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

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. The paper concerns the third grade fluid system with the time-fractional derivative of the order α ∈ (0, 1). We first establish unique existence criterion of weak solutions in the case that the dimension n = 3. Then we prove the sufficient condition of optimal pairs.


Introduction
We consider the significant case of the fluids of grade  introduced by Rivlin-Ericksen [1] in a bounded domain − div (| ()| 2  ()) + ∇ = , where ] > 0 is the constant kinematic viscosity,  1 ,  2 , and  are material constants, and  > 0. The unknown functions ,  are the fluid velocity and the scalar pressure, respectively.The given functions  0 ,  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 |()| 2 denotes ( 2 ()).
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 timefractional 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.

Complexity
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  ∈ (0, 1).
Noting that the system converges formally to the wellknown 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,  : [0, ] → , and the operator  0 :  → H 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).

Preliminaries
In this section, for convenience, we mainly introduce some notions, operators, definitions, and lemmas, which are useful in the sequel.
Let Ω ⊂ R 3 be an open bounded smooth domain; we define = ‖∇⋅‖ , and the dual space V * , , and the dual space W * . ( Define the linear "Stokes operator" and the bilinear operator We also introduce the operators K(), J() from W into W * as Let || < √2].We denote T() from W into W * as Note that (see, e.g., [26][27][28]) By the Hilbert-Schmidt theorem, one can deduce that A has a sequence of orthonormal eigenfunctions   , belonging to  ∞  (Ω) with zero mean in Ω.Since A is a self-adjoint positive operator with compact inverse, {  } ∞ =1 forms a basis of the space H.Moreover, {  } ∞ =1 also forms a basis of the space (A /2 ) = {H  0 (Ω), div = 0}, 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, V : [0, ] → ; the left and right Riemann-Liouville fractional integrals 0    V() and     V() of order  ∈ (0, 1] are defined by provided the integrals are point-wise defined on [0, ∞), where   denotes the Riemann-Liouville kernel Definition 2 (see [29]).The left Caputo and right Riemann-Liouville fractional derivative  0    V() and     V() of order are defined by More generally, for V : [0, ∞) × R  → R  , 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 V : R → ; 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 J is a monotone operator; that is, for any , V ∈ W, we have Lemma 5 (see [8]).The operator T is a monotone operator; that is, for any , V ∈ W, we have Let T 1 : W → W * be defined as Thanks to the above lemmas, T 1 is also a monotone operator.
Assume  0 , ,  1 are Hilbert spaces with Complexity being continuous, and Let () be a function from R to  1 ; we denote by ψ() its Fourier transform The derivative in  of order  is the inverse Fourier transform of (2)  ψ(); that is, For given  > 0, define the space Then,   is a Hilbert space with the norm For any set  ⊂ R, the subspace    of   is defined as the set of functions  ∈   with support contained in : Lemma 10 (see [26]).Assume  0 , ,  1 are Hilbert spaces satisfying (27) and (28).Then, for any bounded set  and ∀ > 0, we have following compact embedding: The following Korn's inequality plays an essential role in our analysis.
Lemma 11 (see [33]).Assume that 1 <  < ∞ and 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.
Now we give the equivalent form of the first equality of (36) as follows: An equivalent form of ( 36) is as follows: (39)

Existence for Time-Fractional Third Grade Fluid System
In this section, we prove the existence of solutions for timefractional third grade fluid system.
Proof.In order to apply the Galerkin procedure.We consider a basis of H 1 0 (Ω) constituted of elements   of D(Ω) and set   : H → H  be the corresponding projection operators.For each , we define   by Applying   to (39), we can obtain where  0 is the orthogonal projection of  0 onto the space spanned by  1 , . . .,   in H.
As done several times before, we extend all functions by 0 outside the interval [0, ] and consider the Fourier transform of the different equations.The following relations then hold on R. where After taking Fourier transforms, (57) yields where û and φ represent, respectively, the Fourier transforms of ũ and φ .We multiply (59) by ĝ () ( ĝ = Fourier transform of g ) and then add these relations for  = 1, . . ., ; we obtain In view of the inequality We can obtain and We have also used the convergence of the infinite integral in the above integral ) .(69) We can conclude that ) .(70) Next, 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  → ∞.
By ( 55)-(56) and using Lebesgue's dominated convergence theorem, we have lim Next, let T 1 : W → W * be defined as

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  1 ,  2 two solutions of problem (39) and set  =  1 −  2 , by subtracting the relation (39) satisfied by  1 and  2 ; we obtain Taking the inner product of (88) with , it yields Since (V, , ) = 0, then Bear in mind that We deduce that ) . (96) Since (0) = 0, the last relation can imply that and the uniqueness is proved.