A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid

Ngoc, Le Thi Phuong; Nhan, Truong Thi; Long, Nguyen Thanh

doi:https://doi.org/10.1155/2016/3875324

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2016 | Article ID 3875324 | https://doi.org/10.1155/2016/3875324

A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid

Le Thi Phuong Ngoc,¹Truong Thi Nhan,^2,3and Nguyen Thanh Long³

Academic Editor: Andrew Pickering

Received18 Sept 2016

Accepted13 Nov 2016

Published29 Dec 2016

Abstract

We study the following initial-boundary value problem − + = , , ; , ; , where are given constants and and are given functions. In Part 1, we use the Galerkin method and compactness method to prove the existence of a unique weak solution of the problem above on for every In Part 2, we investigate asymptotic behavior of the solution as In Part 3, we prove the existence and uniqueness of a weak solution of problem − + = , , ; , associated with a “-periodic condition” , where is given constant.

1. Introduction

In this paper, we consider the following nonlinear pseudoparabolic equation:associated with the boundary conditionsand the initial conditionor the “-periodic condition”where , , and are given constants and , , , , , , and are given functions satisfying conditions specified later.

In the case of , , and being the constants, the initial-boundary value problems (1)–(3) are classical and have a long history of applications and mathematical development. We refer to the monographs of Al’shin et al. [1] and of Carroll and Showalter [2] for references and results on pseudoparabolic or Sobolev type equations. We also refer to [3] for asymptotic behavior and to[4] for nonlinear problems. Problems of this type arise in material science and physics, which have been extensively studied, and several results concerning existence, regularity, and asymptotic behavior have been established.

Equation (1) arises within frameworks of mathematical models in engineering and physical sciences (see [5–11] for references therein and interesting results on second grade fluids or a fourth grade fluid or other unsteady flows). It is well known that fluid solid mixtures are generally considered as second-grade fluids and are modeled as fluids with variable physical parameters; thus, an analysis is performed for a second-grade fluid with space dependent viscosity, elasticity, and density.

In [9], some unsteady flow problems of a second-grade fluid were considered. The flows are generated by the sudden application of a constant pressure gradient or by the impulsive motion of a boundary. Here, the velocities of the flows are described by the partial differential equations and exact analytic solutions of these differential equations are obtained. Suppose that the second-grade fluid is in a circular cylinder and is initially at rest, and the fluid starts suddenly due to the motion of the cylinder parallel to its length. The axis of the cylinder is chosen as the -axis. Using cylindrical polar coordinates, the governing partial differential equation iswhere is the velocity along the -axis, is the kinematic viscosity, is the material parameter, and is the imposed magnetic field. In the boundary and initial conditions, is the constant velocity at and is the radius of the cylinder.

In [6], two types of time-dependent flows were investigated. An eigenfunction expansion method was used to find the velocity distribution. The obtained solutions satisfy the boundary and initial conditions and the governing equation. Remarkably, some exact analytic solutions are possible for flows involving second-grade fluid with variable material properties in terms of trigonometric and Chebyshev functions.

In [5], Mahmood et al. have considered the longitudinal oscillatory motion of second-grade fluid between two infinite coaxial circular cylinders, oscillating along their common axis with given constant angular frequencies and . Velocity field and associated tangential stress of the motion were determined by using Laplace and Hankel transforms. In order to find exact analytic solutions for the flow of second-grade fluid between two longitudinally oscillating cylinders, the following problem was studied:where , , , , , , and are positive constants. The solutions obtained have been presented under series form in terms of Bessel functions , , , , , and , satisfying the governing equation and all imposed initial and boundary conditions.

The nonlinear parabolic problems of the form (1)–(3), with/without the term , were also studied in [12, 13] and references therein. In [12], by using the Galerkin and compactness method in appropriate Sobolev spaces with weight, the authors proved the existence of a unique weak solution of the following initial and boundary value problem for nonlinear parabolic equation:

Furthermore, asymptotic behavior of the solution as was studied. In [13], the following nonlinear heat equation associated with Dirichlet-Robin conditions was investigated:

Condition (4), which we call “-periodic condition,” is known as a drifted periodic condition (see [14]). Indeed, if , , in the case of , then we havewhich meanswith satisfying the condition

Note that (11) holds by the fact that

With , (4) leads to -periodic conditionand with , we have the antiperiodic condition

The present paper is concerned with the second-grade fluid in a circular cylinder associated with the initial condition (3) or a drifted periodic condition (10). The extensive study of such flows is motivated by both their fundamental interest and their practical importance (see [9]).

