Approximation of a Class of Incompressible Third Grade Fluids Equations

Abstract

This paper discusses the approximation of weak solutions for a class of incompressible third grade fluids equations. We first introduce a family of perturbed slightly compressible third grade fluids equations (depending on a positive parameter ) which approximate the incompressible equations as . Then we prove the existence and uniqueness of weak solutions for the slightly compressible equations and establish that the solutions of the slightly compressible equations converge to the solutions of the incompressible equations.

1. Introduction

Fluids of differential type form an important class of non-Newtonian fluids. The fluids of grade , introduced by Rivlin and Ericksen [1], are the fluids for which the stress tensor is a polynomial of degree in the first Rivlin-Ericksen tensor defined recursively by where denotes the material derivative and the transposition of the Jacobian matrix . In [1], the constitutive relation of a particular of fluids of grade is given by where is the identity matrix of degree and is an isotropic polynomial of degree .

There are some references on the existence, uniqueness, and asymptotic behavior of solutions for second and third grade fluids equations; see, for example, [2–10] and the references therein. In [2–4], Azia et al. use the symmetry approach to obtain the analytical solutions of the third grade fluid equations. In [5], Busuioc and Iftime studied the existence of solutions to the following third grade fluids equations: in , with the coefficients , , , and satisfying the following hypotheses: Also, Busuioc and Iftimie in [5] proved that the third grade fluids equations (3) possess a global solution if the initial value is in and established the uniqueness of solutions for (3) when . Recently, Hamza and Paicu studied in [7] a particular case of the third grade fluids equations (3) in , where they assume . Then (3) become with the following assumptions on the coefficients , , , and : Hamza and Paicu in [7] proved the existence and uniqueness, as well as the stability (when ), of global solutions for (5) with natural regularity assumption on the initial data belonging to the energy space . They also proved that if the initial datum belongs to , then the solution belongs to for any positive time and they gave a control of the norm of the solution. Very recently, Paicu et al. proved in [8] the regularity of the global attractor and finite-dimensional behavior for the second grade fluids equations in the two-dimensional torus.

In the present paper, we first discuss the approximation of the incompressible third grade fluids equations (5) in three-dimensional bounded smooth domain, via the following slightly compressible third grade fluids equations depending on a parameter : with initial-boundary value conditions:

Compared with the incompressible third grade fluids equations (5), (7)-(8) overcome the computational difficulties connected with the constraint “.” It is easier to approximate than the original incompressible equations as the constraint “” has been replaced by the evolution equation (8). The third grade fluids equations (7)-(8) are called “slightly compressible” for the parameter will be taken to tend to zero. Obviously, the perturbed compressible third grade fluids equations (7)–(9) turn to be the incompressible third grade fluids equations (5) as . Following this clue, the questions are now the following.(i)Does the solution of the initial-boundary value problem of perturbed compressible third grade fluids equations (7)–(9) uniquely exist?(ii)Does the solution of (7)–(9) converge to the solution of the incompressible third grade fluids equations (5) as ?

The purpose of this paper is to give answers to the above two questions. When using the classical Faedo-Galerkin method to prove the existence of a weak solution for the compressible third grade fluid equations (7)–(9), the main difficulty (compared with the incompressible case) comes from the presence of the term in the slightly compressible third grade fluids equations. Due to the presence of the term , the argument for the incompressible case (see, e.g., [7]) to obtain the bound of the derivative sequence seems not applicable. This is caused essentially by the compressibility of the fluids. We know in the incompressible case and we can take or (see the notation in Section 2) as the phase space. Naturally, the term will disappear under the projection of the Helmhloz-Leray projector from to . While the fluids are compressible, and we shall take or (or other Sobolev spaces) as the phase space. So the term will not disappear. To overcome this difficulty, we will use the Fourier transform (in time ) technique to obtain the boundedness of the fractional derivative in time variable of the sequence .

The paper is organized as follows. In Section 2, we give some notations first and then the existence and uniqueness of weak solutions for (5). In Section 3, we describe the weak formulation of (7)–(9) and then prove the existence and uniqueness of its weak solution. In Section 4, we show how the solutions of the slightly compressible third grade fluids equations converge to the solutions of the corresponding incompressible third grade fluids equations.

2. Preliminaries

In this paper, we denote by the generic constant that can take different values in different places. is the 3D Lebesgue space with norm and is the 3D Sobolev space with norm and dual space , where and are the usual -Lebesgue space and Sobolev space on , respectively. When , we denote and also if there is no confusion. is the usual 3D Sobolev space and is its dual space (see [11]). Write and denote by and the closure space of in norm and in norm, respectively. is the inner product of (or , ), and is the dual pairing between and (the dual space of ), or between and , or between and . Note

To put (5) and (7)–(9) into abstract forms, respectively, we now introduce some operators. Firstly, set (which is taken with zero boundary conditions). Then, using integration by parts, we have Secondly, we define a continuous trilinear form (see, e.g., [12]) on (and in particular on ) by If , one can check For (resp., ), we denote by the element of (resp., ) defined by , for any (resp.,), and set The scalar product of two matrices and is denoted by . We set and for . We now define then the operator maps to via Also define then maps to by Using the notations and operators introduced above, we can express the weak formulation of (5) in the solenoidal field as

Problem 1. Let , for any given , to find satisfying

Lemma 2. Let the coefficients , , and satisfy condition (6). Then Problem 1 possesses a unique solution.

