# Weak Solutions and Optimal Control of Hemivariational Evolutionary Navier-Stokes Equations under Rauch Condition

Authors: Hicham Mahdioui; Sultana Ben Aadi; Khalid Akhlil
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we consider the 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 the Rauch condition, we use the Galerkin approximation method and a weak precompactness criterion 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. At 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.


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## Body

## 1. Introduction

In many engineering situations, one deals with fluid 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 orifice dimensions so that the normal velocity on the boundary of the tube is regulated to reduce the dynamic pressure. The model that usually describes this situation is repesented by the 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 [1–5] 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 [6–8]. From this perspective, the literature has seen a fast emergence of applications in a mathematical and mechanical point of view, see [3, 4, 9–13] for more details. Among the main applications of this theory, we mention the Newtonian and non-Newtonian Navier-Stokes equations and their variants (the Oseen model, heat-conducting fluids, miscible liquids, etc.) with nonstandard boundary conditions ensuing from the multivalued nonmonotone friction law with leak, slip, or nonslip conditions. For recent directions on the hemivariational theory, we refer to [14–17].Over the last two decades, intensive research has been conducted on hemivariational inequalities for the stationary and nonstationary Navier-Stokes equations. For convex functionals, the problem has been studied essentially by Chebotarev [18–20]. We mention also [21] for stationary Boussinesq equations and [22] by Konovalova for nonstationary Boussinesq equations. In all these papers, the considered problems were formulated as variational inequalities. In the nonconvex case, the stationary case was considered by Migórski and Ochal [23] and Migórski [24], and the nonstationary case was considered by Migórski and Ochal in [25]; see also [26]. For an equilibrium approach, one can see for example [27]. On the other hand, the optimal control problem involving hemivariational inequalities attracts more and more attention from researchers in recent years. We refer to the introductions of [28, 29] for a short review on the subject.There are two main conditions that one can impose on the locally Lipschitz function under a subdifferential effect, namely the classical growth condition or the Rauch condition due to J. Rauch [30]. 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 Naniewicz [31]. An advantage of the Rauch condition is that it allows avoiding smallness conditions (i.e., the relationship between the constants of the problem) brought by the classical growth condition. In the case of the 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 the “Stanger” functions at infinity. In fact, the only thing we require from the function is for the essential supremum of the function on the left side to be greater than the essential infimum on the right side.Among the disadvantages of the Rauch condition is that although it ensures the existence of a solution, it does not allow the conclusion that the nonconvex functional is locally Lipschitz or even finite on the whole space. The Aubin-Clarke formula cannot be used, and a slight change in the definition of a solution has to be made. On the other hand, we are looking for the dynamical pressure in a larger space, which makes the question of uniqueness more difficult without a classical growth condition even if a monotonicity type assumption is acquired [32]. Finally, it is worth mentioning 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 [32], where existence and optimal control questions involving the stationary Navier-Stokes problem with the multivalued nonmonotone boundary condition are studied. In this paper, we tackle the nonstationary problem. Always under the 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 take advantage of the techniques used in [25] 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 [33] and Migórski [34].The outline of this paper is as follows. In section2, 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 [25, 32], 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.