This paper is a continuation of paper [15] dealing with the nonlinear pseudoparabolic equation (1) associated with the mixed inhomogeneous condition, in the case of , , being the constants. It consists of five sections. First, preliminaries are done in Section 2. Under appropriate conditions, the existence of a unique weak solution of problems (1)–(3) is proved in Section 3. Next, an asymptotic behavior of the solution of problems (1)–(3), as , is discussed in Section 4. Finally, Section 5 is devoted to the establishment, the existence, and uniqueness of a weak solution of problems (1), (2), and (4).

Because of mathematical context, the results obtained here generalize relatively the ones in [12, 13, 15], by improving the techniques used as before and with appropriate modifications.

2. Preliminaries

Put , , We omit the definitions of the usual function spaces: , , and . We define , and , , . The norm in is denoted by We also denote by the scalar product in . We denote by the norm of a Banach space and by the dual space of We denote by for the Banach space of the real functions measurable, such that

Let , , , and denote , , , , , respectively.

On , we shall use the following norm:

We put

is a closed subspace of and on , two norms and are equivalent.

Note that and are also the Hilbert spaces with respect to the corresponding scalar productsrespectively. The norms in and induced by the corresponding scalar products are denoted by and , respectively. is continuously and densely embedded in . Identifying with (the dual of ), we have ; on the other hand, the notation is used for the pairing between and .

We then have the following lemmas, the proofs of which can be found in [16].

Lemma 1. We have the following inequalities:

Lemma 2. The imbedding is compact.

Lemma 3. The imbedding is compact and

Remark 4. On , two norms and are equivalent. So there are two norms and on and four norms , , , and on

Consider is the symmetric bilinear form on defined by

Then, the symmetric bilinear form is continuous on and coercive on

We have also the following lemma.

Lemma 5. There exists the Hilbert orthonormal base of consisting of the eigenfunctions corresponding to the eigenvalue such thatFurthermore, the sequence is also the Hilbert orthonormal base of with respect to the scalar product
On the other hand, we also have satisfying the following boundary value problem:

The proof of Lemma 5 can be found in [17, p. 87, Theorem ], with and and as defined by (21).

3. The Existence and the Uniqueness

Now, we consider problems (1)–(3) in which is a positive constant and make the following assumptions:, , , , , satisfies the condition that there exists positive constant such that , for all .

In case or , it is clearly that problems (1)–(3) reduce to a problem with homogeneous boundary conditions by the suitable transformation. Indeed, putting , by the transformation , problems (1)–(3) reduce to the following problem:whereand , , and satisfy the condition .

The weak formulation of the initial-boundary value problem (24) can be given in the following manner: Find with , such that satisfies the following variational equation:where is the symmetric bilinear form on defined by (21).

Then, we have the following theorem.

Theorem 6. Let and hold. Then, problem (24) has a unique weak solution such thatMoreover, if is replaced by , then the solution satisfies