The proof of this lemma is similar with that of [7], and we omit it here.

Remark 3. Let , , and satisfy condition (6). Then for any given , for each solution obtained by Lemma 2, there exists a unique pressure corresponding to and, for each , . Moreover, and hence (one can refer to [13, Page 307] for a similar derivation) which satisfies in the sense of distribution that Therefore, for any , there holds and for any ,

Remark 4. If merely belongs to , condition (23) needs not make sense. But if and satisfies (26)-(27), then we will show that By the classical embedding theorem (see [13, 14]), we infer from (21) and (28) that . Therefore, (23) is meaningful.

3. The Existence and Uniqueness of Solutions for Slightly Compressible Third Grade Fluids Equations

In this section, we first give a description of the slightly compressible third grade fluids equations and then prove the existence and uniqueness of its weak solutions.

Since our ultimate purpose is to investigate the convergence of solutions of the slightly compressible third grade fluids equations to the solutions of the incompressible third grade fluids equations, we assume that is given as in Problem 1.

For given , we consider the following initial-boundary value problem.

For any given , set . Find and such that where the function (not appearing in (5)) is independent of and

Remark 5. Equation (29) contains the term which does not appear in (5). This is a stabilization term which corresponds to the substitution of the trilinear form for the form (see (14)), where the trilinear form is given by In fact, if , then . But , . Equations (29)-(30) are the constructed slightly compressible third grade fluids equations. This artificial compressibility method has been used by Zhao and You [15] and Zhao [16] to approximate the incompressible convective Brinkman-Forchheimer equations and a class of incompressible non-Newtonian fluids equations.

We first consider the classical solution of (29)–(33). Assume that and . Then, for any and , multiplying (29) by and (30) by , we obtain The above two equalities are still valid by a continuity argument for any and any .

Let be defined by (35) and set via Then, like , the operator is continuous on .

Lemma 6. If , then

Proof. For almost all , , , and are elements of , and is an element of , and the measurability of the functions , , , and is easy to check. Now for any , we have By Hölder and Gagliardo-Nirenberg inequalities and the embedding , we have Similarly, for any , we have From (42)–(44), we conclude that Therefore, Equations  (46)–(49) give (38)–(41), respectively. The proof is complete.

Now if and satisfy (29) and (30) in the distribution sense, then, for any , we have Since , we see from Lemma 6 that Analogously,

The above analysis leads to the following weak formulation of the problem described by (29)–(33).

Problem 7. Let be fixed. For any given and , find and such that

The rest of this section is devoted to prove the existence of solutions of Problem 7. To this end, we need the compactness theorem (see [13]) involving fractional derivatives in time variable .

Assume , , and are Hilbert spaces with the embedding being continuous and Let be a function from to ; we denote by its Fourier transform: The derivative in of order is the inverse Fourier transform of ; that is, For given , define the space Then 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 8 (see [13]). Assume , , and are Hilbert spaces satisfying (59) and (60). Then, for any bounded set and , the following compact embedding holds:

Theorem 9. Let be fixed. For any given and , Problem 7 possesses a unique solution .