## 2. Problem Statement

LetΩ be a bounded simply connected domain in ℝd with d=2,3 with connected boundary ∂Ω of class C2 and ΩT=0,T×Ω where T>0. We consider the following evolution Navier-Stokes system:
(1)u′−νΔu+u·∇u+∇p=fext,inΩT,divu=0,inΩT,u0=u0,inΩ.This system describes the flux of an incompressible viscous fluid in a domainΩ subjected to an external force fext=fext,kk=1d. u=ukk=1d, 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 ∑j=1duj∂ui/∂xj. As usual, we use the Lamb formulation ([35], chapter I) to rewrite the evolution Navier-Stokes system as follows:
(2)u′+νrotrotu+rotu×u+∇p~=fext,inΩT,(3)divu=0,inΩT,(4)u0=u0,inΩ,where p~=p+1/2u2 is the total head of the fluid, or “total pressure.”We suppose that on boundary∂Ω, the tangential components of the velocity vector are known, and without loss of generality, we put them equal to zero (the nonslip condition):
(5)uτt,x≔ut,x−uNt,xn,on∂ΩT≔0,T×∂Ω,where n=nkk=1d is the unit outward normal on the boundary ∂Ω and uNt,x=ut,x. n=∑1duit,xnidenotes the normal component of the vectoru. Moreover, we assume the following subdifferential boundary condition:
(6)p~t,x∈∂juNt,xon∂ΩT,where ∂jξ is the Clarke subdifferential of j at ξ and is given by
(7)∂jξ=ξ∗∈V∗:j0ξ;h≥ξ∗,hV∗×Vforallh∈V,and j0ξ;h is the generalized derivative of a locally Lipschitz function j at ξ∈V in the direction h∈V defined by
(8)j0ξ;v=limsupν→ξ,λ↓0jν+λv−jνλ.To work conveniently on problems (2), (3), (4), (5), and (6), we need the following functional spaces:
(9)C=u∈C∞Ω;ℝd:divu=0inΩ,uτ=0on∂Ω,V=theclosureofCinthenormofH1Ω;ℝd,H=theclosureofCinthenormofL2Ω;ℝd.Then, we haveV⊂H≃H∗⊂V∗, with all the embedding being continuous and compact. Moreover, for an interval time 0,T, we introduce the following spaces:
(10)V=L20,T;V,H=L20,T;H,W=u∈V:u′∈V∗.Then, we also have the following continuous embedding,W⊂V⊂H⊂V∗.We consider the operatorsA:V⟶V∗ and B:V×V⟶V∗ defined by
(11)Au,v=ν∫Ωrotu·rotvdx,Bu,v,w=∫Ωrotu×v·wdx,for all u,v,w∈V. As usual, we will use the notation B.=B.,.. It is well known (cf. [36]) that if the domain Ω is simply connected, the bilinear form
(12)u,vV=∫Ωrotu·rotvdx,generates a norm in V, uV=u,vV1/2, which is equivalent to the H1Ω,ℝd-norm. Hence, it is clear that the operator A is coercive.In order to give the weak formulation to problems (2), (3), (4), (5), and (6), we multiply it by a certain v∈V and apply the Green formula. We obtain
(13)u′t+Aut+But,v+∫∂Ωp~t,xvNdσx=ft,v,where ft,v=∫Ωfextt·vdx. From relation (6), by using the definition of the Clarke subdifferential, we have
(14)∫∂Ωp~t,xvNxdσx≤∫∂Ωj0x,uNx;vNxdσx.The relations (13) and (14) yield to the following weak formulation:
(15)EHVIFindu∈Wsuchthatforallv∈Vanda.e.t∈0,T,u′t+Aut+But,v+∫∂Ωj0t,x,uNt,x;vNxdσx≥ft,v,u0=u0.The equation above is called an hemivariational inequality.We have already mentioned inIntroduction that the Rauch assumption is not sufficient to make the functional Ju=∫∂Ωjudσ locally lipschitz 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
(16)LN∞∂Ω=u,uN=γu·n∈L∞∂Ω;ℝ,where γ is the trace operator from V in L2∂Ω;ℝd. Now, we are able to give what we mean by a solution to the problem EHVI.Definition 1.
A functionu∈W is said to be the solution of EHVI if there exists κ∈L10,T×∂Ω,ℝ such that for a.e. t∈0,T(17)u′t+Aut+But,v+∫∂Ωκt,zvNzdσz=ft,v,∀v∈V∩LN∞∂Ω,κt,z∈∂jt,z;uNt,z,fora.e.t,z∈0,T×∂Ω,u0=u0.Note that sinceW⊂C0,T;H continuously, the initial condition u0=u0 makes sense in H. To justify the above definition, we refer to [31, 32].