Proof. The proof consists of several steps.
Step 1 (the Faedo-Galerkin approximation (introduced by Lions [18])). Consider the basis for as in Lemma 5. We find the approximate solution of problem (24) in the formwhere the coefficients satisfy the system of linear differential equationswhereThe system of (30) can be rewritten in the formIt is clear that for each there exists a solution in the form of (29) which satisfies (30) almost everywhere on for some , The following estimates allow one to take for all .
Step 2 (a priori estimates)
(a) The First Estimate. Multiplying the th equation of (30) by and summing up with respect to , afterwards, integrating by parts with respect to the time variable from to , we get after some rearrangements:By strongly in , we havewhere always indicates a bound depending on .
PutBy the assumptions , we estimate without difficulty the following terms in (33) as follows:Hence, it follows from (33), (34), and (36) thatwhereBy Gronwall’s lemma, we obtain from (37) thatfor all , for all , ; that is, , where always indicates a bound depending on .
(b) The Second Estimate. Multiplying the th equation of (30) by and summing up with respect to , we haveIntegrating (40), we getWe shall estimate the terms of (41) as follows:On the other hand, we haveand henceIt follows from (41)–(43) and (45) thatfor all , for all , where always indicates a bound depending on
By and (39) and (46), we deduce thatStep 3 (the limiting process). By (39), (46), and (47), we deduce that there exists a subsequence of , still denoted by such thatUsing a compactness lemma ([18], Lions, p. 57), applied to (48), we can extract from the sequence a subsequence still denoted by , such thatBy the Riesz-Fischer theorem, we can extract from a subsequence still denoted by , such thatBecause is continuous, it givesOn the other hand, by , it follows from (44) thatwhere is a constant independent of
Using the dominated convergence theorem, (51) and (52) yieldPassing to the limit in (30) by (31), (48), and (53), we obtainStep 4 (uniqueness of the solution). First, we shall need the following lemma.
Lemma 7. Let be the weak solution of the following problem:Then,Furthermore, if , then the equality in (56) holds.
Lemma 7 is a slight improvement of a lemma used in [12] (or it can be found in Lions’s book [18]).
Now, we will prove the uniqueness of the solutions.
Let and be two weak solutions of (24). Then, is a weak solution of (55) with the right-hand side function replaced by and Using Lemma7, we have equalitywhereBy , we obtainIt follows from (57)–(59) thatBy Gronwall’s lemma,
Assume now that is replaced by ; then we only have to show that is bounded in
Indeed, multiplying the th equation of (30) by and summing up with respect to , afterwards, integrating with respect to the time variable from to , we get after some rearrangementswhereBy the same estimates as above, we obtainThis impliesThen, the sequence is bounded in .
Applying a similar argument used as above, the limit of the sequence in suitable function spaces is a unique weak solution of (24) satisfying (28).
Therefore, Theorem 6 is proved.

4. Asymptotic Behavior of the Solution as

In this part, let , hold. Then, there exists a unique solution of problems (1)–(3) such that

We shall study asymptotic behavior of the solution as .

We make the following supplementary assumptions on the functions , , , , and :, , and there exist the positive constants , , , and , such that, , , ., and there exist the positive constants , , , , and , such that, and there exist the positive constants , and the function , such that , . and there exists a positive constant , with , such that , for all

First, we consider the following stationary problem:

The weak solution of problem (68) is obtained from the following variational problem. Find such thatfor all , where is the symmetric bilinear form on defined by (21).

We then have the following theorem.

Theorem 8. Let hold. Then, there exists a unique solution of the variational problem (69) such that

Proof. Consider the basis for as in Lemma 5. Putwhere satisfies the following nonlinear equation system:By Brouwer’s lemma (see Lions [18], Lemma , p. 53), it follows from the hypotheses that systems (70) and (71) have a solution
Multiplying the th equation of system (71) by , and then summing up with respect to , we haveBy using , we obtainBy using inequalities (20)(iii) and (73), we obtain from (72) that By , choose such that
Hence, we deduce from (74) thatand is a constant independent of
By means of (75) and Lemma 3, the sequence has a subsequence still denoted by such thatOn the other hand, by (76) and the continuity of , we havePassing to the limit in (71), we find without difficulty from (76) and (77) that satisfies the equationEquation (78) holds for every ; that is, (69) is true.
The solution of problem (69) is unique, which can be shown by the same arguments as in the proof of Theorem 6.

Now we consider asymptotic behavior of the solution as

We then have the following theorem.

Theorem 9. Let , , and hold. Let satisfy the following condition, in addition,, , And let in satisfy the following condition, in addition, Then we havewhere , are constants independent of

Proof. Put Let us subtract (30) with (71) to obtainBy multiplying (80) by and summing up in , we obtainBy the assumptions , , , and and using inequality (20)(iii), and with , we estimate without difficulty the following terms in (81) as follows:
(i) Estimate , as(ii) Estimate , as(iii) Estimate , as(iv) Estimate . Note that, from the inequalitiesand , we deduce thatwhereHence,Thus,(v) Estimate . We haveHence,It follows thatThus,(vi) Estimate , asIt follows from (81)–(84), (89), (93), and (94) thatBy , choose such that .
Put , and we have for all , asBywhere , it follows from (96) and (97) thatChoose such that , and then we have from (98) thatHence, we obtain from (99) thatLetting in (100), we obtainorwhereNote thatHence,It follows from (102) and (105) thatThis completes the proof of Theorem 9.

