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

Mahdioui, Hicham; Ben Aadi, Sultana; Akhlil, Khalid

doi:https://doi.org/10.1155/2020/6573219

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Volume 2020 | Article ID 6573219 | https://doi.org/10.1155/2020/6573219

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

Hicham Mahdioui,¹Sultana Ben Aadi,¹and Khalid Akhlil¹

Academic Editor: Yuanfang Ru

Received02 Aug 2020

Revised16 Sept 2020

Accepted24 Sept 2020

Published14 Oct 2020

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.

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 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 [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

Let be a bounded simply connected domain in with with connected boundary of class and where . We consider the following evolution Navier-Stokes system:

This system describes the flux of an incompressible viscous fluid in a domain subjected to an external force . , and denote, respectively, the velocity, the pressure, and the kinematic viscosity of the fluid. The nonlinear term (called the convective term) is the symbolic notation of the vector . As usual, we use the Lamb formulation ([35], chapter I) to rewrite the evolution Navier-Stokes system as follows: where is the total head of the fluid, or “total pressure.”

We suppose that on boundary , the tangential components of the velocity vector are known, and without loss of generality, we put them equal to zero (the nonslip condition): where is the unit outward normal on the boundary and . denotes the normal component of the vector. Moreover, we assume the following subdifferential boundary condition: where is the Clarke subdifferential of at and is given by and is the generalized derivative of a locally Lipschitz function at in the direction defined by

To work conveniently on problems (2), (3), (4), (5), and (6), we need the following functional spaces:

Then, we have , with all the embedding being continuous and compact. Moreover, for an interval time , we introduce the following spaces:

Then, we also have the following continuous embedding,

We consider the operators and defined by for all . As usual, we will use the notation . It is well known (cf. [36]) that if the domain is simply connected, the bilinear form generates a norm in , , which is equivalent to the -norm. Hence, it is clear that the operator is coercive.

In order to give the weak formulation to problems (2), (3), (4), (5), and (6), we multiply it by a certain and apply the Green formula. We obtain where . From relation (6), by using the definition of the Clarke subdifferential, we have

The relations (13) and (14) yield to the following weak formulation:

The equation above is called an hemivariational inequality.

We have already mentioned in Introduction that the Rauch assumption is not sufficient to make the functional locally lipschitz or even finite in the whole space . Because of this reason, a slight modified definition of being a solution should be adopted. Define the functional space where is the trace operator from in . Now, we are able to give what we mean by a solution to the problem .

Definition 1. A function is said to be the solution of if there exists such that for a.e.

Note that since continuously, the initial condition makes sense in . To justify the above definition, we refer to [31, 32].

3. Regularized Problem

In what follows, we restrict our study to superpotentials , which are independent of and which subdifferential is obtained by “filling in the gaps” procedure (cf. [30]). Let , for and , we define

For a fixed , the functions are decreasing and increasing in , respectively. Let and let be a multifunction defined by

From Chang [37], we know that a locally Lipschitz function can be determined up to an additive constant by the relation such that for all . If moreover, the limits exist for every , then .

In order to define the regularized problem, we consider the mollifier and let where denotes the convolution product.

Consider the following auxiliary problem associated to :

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 in , i.e., forms at most a countable sequence of elements of , finitely are linearly independent. Consider , we have and . Moreover, the family satisfies such thatin, as.

Let be an approximation of the given initial value such that for and suppose that

We consider the following regularized Galerkin system of finite dimensional differential equations associated to : where .

The generalized derivative restricted to the subset defines a linear operator given by

For the existence of solutions, we will need the following hypothesis : (1)(Chang assumption) , exists for any (2)(Rauch assumption) There is such that

Remark 2. If one assumes more generally that for some real number , it is possible to come back to the situation where the Rauch assumption is imposed by simply replacing by and by .

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 function defined by where stands for the integer part of . One can prove easily (eventually by a contradiction argument) that the function satisifies the Rauch condition while the growth condition cannot be satisfied. Conversely, one can take a function defined by ; it is clear that it satisfies the growth condition but not the Rauch condition as is positive for negative values.

Lemma 4. Suppose that holds. Then we can determine , such that for every

Proof. This is a classical result in the stationary case (cf. [32], Lemma 3.2). It suffices to integrate over to obtain the result.

Proposition 5. The sequence is weakly precompact in .

Proof. The proof is similar to ([32], Proposition 3.7) with minor changes consisting mainly in replacing by and remarking that for .

Proposition 6. The regularized problem has at leat one solution .