## 3. Regularized Problem

In what follows, we restrict our study to superpotentialsj, which are independent of x and which subdifferential is obtained by “filling in the gaps” procedure (cf. [30]). Let θ∈Lloc∞ℝ, for ε>0 and t∈ℝ, we define
(18)θ¯εt=essinft−s≤εθs,θ¯εt=esssupt−s≤εθs.For a fixedt∈ℝ, the functions θ¯ε,θ¯ε are decreasing and increasing in ε, respectively. Let
(19)θ¯t=limε→0+θ¯εt,θ¯t=limε→0+θ¯εt,and let θ^t:ℝ⟶2ℝ be a multifunction defined by
(20)θ^t=θ¯t,θ¯t.From Chang [37], we know that a locally Lipschitz function j:ℝ⟶ℝ can be determined up to an additive constant by the relation
(21)jt=∫0tθsds,such that ∂jt⊂θ^t for all t∈ℝ. If moreover, the limits θt±0 exist for every t∈ℝ, then ∂jt=θ^t.In order to define the regularized problem, we consider the mollifier(22)h∈C0∞−1,1,h≥0with∫−∞+∞hsds=1,and let
(23)θε=hε⋆θwithhεs=1εhsε,where ⋆ denotes the convolution product.Consider the following auxiliary problem associated toEHVI:
(24)PεFindu∈Wsuchthatforallv∈V∩LN∞∂Ω,a.e.t∈0,T,u′t+Aut+But,v+∫∂ΩθεuNt,xvNdσ=ft,v,u0=u0.Now and in order to define the corresponding finite dimensional problem, we shall use the Faedo-Galerkin approximation approach. Let us consider a Galerkin basisz1,z2,⋯ in V∩LN∞∂Ω, i.e., z1,z2,⋯ forms at most a countable sequence of elements of V∩LN∞∂Ω, finitely z1,z2,⋯,zm are linearly independent. Consider Vm=spanz1,z2,⋯,zm, we have Vm⊂Vm+1 and ∪m≥0Vm¯=V∩LN∞∂Ω. Moreover, the family Vmm satisfies
(25)∀v∈V∩LN∞∂Ω,∃vmm,vm∈Vm,such thatvm⟶vinV∩LN∞∂Ω, asn⟶+∞.Letum0 be an approximation of the given initial value u0 such that um0∈Vm for m∈IN and suppose that
(26)um0⟶u0inH,asm⟶+∞,um0misboundedinV∩LN∞∂Ω.We consider the following regularized Galerkin system of finite dimensional differential equations associated toEHVI:
(27)PεmFindum∈Wmsuchthatforallv∈Vm,a.e.t∈0,T,um′t+Aumt+Bumt,v+∫∂ΩθεmumNtvdσ=ft,v,um0=um0,where Wm=u∈L20,T;Vm:u′∈L20,T;Vm.The generalized derivativeLu=u′ restricted to the subset DL=u∈V:u′∈V∗andu0=u0=u∈W:u0=u0 defines a linear operator L:DL⊂V⟶∗ given by
(28)≪Lu,v≫=∫0Tu′t,vtdtforallv∈V.For the existence of solutions, we will need the following hypothesisHθ:
(1)
(Chang assumption)θ∈Lloc∞ℝ, θt±0 exists for any t∈ℝ(2)
(Rauch assumption) There isδ0>0 such that(29)esssup−∞,−δ0θt≤0≤essinfδ0,+∞θt.Remark 2.
If one assumes more generally that(30)esssup−∞,−δ0θt≤α≤essinfδ0,+∞θt,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.Remark 3.
We point out that the Rauch and growth conditions are completely independent. Indeed, by taking examples, we show that neither of both conditions implies the other. In fact, consider the functionβ:ℝ⟶ℝ defined by
(31)βt=t3,ift≥1,−t,ift<1,where t stands for the integer part of t. One can prove easily (eventually by a contradiction argument) that the function β satisifies the Rauch condition while the growth condition cannot be satisfied. Conversely, one can take a function β:ℝ⟶ℝ defined by βt=1+t; it is clear that it satisfies the growth condition but not the Rauch condition as β is positive for negative values.Lemma 4.
Suppose thatHθ holds. Then we can determine ρ1,ρ2>0, such that for every u∈Vm(32)∫0,T×∂ΩθεuNt,zuNt,zdσzdt≥−ρ1ρ2T·σ∂Ω.Proof.
This is a classical result in the stationary case (cf. [32], Lemma 3.2). It suffices to integrate over t∈0,T to obtain the result.Proposition 5.
The sequenceθεmumm∈IN is weakly precompact in L10,T×∂Ω.Proof.
The proof is similar to ([32], Proposition 3.7) with minor changes consisting mainly in replacing ∂Ω by 0,T×∂Ω and remarking that um′t,umt≥0 for a.e.t∈0,T.Proposition 6.
The regularized problemPεm has at leat one solution um.Proof.
We substituteumt=∑k=1mckmtzk in Pεm to obtain
(33)∑k=1mckm′tzk,zi+∑k=1mckmtAzk,zi+∑k,j=1mckmtcjmtbzk,zj,zi+∫∂Ωθεm∑k=1mckmtzk⋅nzi⋅ndσ=ft,zi,i=1,⋯m,a.e.t∈0,T,(34)cim0=αi0,i=1,⋯,n.
The matrix with elementszk,zi, 1≤i, k≤m is nonsingular (i.e., detzk,zik,i=1m≠0), we invert the matrix, then equation (33) can be written in the usual form:
(35)ckm′t=∑i=1mβkift,zi−∑i=1mαkicimt−∑i,j=1ξkijckmtcjmt−∑i=1mϱki∫∂Ωθεm∑k=1mckmtzk⋅nzi⋅ndσ,where the initial values ckm0, k=1,⋯,m are given, i.e., um0=∑k=1mckm0zk:
(36)ckm0isthekthcomponentofum0.The differential system (35) with the initial condition (36) define uniquely the scalar ckm on the interval 0,tm. Then, the solution um exists on 0,tm, and we can extend it on the closed interval 0,T by using a priori estimates in Lemma 7. Since the scalar function t⟶ft,zi in equation (33) are square integrable, so are the functions ckm; therefore, for each m we have:
(37)um∈L20,T;V,um′∈L20,T;V∗.