Proof. We will use the Faedo-Galerkin method to prove the existence of solutions for Problem 7. Consider an orthonormal basis of constituted of elements and an orthonormal basis of constituted of elements , .
For each positive integer , we define the approximate solution of Problem 7 by which satisfies and the initial conditions where and are the projections of and from the spaces and to the spaces and , respectively.
Equations (68)–(70) form a Cauchy problem of first-order nonlinear ordinary differential equations (ODE) in for the functions and with the following initial conditions: By the standard theory of ODE, we have the existence of a solution defined on , . Moreover, if , then we have the following blow-up criteria and .
Multiplying (68) by () and (69) by () and then adding these resulting equalities, we obtain Set . Using integration by parts and Hölder inequality, we have Also integrating by parts, using the fact that is a symmetric matrix, we get Since and , we have By combining (72)–(75), we deduce that Integrating (76) in from to    shows that Thus and We now need an estimate of the fractional derivative in time of to pass to the limit in the nonlinear terms in (68). Set We write the relations (68) and (69) as Set Then both and have two discontinuities at and . Thus we have on the whole the following where is defined similarly with that of and and are the Dirac distributions at and , respectively. Taking the Fourier transform on both sides of (84), we get Similar with (83), we can define and and let and be the Fourier transforms of and , respectively. Multiplying (85) by () and (86) by () and then adding these resulting equalities, we obtain Since , integration by parts gives It then follows from (87), (88), and then (78) that We next estimate the term . In fact, using the similar derivations as (73) and (74), we get Inserting (90) into (89) gives For some fixed , we have , . Thus Employing Parseval equality, Poincaré inequality, and (79), we obtain By (91) and the fact that , we obtain Equations (79), (80), and Parseval equality imply Using Hölder inequality, (78), and the convergence of the infinite integral we obtain Equations (92)–(97) give Equations (78)–(80), as well as Lemma 8 and (98), imply that we can extract a subsequence (still denoted by ) of such that Taking and multiplying (68) (resp., (69)) by , integrating over , and then integrating the first term by parts, we obtain We now examine the convergence of the nonlinear terms , , and in (104). Firstly, consider Using Hölder inequality, (78)-(79), and (101), we get Similarly, consider Equatuions (106)–(110) imply that Secondly, the following convergence can be proved by using the same way as that in [7], and we omit the details here. It is not difficult to pass to the limit in the linear terms in (104) and (105). Now for fixed and , taking the limit of (104) and (105) as , using (99)–(103) and (111)-(112), we obtain Since , we have the embeddings and use the Hölder inequality to get Using integration by parts, we have From (115)–(117), we find that the nonlinear terms in (113) are continuous with respect to . By a continuity argument, from (113) and (114), we have that, for every and , that is, in the sense of distribution, we obtain We next prove that the functions and satisfy the initial conditions and , respectively. Since , by (47)–(49), we find that Equations (119)–(120) imply that Thus by the classical embedding theorem (see [13, 14]) we have and . To show , we take a test function with and . Then repeating the above limiting process, we obtain On the other hand, multiplying (119) by and then integrating, we obtain We compare (122) with (124) and (123) with (125) to obtain Hence and . The uniqueness of the solutions is proved exactly the same as Lemma 3.4 of [7]. The proof of Theorem 9 now is complete.

4. Convergence of Solutions for the Slightly Compressible Third Grade Fluids to the Solutions of Incompressible Third Grade Fluids

In this section, we shall show that the solutions of the slightly compressible third grade fluids equations converge to the solutions of the incompressible third grade fluids equations. Let be the solutions of the slightly compressible third grade fluids equations. We prove that there exists a sequence converging to such that the sequence is convergent. Let be its limit. We prove that is a solution of the incompressible third grade fluids equations.

Theorem 10. For any given and , there exists a sequence such that as and the solutions of Problem 7 are convergent. Let be its limit. Then is a solution of the incompressible third grade fluids equations (5). Moreover,

Proof. By (78)–(80) and (98), we have for all the following: In terms of (131)–(134), there exist a sequence ( as ) and some and such that when Passing to the limit in (57) in the sense of distribution as , we obtain Equations (57) and (139) give which implies that and thus Now let . Since , (56) shows Let . Multiplying (142) by , integrating in the resulting equality over , and then using integration by parts, we obtain Using the convergent relations (135)–(138) and the similar derivations of (111)-(112), we can obtain since and . Also by the convergent relations (135)–(138) and (144)–(146), we can pass to the limit for in (143) and obtain The relation (147) is the same as (27). Then it is easy to show that is a solution of Problem 1. Correspondingly, satisfies (5).
We next prove the convergent relations (127)–(130). Equation (127) follows easily from (131) and (128) follows easily from (132). Equation (129) can be deduced from (134)–(136) and the compact embedding theorem (see Lemma 8). It remains to prove (130). To this end, we write (56) as The convergent results for show that the right-hand side of (148) converges weakly to in as . We find that (149) is exactly . Therefore, The proof is complete.

5. Concluding Remarks

In this study, we discuss the approximation of a class of incompressible third grade fluids equations by the artificial compressibility method. We first introduce a family of perturbed slightly compressible third grade fluids equations (depending on a positive parameter ) which approximate the incompressible equations as . Then we prove the existence and uniqueness of weak solutions for the slightly compressible equations. The obtained weak solution provides us with the “approximation” of the original incompressible equations. The merit of the solution is that it overcomes the computational difficulties connected with the constraint “.” It is easier to be implemented in the simulation than in the original incompressible equations as the constraint “” has been replaced by the evolution equation (see, e.g., [13]). Lastly, we establish the fact that the solutions of the slightly compressible equations converge to the solutions of the incompressible equations as . This convergence indicates that the approximation is reasonable.

There is still much work to be done concerning the third grade fluids equations. For example, we could consider borrowing the approaches of [17–20], the existence and regularity of the uniform attractors and pullback attractors, and also the existence and limiting behavior of the trajectory attractor for the the third grade fluids equations when its solutions could not be unique. These problems will be the topics of some other papers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank warmly the anonymous referee for his/her pertinent comments and suggestions, which greatly improve the earlier paper. This work was supported by the National Program on Key Basic Research Project of China (no. 2015CB057905) and the National Natural Science Foundation of China (nos. 51279202, 91215301, and U1402231).