5. The Existence and the Uniqueness of a -Periodic Weak Solution

In this section, we shall consider problems (1), (2), and (4) with , as given constants and , , , , , and as given functions satisfying the following assumptions:, and and are -periodic; that is, , , , ;, , , , a.e., ;, is -periodic, ; and there exists a positive constant , with

Remark 10. An example of the functions satisfying iswhere , , and are constants. It is obvious that holds, because with and Similarly, by the transformation , with , , problems (1), (2), and (4) reduce to the following problem:where is defined by (25)
The weak formulation of problem (112) can be given in the following manner: Find with , such that satisfies the following variational equation:where is the symmetric bilinear form on defined by (21).

Then, we have the following theorem.

Theorem 11. Let and hold. Then, problem (112) has a unique weak solution such that

Proof. The proof consists of several steps.
Step 1 (the Faedo-Galerkin approximation (introduced by Lions [18])). Consider the basis for as in Lemma 5. Let be the linear space generated by We consider the following problem.
Find a function in the form (29) satisfying the nonlinear differential equation system (30) and the - periodic condition:We consider an initial value problem given by (30), where is given in
It is clear that, for each , there exists a solution in the form (29) which satisfies (30) almost everywhere on for some , The following a priori estimates allow us to take for all .
Step 2 (a priori estimates). Multiplying the th equation of (30) by and summing up with respect to , we getBy the same estimates as in Section 3, and with , we obtainHence, it follows from (116) and (117) thatBy , choose such that .
It is similar to (97); we getwhere
From (118) and (119), it leads toin which
Integrating (120), we havewhere , withTherefore, if we choose such that , we obtain from (121) thatLet be a closed ball in the space with the norm
By , we obtainLet us defineWe prove that is a contraction. Let and let , where and are solutions of system (30) on satisfying the initial conditions and , respectively. Then, satisfies the following differential equation system:where , with initial conditionBy using the same arguments as before, we can show thatwhere ,
Integrating inequality (128), we obtain By , it follows that orthat is, is a contraction.
Therefore, there exists a unique function such that the solution of the initial value problem (30) is a solution of system (30), (115). This solution satisfies inequality (124) a.e., in
On the other hand, multiplying the th equation of (30) by and summing up with respect to , afterwards, integrating with respect to the time variable from to , we get after some rearrangementsWe estimate without difficulty the following terms in (132):Note thatHence,where Moreover,It follows from (132), (133), (136), and (138) thatfor all , for all , where always indicates a bound depending on .
Step 3 (the limiting process). By (123) and (139), we deduce that there exists a subsequence of , still denoted by such thatFrom (115), we obtainIndeed, we prove (141) as follows.
By , and by the imbedding being compact, there exists a subsequence of , still denoted by such thatBy , we deduce from (140) and (142) thatThis impliesFrom (115), we obtainBy (140), (145), and (146), it yieldsTherefore,Therefore, (141) is proved.
Using a compactness lemma ([18], Lions, p. 57), applied to (140), we can extract from the sequence a subsequence still denoted by , such thatBy the Riesz-Fischer theorem, we can extract from a subsequence still denoted by , such that Because is continuous, thenOn the other hand,where is the constant defined by (135).
Using the dominated convergence theorem, (151) and (152) yieldDenote by the orthonormal base in the real Hilbert space . The set forms an orthonormal base in From (30) we havefor all , ,
For fixed, passing to the limit in (154) by (140) and (153), we getNote that (155) holds for every ; that is, the equalityis fulfilled.
Step 4 (uniqueness of the solutions). Let and be two solutions of (113). Then satisfies the following problem:Taking in (157) and using (157), we getBy , we haveOn the other hand,Hence,By , implying , we deduce from (161) that ; that is,
This completes the proof of Theorem 11.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to this article. They read and approved the final manuscript.

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Copyright © 2016 Le Thi Phuong Ngoc 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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