## 4. Existence Result

In this section, we will prove the existence of solutions to the problemEHVI by analysing the convergence of the sequence umm solutions to Pεm. To do so, we need some a priori estimates.Lemma 7.
The solutionumm is bounded in L20,T;V∩L∞0,T;H.Proof.
From Proposition5, the regularized problem Pεm has at least one solution umm. By replacing v by umt in Pεm, we get for a.e. t∈0,T(38)um′t+Aumt+Bumt,umt+∫∂ΩθεmumNt,xumNt,xdσ=ft,umt.
Because of (37) we have
(39)um′t,umt=12ddtumt2.
Then, equation (38) becomes
(40)ddtumt2+2Aumt,umt+2∫∂ΩθεmumNt,xumNt,xdσ=2ft,umtV∗fora.e.t∈0,T.
By the coerciveness ofA, the Cauchy-Schwartz inequality, and the Young inequality, we obtain
(41)ddtumt2+Mumt2+2∫∂ΩθεmumNt,xumNt,xdσ≤2MftV∗,for a.e. t∈0,T (M is the constant of coercivity). Integrating equation (40) from 0 to s, 0≤s≤T, and using Lemma 4, one has
(42)ums2+M∫0sumτV2dτ≤2M∫0sfτV∗2dτ+2ρ1ρ2s·σ∂Ω+um02.
Hence(43)sups∈0,Tums2≤2MfV∗2+2ρ1ρ2T⋅σ∂Ω+um02.
The right-hand side of the previous inequality is finite and independent ofm. We deduce that umm is bounded in L∞0,T;H.
Again, from (42) we have
(44)M∫0TumtV2dt≤2MfV∗2+2ρ1ρ2T·σ∂Ω+um02.
Then(45)umV2≤2M2fV∗2+2Mρ1ρ2T·σ∂Ω+1Mum02.
Then,umm remains in a bounded subset of V.Theorem 8.
Under assumptionHθ, the problem EHVI has at least one solution.Proof.
From Proposition5 and Proposition 6, we get
(46)um⟶uweaklyinV,um⟶uweakly−starinL∞0,T;H,θεmumN⟶κweaklyinL10,T×∂Ω.
Now, we focus on the weak convergence of the nonlinear termBum by using exactly the same procedure as in [25]. For the case d=2, we obtain from Temam [38] the following:
(47)BumV∗≤cumL∞0,T;HumVwithc>0.
Moreover, operatorA is continuous. Hence um′ is bounded in V∗. Thus, by passing to a next subsequence, if necessary, we have the following:
(48)u′m⟶u′weaklyinW.
Using the facts thatW⊂C0,T;H continuously, W⊂H compactly, and W⊂L20,T;L2Γ,ℝn compactly, we have u∈C0,T;H and
(49)um⟶uinH,γum⟶γuinL20,T;L2Γ;ℝd.
Sinceum⟶u weakly in V and in H, analogously as in Ahmed [39], we have Bum⟶Bu weakly in V∗. We remark that if d=3, we also have the convergence of Bum⟶Bu weakly in V∗ by a compactness embedding theorem as in [39].
Letϕ∈C0∞0,T and v∈V∩LN∞∂Ω. Then, there exists vmm∈ℕ such that vm∈Vm and vm⟶v in V, as m⟶∞. Denoting ψmx,t=ϕtvmx and ψx,t=ϕtvx, we have ψn⟶ψ in W. From Pεm, we have the following:
(50)∫0Tum′t+Aumt+Bumt,ψmt+∫0,T×∂ΩθεmumNtψNmtdσdt=∫0Tft,ψmtdt.
Using the above convergences, lettingm⟶+∞, we obtain
(51)∫0Tu′t+Aut+But,vϕtdt+∫0,T×∂Ωκ·vNϕtdσdt=∫0Tft,vϕtdt.
Sinceϕ is arbitrary, we deduce that
(52)u′t+Aut+But,v+∫∂Ωκ·vNdσ=ft,v,for a.e. t∈0,T and for all v∈V∩LN∞∂Ω.
In order to complete the proof, it will be shown that(53)κt,z∈θ^uNt,z=∂cjuNt,z,fora.e.t,z∈0,T×∂Ω.
Sinceγum⟶γu in L20,T;L2∂Ω, we obtain umN⟶uN in L20,T;L2∂Ω, and consequently, umNt,x⟶uNt,x for a.e. t,x∈0,T×∂Ω; then, by applying Egoroff’s theorem, we can find that for any α>0, we can determine ω⊂0,T×∂Ω with σω<α, such that
(54)umN⟶uN,uniformlyon0,T×∂Ω\ω,with uN∈L∞0,T×∂Ω\ω. Thus, for any α>0, we can find ω⊂0,T×∂Ω with σω<α, such that for any μ>0 and for ε<ε0<μ/2 and n>n0>2/μ, we have
(55)umN−uN<μ2,on0,T×∂Ω\ω.
Consequently, one obtains(56)θεumN≤esssupumN−ξ≤εθξ≤esssupumN−ξ≤μ2θξ≤esssupumN−ξ≤μθξ=θ¯μuN.
Analogously, we prove the inequality(57)θ¯μuN=essinfuN−ξ≤μθξ≤θεumN.
We now takev≥0 a.e. on 0,T×∂Ω\ω with v∈L∞0,T×∂Ω\ω. This implies
(58)∫0,T×∂Ω\ωθ¯μuNvdσ≤∫0,T×∂Ω\ωθεumNvdσ≤∫0,T×∂Ω\ωθ¯μuNvdσ.
Taking the limits asε⟶0 and m⟶∞, we obtain
(59)∫0,T×∂Ω\ωθ¯μuNvdσ≤∫0,T×∂Ω\ωκvdσ≤∫∂0,T×Ωωθ¯μuNvdσ,and as μ⟶0+, we obtain
(60)∫0,T×∂Ω\ωθ¯uNvdσ≤∫0,T×∂Ω\ωκvdσ≤∫0,T×∂Ω\ωθ¯uNvdσ.
Sincev is arbitrary, we have
(61)κ∈θ¯uN,θ¯uN=θ^uN,where σ∂Ω<α. For α as small as possible, we obtain the result.