Proof. We substitute in to obtain The matrix with elements , , is nonsingular (i.e., ), we invert the matrix, then equation (33) can be written in the usual form: where the initial values , are given, i.e., :

The differential system (35) with the initial condition (36) define uniquely the scalar on the interval . Then, the solution exists on , and we can extend it on the closed interval by using a priori estimates in Lemma 7. Since the scalar function in equation (33) are square integrable, so are the functions ; therefore, for each we have:

4. Existence Result

In this section, we will prove the existence of solutions to the problem by analysing the convergence of the sequence solutions to . To do so, we need some a priori estimates.

Lemma 7. The solution is bounded in .

Proof. From Proposition 5, the regularized problem has at least one solution . By replacing by in , we get for a.e. Because of (37) we have Then, equation (38) becomes By the coerciveness of , the Cauchy-Schwartz inequality, and the Young inequality, we obtain for a.e. ( is the constant of coercivity). Integrating equation (40) from 0 to , , and using Lemma 4, one has Hence The right-hand side of the previous inequality is finite and independent of . We deduce that is bounded in .
Again, from (42) we have Then Then, remains in a bounded subset of .

Theorem 8. Under assumption , the problem has at least one solution.

Proof. From Proposition 5 and Proposition 6, we get Now, we focus on the weak convergence of the nonlinear term by using exactly the same procedure as in [25]. For the case , we obtain from Temam [38] the following: Moreover, operator is continuous. Hence is bounded in . Thus, by passing to a next subsequence, if necessary, we have the following: Using the facts that continuously, compactly, and compactly, we have and Since weakly in and in , analogously as in Ahmed [39], we have weakly in . We remark that if , we also have the convergence of weakly in by a compactness embedding theorem as in [39].
Let and . Then, there exists such that and in , as . Denoting and , we have in . From , we have the following: Using the above convergences, letting , we obtain Since is arbitrary, we deduce that for a.e. and for all .
In order to complete the proof, it will be shown that Since in , we obtain in , and consequently, for a.e. ; then, by applying Egoroff’s theorem, we can find that for any , we can determine with , such that with . Thus, for any , we can find with , such that for any and for and , we have Consequently, one obtains Analogously, we prove the inequality We now take a.e. on with . This implies Taking the limits as and , we obtain and as , we obtain Since is arbitrary, we have where . For 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.

Let . Under , we denote by the solution set corresponding to of the problem . That is, and there exists such that , and for a.e. and all .

Lemma 9. Let . For every , there exist such that where and are independent from .

Proof. By definition of and , we have for every , there exists with such that and there exists with , such that It follows that That is, Enlarging the bounds, we obtain For small enough and , one has Consequently, where is defined in . Thus, from we obtain It results in if and if . We let to arrive at if and if . Consequently, as and , then in the case of , we have It follows that

Theorem 10. Under , assume that , such that weakly in . Let be a sequence such that for each ; then, we can find a subsequence (still denoted with the same symbol) such that weakly in and .

Proof. Let , with weakly in . Let be a sequence such that for each ; then, by Theorem 8, there exists , such that for a.e. and for a.e. and all . With the same calculations as in the last section, one obtains It follows that Integrating over , we get It follows that is bounded in . Hence, by passing to a subsequence if necessary, there exists such that converges to weakly in and weakly – in . Using the compactness of the trace operator , we may assume that in and then for a.e. . Consequently, for a.e. . Let us show that there exists such that As for a.e. , we can find by Egoroff’s theorem that for any , we can determine with , such that with . Thus for any , there exists , such that for all , we have By using the triangle inequality, we have Analogously, we prove the inequality Taking the limit , we obtain for each where . For as small as possible, we obtain from which we can conclude that thethat follows is bounded. By the Dunford-Pettis theorem ([40], p. 239), we will show that the sequence is weakly precompact in . For this end, we show that for each , there exists , such that for , : For some and remarking that in , , one has From one hand, one has From equation (74) with On the other hand, for each there is , such that for one has This implies We choose for example , which leads to Now we choose such that and such that With this choice of one have It follows Consequently, we can extract from a subsequence (denoted with the same symbol) that converges in to some . By passing the limit in (74), we get with

Remark 11. We will need Theorem 10 just for external forces in . As in this situation, the duality between and coincides with the one on , 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 are stable under the perturbation of .

