The Unique Existence of Weak Solution and the Optimal Control for Time-Fractional Third Grade Fluid System

Shao, Guangming; Liu, Biao; Liu, Yueying

doi:https://doi.org/10.1155/2018/7941012

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Iterative Learning and Fractional Order Control for Complex Systems

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Volume 2018 | Article ID 7941012 | https://doi.org/10.1155/2018/7941012

The Unique Existence of Weak Solution and the Optimal Control for Time-Fractional Third Grade Fluid System

Guangming Shao,¹Biao Liu,²and Yueying Liu³

Guest Editor: Viet-Thanh Pham

Received10 Jun 2018

Revised11 Oct 2018

Accepted22 Oct 2018

Published13 Nov 2018

Abstract

The paper concerns the third grade fluid system with the time-fractional derivative of the order . We first establish unique existence criterion of weak solutions in the case that the dimension . Then we prove the sufficient condition of optimal pairs.

1. Introduction

We consider the significant case of the fluids of grade introduced by Rivlin-Ericksen [1] in a bounded domain where is the constant kinematic viscosity, , , and are material constants, and . The unknown functions are the fluid velocity and the scalar pressure, respectively. The given functions represent the initial fluid velocity and the forcing term, respectively. In fact, is the tensor whose component is Here, is the transposition of the Jacobian matrix , and denotes

In [2], Ladyzhenskaya studied a particular case of the third grade fluids (1) in , where they assumed . In this case, system (1) becomes

In the past years, some nice works on the third grade fluid system have already been available. We refer to Fosdick and Rajagopal [3], Amrouche and Cioranescu [4], Sequeira and Videman [5], Busuioc and Iftimie [6], Paicu [7], Hamza and Paicu [8], Zhao et al. [9], and Chai et al. [10]; much is yet to be carried out.

Unlike the boundary value problems of the classical third grade fluid system, the theoretical analysis of time-fractional third grade fluid system is not so rich, so we need to develop many aspects of these problems. It is significant to study this type of system when it is recognized that the time-fractional third grade fluid system can be used to describe diffusion phenomenon in fractal media. So the scholars have been more and more interested in investigating the fractional calculus.

The fractional calculus was introduced by Herrmann [11] and Hilfer [12], especially for the classical time-fractional Navier-Stokes equations where is the Caputo fractional derivative of order .

For some recent work on analytical solutions of these systems, we refer to El-Shahed et al. [13], Ganji et al. [14], Zhou [15], Momani and Zaid [16], Carvalho-Neto and Gabriela [17], Zhou and Peng [18, 19], Zhou et al. [20], Al-Mdallal et al. [21], Jarad [22–25], and further papers cited therein.

In this paper, we consider mainly the weak solutions of the time-fractional third grade fluid system and optimal control in an open set with smooth boundary with the initial and boundary value conditions

Noting that the system converges formally to the well-known time-fractional Navier-Stokes equations when the material constants in (5) tend to zero. So it is reasonable to propose the time-fractional third grade fluid system.

In this paper, we firstly investigate the existence and uniqueness of solutions for the time-fractional third grade fluid equations (5) (associated with appropriate initial and boundary conditions). Then we proceed the systems with control in where is a real Hilbert space, , and the operator is linear continuous.

Now let us talk about the organization of this paper. In Section 2, we provide some preliminaries about the function spaces. In Section 3, we prove the existence of solutions for system (5). The uniqueness of solutions is established in Section 4, while in Section 5 we show the existence of the optimal control of system (7).

2. Preliminaries

In this section, for convenience, we mainly introduce some notions, operators, definitions, and lemmas, which are useful in the sequel.

Let be an open bounded smooth domain; we define Define the linear “Stokes operator” from to byand the bilinear operator from into asWe also introduce the operators , from into asLet . We denote from into asNote that (see, e.g., [26–28])By the Hilbert-Schmidt theorem, one can deduce that has a sequence of orthonormal eigenfunctions , belonging to with zero mean in . Since is a self-adjoint positive operator with compact inverse, forms a basis of the space . Moreover, also forms a basis of the space , for any positive integer [28].

Next, we will introduce some definitions and lemmas, which are used throughout this paper.

Definition 1 (see [18, 29]). Let be a Banach space, ; the left and right Riemann-Liouville fractional integrals and of order are defined byprovided the integrals are point-wise defined on , where denotes the Riemann-Liouville kernel

Definition 2 (see [29]). The left Caputo and right Riemann-Liouville fractional derivative and of order are defined by More generally, for , the left Caputo fractional derivative with respect to time can be defined by

For more detail, we can refer to [30].

Definition 3 (see [18, 29]). Let ; the Liouville-Weyl fractional integral and the Caputo fractional derivative on the real axis are defined, respectively, as follows:

Lemma 4 (see [8]). The operator is a monotone operator; that is, for any , we have

Lemma 5 (see [8]). The operator is a monotone operator; that is, for any , we have

Let be defined asThanks to the above lemmas, is also a monotone operator.

Lemma 6 (see [31]). The fractional integration by parts in the formulaSince for , then

For more details, we refer to relations (16)-(21) in [31].

Lemma 7 (see [19]). in for , , where .

Lemma 8 (see [32]). Suppose that is a real Hilbert space and have a derivative, then there holds

Lemma 9 (see [19]). Suppose that a nonnegative function satisfiesfor almost all , where and the function is nonnegative and integrable for . Then

We need the compactness theorem [26] involving fractional derivative.

