations. For convex functionals, the problem has been studied essentially by Chebotarev [7,8,10]. We mention also [9] for stationary Boussinesq equations and [17] by Konovalova for non-stationary Boussinesq equations. In all these papers the considered problems was formulated as variational inequalities. In the nonconvex case, the stationary case was considered by Migorski and Ochal [22] Migroski [19], for nonstationary case Migorski and Ochal [21]. For an equilibrium approach one can see for example [3]. On the other hand, optimal control problem involving hemivariational inequalities attracts more and more attention of researchers in recent years, we refer to the introductions of [37] and [13] for a short review on the subject.
There are two main conditions that one can impose on the locally Lipschitz function under subdifferential effect, namely the classical growth condition or the Rauch condition due to J. Rauch [38]. The last one is less popular even if it was the main assumption in the beginning of the theory of hemivariational inequalities. The Rauch condition expresses actually the ultimate increase of the graph of a certain locally bounded function and is, in fact a special case of another unpopular condition, namely the directional growth condition due to Z. Naniewicz [26]. As an advantage of the Rauch condition is that it allows to avoid smallness conditions(i.e. the relationship between the constants of the problem) brought by the classical growth condition. In the case of Navier-Stokes equations, the smallness condition links the growth condition constant, the coercivity constant and the norm of the trace operator. It is however not clear how it can be checked in a concrete situation. Another advantage, is that it allows us to consider "stanger" functions at infinity. In fact the only thing we require to the function is that the essential supremum of the function on the left side to be greater that the essential infimum on the right side.
Among the disadvantages of Rauch condition is that although it insure the existence of a solution, does not allow the conclusion that the non-convex functional is locally Lipschitz or even finite on the whole space. The Aubin-Clarke formula can't be used and a slight change on the definition of a solution have to be made. On the other hand we are looking for the dynamical pressure in a larger space which makes harder the question of uniqueness without a classical growth condition even if a monotonicity type assumption is acquired [18]. Finally it is worth to mention that there is no direct link between the Rauch condition and the classical growth condition and the choice depends mainly on the concrete situation.
The present paper represents a continuation of our previous paper [18] where existence and optimal control questions involving stationary Navier-Stokes problem with multivalued nonmonotone boundary condition are studied. In this paper we tackle the non-stationary problem. Always under Rauch condition, we use the Faedo-Galerkin approximation to regularize the system at the level of the multivalued boundary condition and we use the fact that the approximation sequence so-obtained is weakly precompact in the space of integrable functions. We also make advantage of the techniques used in [21] at the level of the nonlinear term to ensure the convergence of the approximate sequence to the desired solution. This study can be also done with the directional growth condition as a generalization. The question of the existence of an optimal control is important in applications. We tackle this subject in the spirit of the works of Barbu [2] and Migorski [23].
The outline of this paper is as follows. In section 2 we state the problem and give its hemivariational form by using the Lamb formulation. In section 3 we regularize our problem by using the Faedo-Galerkin approximation method and prove the existence of solutions to the regularized problem. By combining techniques from [21] and [18] we will provide an existence result in section 4. Section 5 is devoted to the optimal control problem subjected to our evolutionary hemivariational inequality, while section 6 is dedicated to the directional growth condition as a generalization of the Rauch condition.

Problem statement
Let Ω be a bounded simply connected domain in R d with d = 2, 3 with connected boundary ∂Ω of class C 2 and Ω T = (0, T ) × Ω where T > 0. We consider the following evolution Navier-Stokes system: This system describes the flux of an incompressible viscous fluid in a domain Ω subjected to an external forces fext = {f ext,k } d k=1 . u = {u k } d k=1 , p and ν denote respectively the velocity, the pressure and the kinematic viscosity of the fluid. The nonlinear term (u.∇)u (called the convective term) is the symbolic notation of the vector d j=1 u j ∂ui ∂xj . As usual we use the Lamb formulation [15, Chapter I] to rewrite the evolution Navier-Stokes system as follows: where p = p + 1 2 |u| 2 is the total head of the fluid, or "total pressure" .
We suppose that on the boundary ∂Ω the tangential component of the velocity vector are known, and without loss of generality we put them equal to zero (the nonslip condition): where n = {n k } d k=1 is the unit outward normal on the boundary ∂Ω and u N (t, x) = u(t, x).n = d 1 u i (t, x)n i denotes the normal component of the vector u. Moreover, we assume the following subdifferential boundary condition: where ∂j(ξ) is the Clarke subdifferential of j at ξ and is given by and j 0 (ξ; h) is the generalized derivative of a locally Lipschitz function j at ξ ∈ V in the direction h ∈ V defined by: To work conveniently on the problem (2.4)-(2.8), we need the following functional spaces: Then we have V ⊂ H ≃ H * ⊂ V * , with all the embedding being continuous and compact. Moreover, for an interval time [0, T ], we introduce the following spaces: Then, we have also the following continuous embedding, W ⊂ V ⊂ H ⊂ V * .