## 5. Optimal Control

In this section, we provide a result on the 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.Letf∈L20,T;V∗. Under Hθ, we denote by Su0θf⊂V the solution set corresponding to f of the problem EHVI. That is, u∈W and there exists κ∈L10,T×∂Ω such that u0=u0∈H, κ∈∂juN=θ^uN and
(62)u′t+Aut+But,v+∫∂Ωκ·vNzdσz=ft,v,for a.e. t∈0,T and all v∈V∩LN∞∂Ω.Lemma 9.
Letf∈L20,T;V∗. For every u∈Su0θf, there exist δ0,δ1>0 such that
(63)∫0,T×∂Ωκ·uNdσdt≥−δ0δ1Tσ∂Ω,where δ0 and δ1 are independent from u.Proof.
By definition ofθ¯uN and θ¯uN, we have for every ε>0, there exists δ¯ with μ<δ¯ such that
(64)θ¯μuN−ε≤θ¯uN≤θ¯μuN+ε,and there exists δ¯ with μ<δ¯, such that
(65)θ¯μuN−ε≤θ¯uN≤θ¯μuN+ε.
It follows that(66)θ¯μuN−ε≤κ≤θ¯μuN+ε.
That is,(67)essinfuN−s<μθs−ε≤κ≤esssupuN−s<μθs+ε.
Enlarging the bounds, we obtain(68)essinfuN−μ≤s<+∞θs−ε≤κ≤esssup−∞<s≤uN+μθs+ε.
Forε small enough and μ<δ¯,δ¯, one has
(69)essinfuN−δ¯∧δ¯≤s<+∞θs−ε≤κ≤esssup−∞<s≤uN+δ¯∧δ¯θs+ε.
Consequently,(70)supuN∈−∞,−δ0−δ¯∧δ¯κ≤esssup−∞,−δ0θs+ε,infuN∈δ0+δ¯∧δ¯,+∞κ≥essinfδ0,+∞θs−ε,where δ0 is defined in Hθ. Thus, from Hθ we obtain
(71)supuN∈−∞,−δ0−δ¯∧δ¯κ≤ε,infuN∈δ0+δ¯∧δ¯,+∞κ≥−ε.
It results inκ≤ε if uN<−δ0−δ¯∧δ¯<−δ0 and κ≥−ε if uN>δ0+δ¯∧δ¯>δ0. We let ε⟶0+ to arrive at κ≤0 if uN<−δ0 and κ≥0 if uN>δ0. Consequently, as θ∈Lloc∞ℝ and κ∈θ^uN, then in the case of uN≤δ0, we have
(72)supuN≤δ0κ≤esssups≤δ0θs≔δ1.
It follows that(73)∫0,T×∂Ωκ·uNdσdt=∫uN<−δ0κ·uNdσdt+∫uN>δ0κ·uNdσdt+∫uN≤δ0κ·uNdσdt≥0−δ0∫uN≤δ0κdσdt≥−δ0δ1Tσ∂Ω.Theorem 10.
UnderHθ, assume that fm, f∈L20,T;V∗ such that fm⟶f weakly in V∗. Let umm⊂W be a sequence such that um∈Su0θfm 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∈Su0θf.Proof.
Letfm, f∈V∗ with fm⟶f weakly in V∗. Let umm be a sequence such that um∈Su0θfm for each m∈ℕ; then, by Theorem 8, there exists κm∈L10,T×∂Ω, such that κmt,x∈∂jumNt,x for a.e. t,x∈0,T×∂Ω and
(74)um′t+Aumt+Bumt,v+∫∂ΩκmvNdσ=fmt,v,for a.e. t∈0,T and all v∈V∩LN∞∂Ω. With the same calculations as in the last section, one obtains
(75)12ddtumt2+MumtV2≤2MfmtV∗2+M2utV2−2∫∂ΩκmumNdσ.
It follows that(76)ddtumt2+MumtV2≤4MfmtV∗2−2∫∂ΩκmumNdσ.
Integrating over0,t, we get
(77)umt2+M∫0tumsV2ds≤4MfmV∗+u02+2δ0δ1Tσ∂Ω.
It follows thatumm is bounded in L∞0,T;H∩L20,T;V. Hence, by passing to a subsequence if necessary, there exists u such that umm converges to u weakly in L20,T;V and weakly –∗ in L∞0,T;V. Using the compactness of the trace operator γ, we may assume that γum⟶γu in L20,T;H and then γumt,z⟶γut,z for a.e. t,z∈0,T×∂Ω. Consequently, umNt,z⟶uNt,z for a.e. t,z∈0,T×∂Ω. Let us show that there exists m0∈ℕ such that
(78)∂jumN⊂∂juN,forallm≥m0.
AsumN⟶uN for a.e. t,z∈0,T×∂Ω, we can find by Egoroff’s theorem that for any α>0, we can determine ω⊂0,T×∂Ω with dt×σω<α, such that
(79)umN⟶uN,uniformlyon0,T×∂Ω\ω,with uN∈L∞0,T×∂Ω\ω. Thus for any μ>0, there exists m0, such that for all m>m0, we have
(80)umN−uN<μ2,a.e.on0,T×∂Ω\ω.