Assume are Hilbert spaces withbeing continuous, andLet be a function from to ; we denote by its Fourier transformThe derivative in of order is the inverse Fourier transform of ; that is,For given , define the spaceThen, is a Hilbert space with the norm

For any set , the subspace of is defined as the set of functions with support contained in :

Lemma 10 (see [26]). Assume are Hilbert spaces satisfying (27) and (28). Then, for any bounded set and , we have following compact embedding:

The following Korn’s inequality plays an essential role in our analysis.

Lemma 11 (see [33]). Assume that and , . Let . Then there exist two positive constants , such that

Using the notation and operators introduced earlier, we can express the weak formulation of the time-fractional third grade fluid system (5) in the solenoidal vector fields as follows.

Definition 12. Let and () be given. A weak solution of system (5) is a function such that for any

Now we give the equivalent form of the first equality of (36) as follows:Since , then . ThusAn equivalent form of (36) is as follows:

3. Existence for Time-Fractional Third Grade Fluid System

In this section, we prove the existence of solutions for time-fractional third grade fluid system.

Theorem 13. Let and , . Then there exists at least one solution .

Proof. In order to apply the Galerkin procedure. We consider a basis of constituted of elements of and set be the corresponding projection operators. For each , we define byApplying to (39), we can obtainwhere is the orthogonal projection of onto the space spanned by in .
Taking the scalar product of (41) with , we haveEquations (42) form a nonlinear differential system for the functions . By the standard theory of ODE, we have the existence of a solution defined at least on some interval , . Moreover, if , then . Otherwise, the following a priori estimates show that in fact .
Indeed, if we multiply (42) by and add all these equations for , Since , according to Lemma 5 and Young’s inequality, we haveNote that is symmetric; we get From the Cauchy-Schwartz inequality and the fact that we deduce that Since , we can choose , such that We deduce thatIntegration of (49) from 0 to shows that Hence andwhere MoreoverHence, is bounded in .
As done several times before, we extend all functions by 0 outside the interval and consider the Fourier transform of the different equations. The following relations then hold on .where After taking Fourier transforms, (57) yieldswhere and represent, respectively, the Fourier transforms of and .
We multiply (59) by ( Fourier transform of ) and then add these relations for ; we obtainIn view of the inequality We can obtain Therefore And we can conclude from (60) that namely For some fixed , we have . ThusBy Parseval equality and Poincaré inequality, we haveand We have also used the convergence of the infinite integral in the above integral We can conclude thatNext, we want to pass to the limit as in (42) using the estimates (51), (55)-(56), and (70). We are only concerned with a passage to the limit as .
Considering the Lemma 7, there exist a sequence , and one function such that Taking , we multiply (41) by , integrate over , and we haveIn view of , we can assume that in for all , where . Therefore, in for .
Finally, we look at the convergence of the nonlinear terms in (72).
By (55)-(56) and using Lebesgue’s dominated convergence theorem, we have Next, let be defined asIt is clear that is a monotone operator. We show that This can be achieved by the monotony method as in [8, 34]; see also [35, 36]. Indeed, since is uniformly (in ) bounded in , we may assume that Due to (43) and taking a sequence , we haveand integration of (77) from 0 to red shows thatwhere is the duality product between the spaces and .
By (42), we havePassing to the limit in (79),Taking , we can obtainDue to the monotonicity of , we have for any Combining (78), (81), and (82), we obtain by passing to the limit in Take for and Then after division by and tending to , we deduce that Hence, Then A simple argument shows that Taking the limit of (72) as , we find thatIt is clear that , as
Set in (87); we can obtain (36). The proof is complete.

4. Uniqueness of Solutions for the Time-Fractional Third Grade Fluid System

In this section, we prove the uniqueness of solutions of problem (39).

Theorem 14. The solution of problems (39) given by Theorem 13 is unique.

Proof. We denote by two solutions of problem (39) and set , by subtracting the relation (39) satisfied by and ; we obtainTaking the inner product of (88) with , it yields Since , then Bear in mind that We deduce that Noting that and implies that Thus By Gronwall inequality and (51), (53), we have Since , the last relation can imply that and the uniqueness is proved.

5. Optimal Control

In this section, we will consider the optimal control problemswhere

We will consider the systems with control in

Definition 15. Let and ; a function is called a weak solution of system (7) if

Similarly we give the equivalent form as follows:If , rewriting problem (98) in the abstract form, we have over subject to (101).

We assume that(i);(ii)the function is convex and lower semicontinuous and satisfies for some ,

Theorem 16. If hypotheses (i) and (ii) hold, then problem (P) has at least one solution

Proof. Replacing with in Theorems 13 and 14, we show that system (101) has an weak solution corresponding to . Suppose that () are a minimizing sequence of problem (P); namely,According to hypothesis (ii), we can conclude that is bounded in . Thus, there exists such that Similar to (49), we obtain This yields Hence Therefore, is bounded in . Thus, there exists an element such that We need to have some uniform bound on , , and .
We firstly estimate the term . In fact Then Therefore This implies that there exists such that For the similar derivations of relation (I), we deduce that Now let ; take the scalar product of (104) with and integrate; it follows By similar arguments in Theorem 13 and taking the limits as , we can conclude Applying the Caputo fractional derivative of order on the sides of the above equality for , we obtain Namely, an equivalent form of the last equality is Hence, we can see that () satisfies system (104).
From weak lower semicontinuity of function , we deduce Then is an optimal pair for problem (P). The proof of Theorem 16 is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was partially supported by the Scientific Research Project of Anhui University of Chinese Medicine (no. 2018zryb04).

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Copyright © 2018 Guangming Shao 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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