We consider the operators A : V → V * and B : V × V → V * defined by: for all u, v, w ∈ V . As usual, we will use the notation B[.] = B(., .). It is well known (cf. [4]) that if the domain Ω is simply connected, the bilinear form V which is equivalent to the H 1 (Ω, R d )-norm. Hence, it is clear that the operator A is coercive.
In order to give the weak formulation to the problem (2.4)-(2.8), we multiply it by a certain v ∈ V and apply the Green formula. We obtain: .v dx. From the relation (2.8), by using the definition of the Clarke subdifferential, we have The relations (2.12)-(2.13) yields to the following weak formulation The equation above is called an hemivariational inequality.
We have already mentioned in the introduction that the Rauch assumption is not sufficient to make the functional J(u) = ∂Ω j(u) dσ locally lipchitz or even finite in the whole space V. Because of this reason, a slight modified definition of being a solution should be adopted. Define the functional space where γ is the trace operator from V in L 2 (∂Ω; R d ). Now, we are able to give what we mean by a solution to the problem (EHV I).
Note that since W ⊂ C(0, T ; H) continuously the initial condition u(0) = u 0 makes sense in H. To justify the above definition we refer to [18] and [26].

Regularized Problem
In what follows we restrict our study to superpotentials j which are independent of x and which subdifferential is obtained by "filling in the gaps" procedure (cf. [38]). Let θ ∈ L ∞ loc (R), for ε > 0 and t ∈ R, we define: For a fixed t ∈ R, the functions θ ε , θε are decreasing and increasing in ε, respectively. Let and letθ(t) : R → 2 R be a multifunction defined by From Chang [5] we know that a locally Lipschitz function j : R → R can be determined up to an additive constant by the relation such that ∂j(t) ⊂θ(t) for all t ∈ R. If moreover, the limits θ(t ± 0) exist for every t ∈ R, then ∂j(t) =θ(t).
In order to define the regularized problem, we consider the mollifier where ⋆ denotes the convolution product. Consider the following auxiliary problem associated to (EHVI): Now and in order to define the corresponding finite dimensional problem, we shall use the Faedo-Galerkin approximation approach. Let us consider a Galerkin basis .., zm} are linearly independent. Consider Vm = span{z 1 , z 2 , . . . , zm), we have Vm Let {um(0)} be an approximation of the given initial value u 0 such that um(0) ∈ Vm for m ∈ IN and suppose that and, . We consider the following regularized Galerkin system of finite dimensional differential equations associated to (EHVI): For the existence of solutions we will need the following hypothesis H(θ) : Remark 1 If one assume more generally that for some real number α, it is possible to come back to the situation where the Rauch assumption is imposed by simply replacing θ by θ − α and f by f − α.
Proof This is a classical result in the stationary case(cf. [18,Lemma 3.2]. It suffices to integrate over t ∈ (0, T ) to obtain the result.

Proposition 2
The regularized problem (P m ε ) has at leat one solution um.
The matrix with elements z k , z i , 1 ≤ i, k ≤ m is nonsingular (i.e det{ z k , z i } m k,i=1 = 0), we invert the matrix, then the equation (3.3) can be written in the usual form: (3.6) The differential system (3.5) with the initial condition (3.6) define uniquely the scalar c km on the interval [0, tm). Then the solution um exists on [0, tm), we can extended it on the closed interval [0, T ] by using a priori estimates in Lemma 2.
Since the scalar function t → f (t), z i in the equation (3.3) are square integrable, so are the functions c km and therefore, for each m we have: In this section we will prove the existence of solutions to the problem (EHVI) by analysing the convergence of the sequence (um)m solutions to (P m ε ). To do so we need some a priori estimates Proof From Proposition 1, the regularized problem (P m ε ) has at least one solution {um}m. By replacing v by um(t) in (P m ε ), we get for a.e t ∈ [0, T ]: (4.1) Because of (3.7) we have: Then the equation (4.1) becomes: By the coerciveness of A , Cauchy-Schwartz inequality and Young inequality we obtain for a.e. t ∈ (0, T )( M is the constant of coercivity). Integrating the previous equation from 0 to s, 0 ≤ s ≤ T and using Lemma 1, one has: The right hand side of the previous inequality, is finite and independent of m. We deduce that {um}m is bounded in L ∞ (0, T ; H). Again from (4.4) we have: Then: then {um}m remains in a bounded subset of V.