By using the triangle inequality, we have(81)θ¯μ2umN=esssupumN−ξ≤μ2θξ≤esssupuN−ξ≤μθξ=θ¯μuN.
Analogously, we prove the inequality(82)θ¯μuN≤θ¯μ2umN.
Taking the limitμ⟶0+, we obtain for each m≥m0(83)κm∈θ^umN⊂θ^uN,a.e.on0,T×∂Ω\ω,where dt×σω<α. For α as small as possible, we obtain
(84)κm∈θ^umN⊂θ^uN,a.e.on0,T×∂Ω,from which we can conclude that theκmmthat follows is bounded. By the Dunford-Pettis theorem ([40], p. 239), we will show that the sequence κmm is weakly precompact in L10,T×∂Ω. For this end, we show that for each μ>0, there exists δ>0, such that for ω⊂0,T×∂Ω, dt×σω<δ:
(85)∫ωκmdσdt<μ.
For somea>0 and remarking that in umN>a, 1<umN/a, one has
(86)∫ωκmdσdt=∫ωκm1umN>adσdt+∫ωκm1umN≤adσdt≤1a∫0,T×∂ΩκmumNdσdt+∫ωκm1umN≤adσdt.
From one hand, one has(87)∫0,T×∂ΩκmumNσdt=∫umN>δ0κmumNdσdt+∫umN≤δ0κmumNdσdt=∫umN>δ0κmumNdσdt−∫umN≤δ0κmumNdσ+2∫umN≤δ0κmumNdσdt≤∫umN>δ0κmumNdσdt+∫umN≤δ0κmumNdσdt+2∫umN≤δ0κmumNdσdt=∫0,T×∂ΩκmumNdσdt+2∫umN≤δ0κmumNdσdt.
From equation (74) with v=umt(88)∫0,T×∂ΩκmumNσdt≤∫0Tfmt,umtdt−∫0Tum′t,umtdt−∫0TAumt,umtdt+2∫umN≤δ0κmumNdσdt≤∫0TfmtV∗umtVdt−12uT2+12u02−M∫0TumtV2dt2δ0∫umN≤δ0κmdσdt≤12∫0TfmtV∗2dt+12∫0TumV2dt+12u02+2δ0dt×σumN≤δ0supumN≤δ0κm≤c+2δ0δ1Tσ∂Ω.
On the other hand, for eachε>0 there is aε>0, such that for ∣ν∣<aε one has
(89)κm≤esssups−umN<νθs+ε≤esssups−umN<aεθs+ε.
This implies(90)supumN≤aκm≤esssups<a+aεθs+ε.
We choose for exampleε=1, which leads to
(91)supumN≤aκm≤esssups<a+a1θs+1.
Now we choosea such that
(92)1a∫0,T×∂ΩκmumNdσdt≤1ac+2δ0δ1Tσ∂Ω<μ2,and δ such that
(93)esssups<a+a1θs+1<μ2δ.
With this choice ofδ one have
(94)∫ωκm1umN≤adσdt≤supumN≤aκmdt×σω≤esssups<a+a1θs+1dt×σω<μ2δδ=μ2.
It follows(95)∫ωκmdσ≤1a∫∂ΩκmumNdσ+∫ωκm1umN≤adσ<μ2+μ2=μ.
Consequently, we can extract fromκmm a subsequence (denoted with the same symbol) that converges in L10,T×∂Ω to some κ∈L10,T×∂Ω. By passing the limit in (74), we get
(96)u′t+Aut+But,v+∫∂ΩκvNdσ=ft,v,with
(97)κ∈conv¯θ^uN=θ^uN,a.e.on0,T×∂Ω.Remark 11.
We will need Theorem10 just for external forces in L20,T;H. As in this situation, the duality between V and V∗ coincides with the one on H, and this will bring no more difficulties.Remark 12.
One can prove in the same way as in ([32], Theorem 5.1) that the solutions of EHVI are stable under the perturbation of θ.In the remaining of this section, we will use the notationSf instead of Su0θf. We follow Migórski [34], and we let U=L20,T;H be the space of controls and Uad 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:
(98)Findacontrolf^∈Uadandastateu^∈Sf^suchthat:Ff^,u^=infFf,u:f∈Uad,u∈Sf.A pairf^,u^ which solves (98) is called an optimal solution. The existence of such optimal solutions can be proved by using Theorem 10. To do so, we need the following additional hypotheses:
(1)
HUad: Uad is a bounded and weakly closed subset of U(2)
HF: F is lower semicontinuous with respect to U×V endowed with weak topologyTheorem 13.