In the remaining of this section, we will use the notation instead of . We follow Migórski [34], and we let be the space of controls and a nonempty subset of consisting of admissible controls. Let be the objective functional we want to minimize. The control problem reads as follows:

A pair 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): is a bounded and weakly closed subset of (2): is lower semicontinuous with respect to endowed with weak topology

Theorem 13. Assume that , , and are fulfilled. Then problem (98) has an optimal solution.

Proof. Let be a minimizing sequence for problem (98), i.e., and , such that It follows that the sequence belongs to a bounded subset of the reflexive Banach space . We may then assume that weakly in (by passing to a subsequence if necessary). By , we have . From Theorem 10, we obtain, by again passing to a subsequence if necessary, that weakly in with . By , we have . Which completes the proof.
Next we apply Theorem 13 in a concrete example. Let be another Hilbert space, , the set of admissible controls, and a bounded linear operator from to . Let , as we aim to study the following optimal control problem where the objective functional is given by for some function and . 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 . Our goal is to minimize the discrepancy between the ideal velocity profile and the actual flow . 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 Theorem 10.

Corollary 14. Under assume that , , and , such that weakly in . Then, for every , we can find a subsequence (still denoted with the same symbol), such that in and .

Proof. It suffices to take in Theorem 10.

Corollary 15. Under , assume that and , are such that converges weakly to in . Then, for every sequence , such that , we can find a subsequence that converges weakly in to .

Proof. It suffices to take in Corollary 14.
Assume the following: (i) and (ii) is a weakly compact subset of (iii)The function is convex, lower semicontinuous, and satisfies the coercivity conditionfor some and . stands for the norm of the Hilbert space

Theorem 16. If hypotheses (i)–(iii) and hold, then problem (100) has an optimal solution.

Proof. Let be a minimizing sequence to problem (100), i.e., and such that Denote , , and . It suffices now to apply Theorem 13 for and .

6. Directional Growth Condition

As mentioned in the Introduction, 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 . 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.

Let be a measurable function with respect to the first argument and locally Lipschitz with respect to the second argument. We assume the following: (1): there exists integrable with respect to the first argument and nondecreasing with respect to the second argument such that(2): there exists a function square-integrable with respect to the first argument and nondecreasing with respect to the second argument such that the following estimate holdsfor almost every and for any with , .

Remark that if does not depend on , then it satisfies automatically. The hypothesis is called the directional growth condition.

For and , define and denote by the derivative of with respect to the second argument. As usual, let be a basis in and . We then consider the following regularized problem of the Galerkin type associated with , noted : find such that and for all and all .

Note that due to the integrability of with respect to the first argument and , the integral above is finite for each . In fact, we have

Since , the integrability of over follows immediately for any . We have the following lemma (cf. [28], Lemma 3.1).

Lemma 17. Suppose that holds. Then the estimate is valid for any with , , and almost all , where .

The problem has at least one solution in . In fact, substitution of gives an initial value problem for a system of first order ordinary differential equations for , . Its solvability on some interval follows from the Carathéodory theorem. This solution can be extended on the closed interval by using the a priori estimates below.

Using the coercivity of , the properties of , and the Young inequality, we have

From Lemma 17, we have

Then

Integrating over we get

It follows

It follows that is a bounded subset of and , so passing to subsequence, if necessary, we have

Following the same procedure as in section 4, see also [25], we have weakly in and weakly in . Using the same proof as in ([31], Lemma 3.3), one can prove that the sequence is weakly precompact in . This means that

Moreover, the following equality holds for almost every and . We pass to the limit as usual to obtain for almost every and .

We still need to prove that for almost every and . Since in , we obtain that in and consequently for almost every :

By Egoroff’s theorem, with respect to , we have for any , a subset of with with . Let be arbitrarily given. Due to Fatou’s lemma, for any positive there exists and such that provided , , and . This inequality multiplied by and integrated over yields

But as , we get which is valid for . Now letting , we are led to

Since was chosen arbitrarily

But the last inequality easily implies that where . Now since was chosen arbitrarily which completes the proof.

Remark 18. The directional growth condition is meant to study problems involving vector valued functions, i.e., functions on . Our situation is simpler as . In this case, the direction growth condition can be simplified to the following condition for some nonnegative function (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 section 5, regarding optimal solution, are also valid if one replaces the assumption by the more general assumption .

Data Availability

There is no data needed in our manuscript.

Conflicts of Interest

The authors declare that they have no conflict of interest.

Acknowledgments

We thank Prof. S. Migórski for pointing out that the Rauch and the growth conditions are completely independent.

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Copyright © 2020 Hicham Mahdioui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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