Theorem 1
Under assumption H(θ), the problem (EHVI) has at least one solution.
Proof From Proposition 1 and Proposition 2, we get Now, we focus on the weak convergence of the nonlinear term B[um] by using exactly the same procedure as in [21]. For the case d = 2, we obtain from Temam [40] that Moreover, the operator A is continuous. Hence {u ′ m } is bounded in V * . Thus, by passing to a next subsequence, if necessary, it follows Using the facts that W ⊂ C(0, T ; H) continuously, W ⊂ H compactly and W ⊂ L 2 (0, T ; L 2 (Γ, R n )) compactly, we have u ∈ C(0, T ; H) and Since um → u weakly in V and in H, analogously as in Ahmed [1], we have B[um] → B[u] weakly in V * . We remark that if d = 3 we also have the convergence of B[um] → B[u] weakly in V * by a compactness embedding theorem as in [1].

Optimal Control
In this section, we provide a result on dependence of solutions with respect to the density of the external forces and use it to study the distributed parameter optimal control problem corresponding to it.
Let f ∈ L 2 (0, T ; V * ). Under H(θ), we denote by S θ u0 (f) ⊂ V the solution set corresponding to f of the problem (EHVI). That is, u ∈ W and there exists κ ∈ where δ 0 and δ 1 are independent from u.
Proof By definition of θ(u N ) and θ(u N ) we have for every ε > 0, there exists δ with |µ| < δ such that and there exists δ with |µ| < δ such that Theorem 2 Under H(θ) assume that fm, f ∈ L 2 (0, T ; V * ) such that fm → f weakly in V * . Let {um}m ⊂ W be a sequence such that um ∈ S θ u0 (fm) for each m ∈ N, then we can find a subsequence (still denoted with the same symbol) such that um → u weakly in V and u ∈ S θ u0 (f).
Proof Let fm, f ∈ V * with fm → f weakly in V * . Let {um}m be a sequence such that um ∈ S θ u0 (fm) for each m ∈ N, then by Theorem 1, there exists κm ∈ for a.e t ∈ [0, T ] and all v ∈ V ∩ L ∞ N (∂Ω). With the same calculations as in the last section one obtains Integrating over (0, t) we get It follows that {um}m is bounded in L ∞ (0, T ; H)∩L 2 (0, T ; V ). Hence, by passing to a subsequence if necessary, there exists u such that {um}m converges to u weakly in L 2 (0, T ; V ) and weak− * in L ∞ (0, T ; V ). Using the compactness of the trace operator γ, we may assume that γ um → γ u in L 2 (0, T ; H) and then γ um(t, z) → γ u(t, z) for a.e. (t, z) Thus for any µ > 0 there exists m 0 such that for all m > m 0 we have a.e on [0, T ] × ∂Ω \ ω By using triangle inequality, we have that Analogously we prove the inequality Taking the limit µ → 0 + , we obtain for each m ≥ m 0 For α as small as possible, we obtain On the other hand, for each ε > 0 there is aε > 0, such that for |ν| < aε one have Consequently, we can extract from {κm}m a subsequence( denoted with the same symbol) that converges in L 1 ((0, T ) × ∂Ω) to some κ ∈ L 1 ((0, T ) × ∂Ω). By passing to the limit in (5.1), we get Remark 2 We will need Theorem 2 just for external forces in L 2 (0, T ; H). As in this situation the duality between V and V * coincides with the one on H, this will bring no more difficulties. Remark 3 One can prove in the same way as in [18,Theorem 5.1] that the solutions of (EHVI) are stable under perturbation of θ.
In the remaining of this section, we will use the notation S(f ) instead of S θ u0 (f). We follow Migorski [23] and we let U = L 2 (0, T ; H) be the space of controls and U ad a nonempty subset of U consisting of admissible controls. Let F : U × V → R be the objective functional we want to minimize. The control problem reads as follows: Find a controlf ∈ U ad and a stateû ∈ S(f ) such that : A pair (f,û) which solves (5.3) is called an optimal solution. The existence of such optimal solutions can be proved by using Theorem 2. To do so, we need the following additional hypotheses: H(U ad ) U ad is a bounded and weakly closed subset of U.