Assume thatHθ, HUad, and HF are fulfilled. Then problem (98) has an optimal solution.Proof.
Letfm,um be a minimizing sequence for problem (98), i.e., fm∈Uad and um∈Sfm, such that
(99)limm→∞Ffm,um=infFf,u:f∈Uad,u∈Sf≕ϑ.
It follows that the sequencefm 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 HUad, we have f^∈Uad. From Theorem 10, we obtain, by again passing to a subsequence if necessary, that um⟶u^ weakly in V with u^∈Sf^. By HF, we have ϑ≤Ff^,u^≤liminfm→∞Ffm,um=ϑ. Which completes the proof.
Next we apply Theorem13 in a concrete example. Let X be another Hilbert space, X≔L20,T;X, Xad⊂X the set of admissible controls, and C∈LX,U a bounded linear operator from X to U. Let f∈L20,T;V∗, as we aim to study the following optimal control problem
(100)Findacontrolw∈Xadandastateu^∈SCw^+fsuchthat:Jw^,u^=infJw,u:w∈Xad,u∈SCw+f,where the objective functional is given by
(101)Jw,u=∫0T∫Ωut,x−zt,x2dxdt+∫0Thwtdt,for some function h:X⟶ℝ and z∈L20,T;H. Such optimal control problems arise in a wide range of applications, particularly in fluid flow control. More specifically, one tries to act on the flow in such a way that a certain flow profile is stabilized or enforced by devices like actuators. Also sensors are used to provide necessary information for the actuation measured here by the control input operator C. Our goal is to minimize the discrepancy between the ideal velocity profile z and the actual flow u. Moreover, the cost related to the actuators and the sensors should be also minimized. A more sophisticated example of this framework is the blood flow in an artificial heart. The goal will be to avoid, among other things, the stagnation causing some serious hymodynamic problems.Let us first announce the following corollaries of Theorem10.Corollary 14.
UnderHθ assume that φm, φ, and f∈V∗, such that φm⟶φ weakly in V∗. Then, for every um∈Sφm+f, we can find a subsequence (still denoted with the same symbol), such that um⟶u in V and u∈Sφ+f.Proof.
It suffices to takefm=φm+f in Theorem 10.Corollary 15.
UnderHθ, assume that f∈V∗ and wm, w∈X are such that wm converges weakly to w in X. Then, for every sequence umm, such that um∈SCwm+f, we can find a subsequence that converges weakly in L20,T;V to u∈SCw+f.Proof.
It suffices to takeφm=Cwm in Corollary 14.
Assume the following:(i)
f∈L20,T,V∗ and z∈L20,T;H(ii)
Xad is a weakly compact subset of X(iii)
The functionh:X⟶ℝ is convex, lower semicontinuous, and satisfies the coercivity condition(102)hw≥αwX2+β,for some α>0 and β∈ℝ. .X stands for the norm of the Hilbert space XTheorem 16.
If hypotheses (i)–(iii) andHθ hold, then problem (100) has an optimal solution.Proof.
Letwn,un be a minimizing sequence to problem (100), i.e., wn∈Xad and um∈SCwn+f such that
(103)limm→∞Jwm,um=infJw,u:w∈Xad,u∈SCw+f.
Denotefm=Cwm+f, Ffm,um=Jwm,um, and Uad=CXad. It suffices now to apply Theorem 13 for Uad and F.