H(F ) F is lower semicontinuous with respect to U × V endowed with the weak topology. Proof Let (fm, um) be a minimizing sequence for the problem (5.3), i.e fm ∈ U ad and um ∈ S(fm) such that It follows that the sequence fm belongs to a bounded subset of the reflexive Banach space V. We may then assume that fm →f weakly in V (by passing to a subsequence if necessary). By H(U ad ), we havef ∈ U ad . From Theorem 2, we obtain, by again passing to a subsequence if necessary, that um →û weakly in V withû ∈ S(f ). By H(F ), we have ϑ ≤ F (f,û) ≤ lim inf m→∞ F (fm, um) = ϑ. Which completes the proof.
Next we apply Theorem 3 in a concrete example. Let X be another Hilbert space, X := L 2 (0, T ; X), X ad ⊂ X the set of admissible controls and C ∈ L (X , U) a bounded linear operator from X to U. Let f ∈ L 2 (0, T ; V * ), we aim to study the following optimal control problem Find a control w ∈ X ad and a stateû ∈ S(Cŵ + f ) such that : J (ŵ,û) = inf {J (w, u) : w ∈ X ad , u ∈ S(Cw + f )} (5.4) where the objective functional is given by for some function h : X → R and z ∈ L 2 (0, T ; H). Let us first announce the following corollaries of Theorem 2: Corollary 1 Under H(θ) assume that ϕm, ϕ, f ∈ V * such that ϕm → ϕ weakly in V * . Then for every um ∈ S(ϕm + f ), we can find a subsequence (still denoted with the same symbol) such that um → u in V and u ∈ S(ϕ + f ).
Proof It suffices to take fm = ϕm + f in Theorem 2.
Corollary 2 Under H(θ), assume that f ∈ V * and wm, w ∈ X are such that wm converges weakly to w in X . Then for every sequence {um}m such that um ∈ S(Cwm + f ), we can find a subsequence that converges weakly in L 2 (0, T ; V ) to u ∈ S(Cw + f ).
Proof It suffices to take ϕm = Cwm in Corollary 1.
(ii) X ad is a weakly compact subset of X .
(iii) the function h : X → R is convex, lower semi-continuous and satisfies the coercivity condition |h(w)| ≥ α|w| 2 X + β for some α > 0 and β ∈ R. |.| X stands for the norm of the Hilbert space X. Proof Let (wn, un) be a minimizing sequence to the problem (5.4), i.e wn ∈ X ad and um ∈ S(Cwn + f ) such that lim m→∞ J (wm, um) = inf {J (w, u) : w ∈ X ad , u ∈ S(Cw + f )} Denote fm = Cwm + f , F (fm, um) = J (wm, um) and U ad = CX ad . It suffices now to apply Theorem 5.3 for U ad and F .

Directional growth condition
As mentioned in the introduction, Rauch condition is a particular case of the directional growth condition due to Z. Naniwiecz [26]. It is of common knowledge that the foregoing mentioned conditions are sufficient to establish the existence of solution without any additional growth hypothesis on j. The notion of being solution needs only to be modified. Here we will reconsider the same problem of evolutionary hemivariational Navier-Stokes equations but with the more general condition of directional growth.
Let j : ∂Ω × R → R be a measurable function with respect to the first argument and locally Lipschitz with respect to the second argument. We assume the following: H(j) there exists β : ∂Ω × R + → R integrable with respect to the first argument and nondecreasing with respect to the second argument such that |j(x, ξ) − j(x, η)| ≤ β(x, r)|ξ − η|, ∀ξ, η ∈ B(0, r), r ≥ 0 H(j 0 ) there exists a function α : ∂Ω × R + → R square-integrable with respect to the first argument and nondecreasing with respect to the second argument such that the following estimate holds for almost every x ∈ ∂Ω and for any ξ, η ∈ R with −r ≤ η ≤ r, r ≥ 0.
Note that due to the integrability of β with respect to the first argument and H(j) , the integral above is finite for each um, v ∈ Vm. In fact, we have Since β(., u N L ∞ (∂Ω) + 1) ∈ L 1 (∂Ω) the integrability of j ′ εm (u mN ).v N over ∂Ω follows immediately for any v ∈ Vm.
The problem (P m ε ) has at least one solution in Vm. In fact, substitution of um(t) = m k=1 c km (t)ϕ k gives an initial value problem for a system of first order ordinary differential equations for c km (.), k = 1, 2, . . . , m. Its solvability on some interval [0, tm) follows from Carathéodory theorem. This solution can be extended on the closed interval [0, T ] by using the a priori estimates below.
Remark 5 It is an easy task to check that the results in section 5, regarding optimal solution, are also valid if one replace the assumption H(θ) by the more general assumption H(j 0 ).