## 6. Directional Growth Condition

As mentioned in theIntroduction, the Rauch condition is a particular case of the directional growth condition due to Naniwiecz [31]. 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 a solution needs only to be modified. Here, we will reconsider the same problem of the evolutionary hemivariational Navier-Stokes equations but with the more general condition of directional growth.Letj:∂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:
(1)
Hj: there exists β:∂Ω×ℝ+⟶ℝ integrable with respect to the first argument and nondecreasing with respect to the second argument such that(104)jx,ξ−jx,η≤βx,rξ−η,∀ξ,η∈B0,r,r≥0.(2)
Hj0: 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(105)j0x;ξ,η−ξ≤αx,r1+η,for almost every x∈∂Ω and for any ξ,η∈ℝ with −r≤η≤r, r≥0.Remark that ifj does not depend on x∈∂Ω, then it satisfies Hj automatically. The hypothesis Hj0 is called the directional growth condition.Forx∈∂Ω and ξ∈ℝ, define jεx,ξ=hε⋆jx,.ξ and denote by j′ε the derivative of jε with respect to the second argument. As usual, let φ1,φ2,⋯ be a basis in V∩LN∞∂Ω and Vm=spanφ1,φ1,⋯,φm. We then consider the following regularized problem of the Galerkin type associated with EHVI, noted Pεm: find um∈Wm such that um0=u0m and
(106)u′mt+Aumt+Bumt,v+∫∂Ωjεm′umN·vNdσ=ft,v,for all t∈0,T and all v∈Vm.Note that due to the integrability ofβ with respect to the first argument and Hj, the integral above is finite for each um,v∈Vm. In fact, we have
(107)jεm′x;umNx·vNx≤βx,uNL∞∂Ω+1vNL∞∂Ω.Sinceβ.,uNL∞∂Ω+1∈L1∂Ω, the integrability of jεm′umN·vN over ∂Ω follows immediately for any v∈Vm. We have the following lemma (cf. [28], Lemma 3.1).Lemma 17.
Suppose thatHj0 holds. Then the estimate
(108)jε′x;ξη−ξ≤α¯x,r1+ξ,0<ε<1,is valid for any ξ,η∈ℝ with −r≤η≤r, r≥0, and almost all x∈∂Ω, where α¯x,r=2αx,r+1.The problemPεm has at least one solution in Vm. In fact, substitution of umt=∑k=1mckmtφk gives an initial value problem for a system of first order ordinary differential equations for ckm., k=1,2,⋯,m. Its solvability on some interval 0,tm follows from the Carathéodory theorem. This solution can be extended on the closed interval 0,T by using the a priori estimates below.Using the coercivity ofA, the properties of B., and the Young inequality, we have
(109)12ddtumtH+MumtV2+∫∂Ωjεm′umNt·umNtdσ≤M2umtV2+2MftV∗2.From Lemma17, we have
(110)jεm′umN·umN≥−α¯x,01+umN.Then(111)12ddtumtH+MumtV2≤2MftV∗2+∫∂Ωα¯x,01+umNdσ≤2MftV∗2+σ∂Ω1/2α¯0L2∂Ω+α¯0L2∂ΩumNtL2∂Ω≤2MftV∗2+σ∂Ω1/2α¯0L2∂Ω+α¯0L2∂ΩγumtV.Integrating over0,t we get
(112)12umtH−12u0mH2+M2∫0t∥ums∥V2ds≤2M∥f∥V∗2+σ∂Ω1/2∥α¯0∥L2∂Ωt+∥α¯0∥L2∂Ωγ∫0tumNsVds.It follows(113)12umtH+M2∥um∥V2≤12u0mH2+2M∥f∥V∗2+σ∂Ω1/2T∥α¯0∥L2∂Ω+T∥α¯0∥L2∂ΩγumV.It follows thatumm is a bounded subset of V and L∞0,T;H, so passing to subsequence, if necessary, we have
(114)um⟶uweaklyinVandweakly−∗inL∞0,T;H.Following the same procedure as in section4, see also [25], we have um′⟶u′ weakly in W and Bum⟶Bu weakly in V∗. Using the same proof as in ([31], Lemma 3.3), one can prove that the sequence jεm′umN is weakly precompact in L10,T×∂Ω. This means that
(115)jεm′umN⟶κweaklyinL10,T×∂Ω.Moreover, the following equality holds(116)um′t+Aumt+Bumt,v+∫∂Ωjεm′umN·vNdσ=ft,v,for almost every t∈0,T and v∈V∩LN∞∂Ω. We pass to the limit as usual to obtain
(117)u′t+Aut+Bunt,v+∫∂Ωκt·vNdσ=ft,v,for almost every t∈0,T and v∈V∩LN∞∂Ω.We still need to prove thatκt,x∈∂jx,uNt,x for almost every t∈0,T and x∈∂Ω. Since γun⟶γu in L20,T;L2∂Ω;ℝd, we obtain that umN⟶uN in L20,T;L2∂Ω and consequently for almost every t∈0,T:
(118)umNt,x⟶uNt,xa.e.x∈∂Ω.By Egoroff’s theorem, with respect tox∈∂Ω, we have for any ρ>0, a subset ω of Γ with
(119)umN⟶uNuniformlyon∂Ω\ω,with uN∈L∞∂Ω\ω. Let v∈LN∞∂Ω\ω be arbitrarily given. Due to Fatou’s lemma, for any positive ε>0 there exists δε>0 and Nε such that
(120)∫∂Ω\ωjx,umNx−τ+νvNx−jx,umx−τνdσ≤∫∂Ω\ωj0x,uNx;vNxdσ+ε,provided m>Nε, τ<δε, and 0<ν<δε. This inequality multiplied by hεm and integrated over ℝ yields
(121)∫∂Ω\ωjεmx,umNx+νvNx−jεmx,umNxνdσx=∫ℝhεmτ∫∂Ω\ωjx,umNx−τ+νvNx−jx,unx−τνdσdτ≤∫∂Ω\ωj0x,uNx;vNxdσ+ε.But asν⟶0, we get
(122)∫∂Ω\ωjεm′x,uNx·vNxdσ≤∫∂Ω\ωj0x,uNx;vNxdσ+ε,which is valid for m>Nε. Now letting m⟶∞, we are led to
(123)∫∂Ω\ωκ·vNdσ≤∫∂Ω\ωj0x,uNx;vNxdσ+ε.Sinceε>0 was chosen arbitrarily
(124)∫∂Ω\ωκ·vNdσ≤∫∂Ω\ωj0x,uNx;vNxdσ,forallv∈LN∞∂Ω\ω.But the last inequality easily implies that(125)κx∈∂jx,uNx,fora.e.x∈∂Ω\ω,where σω<ρ. Now since ρ was chosen arbitrarily
(126)κx∈∂jx,uNx,fora.e.x∈∂Ω,which completes the proof.Remark 18.
The directional growth condition is meant to study problems involving vector valued functions, i.e., functions onℝN. Our situation is simpler as N=1. In this case, the direction growth condition can be simplified to the following condition
(127)j0x,ξ,−ξ≤kxξ,∀ξ∈ℝfora.e.x∈∂Ω,for some nonnegative function k∈L2∂Ω (cf. [31], Remark 4.1). Moreover, the Rauch condition and sign condition also fulfil the estimate (127) (cf. [31], Remark 4.7).Remark 19.
It is an easy task to check that the results in section5, regarding optimal solution, are also valid if one replaces the assumption Hθ by the more general assumption Hj0.



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