Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 5036048, 11 pages
http://dx.doi.org/10.1155/2016/5036048

Research Article

Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition

Fan Geng,¹ Ruizhai Li,¹ Xiaojun Zhang,² and Xiangyu Ge²

¹School of Arts and Science, Sias International University, Zhengzhou 451150, China
²Wuhan Technology and Business University, Wuhan 430065, China

Received 29 November 2015; Accepted 7 February 2016

Academic Editor: Taher S. Hassan

Copyright © 2016 Fan Geng 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.

Abstract.

The paper studies the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary condition . The main result is concerned with nonlinearities with supercritical growth. In that case, we construct a bounded absorbing set with further regularity and obtain quasi-stability estimates. Then the exponential attractor is established in natural energy space .

1. Introduction

In this paper, we are concerned with the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary condition

where is a bounded domain in with the smooth boundary , on which we consider the clamped boundary condition

where is the unit outward normal on , and the initial condition

and the assumptions on and will be specified later.

In 1872, Boussinesq [1] established the equation

to describe the longitudinal displacement of the shallow water wave. Here and are some constants depending on the depth of the fluid and characteristic velocity of the water wave. When , (4) is called “good” Boussinesq equation, when , (4) is called “bad” Boussinesq equation. There have been lots of research on the well-posedness, blowup, and other properties of solutions for both the “good” and the “bad” Boussinesq equation of type (1) (see [2–14] and references therein). While for the investigation on the global attractor to (1), one can see [15–19] and references therein.

Chueshov and Lasiecka [20, 21] studied the longtime behavior of solutions to the Kirchhoff-Boussinesq plate equation

with and the clamped boundary condition (2). Here is the damping parameter and the mapping and the smooth functions and represent (nonlinear) feedback forces acting upon the plate, in particular,

Ignoring both restoring force and feedback force and replacing the inertial term by , with (the relaxation time) sufficiently small, (5) becomes the modified Cohn-Hilliard equation

which is proposed by Galenko et al. [22–24] to model rapid spinodal decomposition in nonequilibrium phase separation processes. Grasselli et al. [25–27] studied the well-posedness and the longtime dynamics of (7) in both and cases, with hinged boundary condition. They established the existence of the global and exponential attractor for in case, and for sufficiently small in case. Taking in (7) or taking in (5), and taking into account the inertial force represented by and replacing the weak damping by a strong one , (1) arises.

In case, Dai and Guo [15, 16] studied the “bad” Boussinesq equation with strong damping

They got global solution , where , , .

For the multidimensional case, Yang [17] proved the IVP of the Boussinesq equation

There existed the global weak solution, where , , , , , and , , . Here the growth exponent () is called critical. The growth exponent (≥) is called supercritical. However, there is little research on the the higher global regularity of a bounded absorbing set, the global attractor and an exponential attractor in natural energy space for the dynamical system. We try to solve those problems in this paper.

Global attractor is a basic concept in the research studies of the asymptotic behavior of the dissipative system. From the physical point of view, the global attractor of the dissipative equation (1) represents the permanent regime that can be observed when the excitation starts from any point in natural energy space, and its dimension represents the number of degrees of freedom of the related turbulent phenomenon and thus the level of complexity concerning the flow. All the information concerning the attractor and its dimension from the qualitative nature to the quantitative nature then yields valuable information concerning the flows that this physical system can generate. On the physical and numerical sides, this dimension gives one an idea of the number of parameters and the size of the computations needed in numerical simulations. However, the global attractor may possess an essential drawback; namely, the rate of attraction may be arbitrarily slow and it can not be estimated in terms of physical parameters of the system under consideration. While the exponential attractor overcomes the drawback because not only it has finite fractal dimension but also its contractive rate is exponential and measurable in terms of the physical parameters, the purpose of the present paper is to establish the existence of an exponential attractor in supercritical case. Our result (see Theorem 8 below) in this paper extends the corresponding result in [28].

In comparison with the results in [17, 18], the contribution of the paper lies in that

(1)the exponential attractor is established in natural energy space in supercritical case. See Theorem 8;(2)the critical case is solved in . In the concrete, when , the global and exponential attractor in is established, and the higher regularity of the global attractor is obtained. See Theorem 15;(3)the restriction is removed in subcritical case. See Theorem 15.

The plan of the paper is as follows. In Section 2, the global existence of the weak solutions is discussed by the energy method and the existence of global attractor is established. In Section 3, the exponential attractor is established for supercritical case. In Section 4, global attractor and the exponential attractor are established for nonsupercritical case.

2. Global Existence of Weak Solutions

For brevity, we use the following abbreviations:

with , where are the -based Sobolev spaces and are the completion of in for . The notation for the -inner product will also be used for the notation of duality pairing between dual spaces and denotes positive constants depending on the quantities appearing in the parenthesis.

We define the operator (the dual space of ),

Then, the operators () are strictly positive and the spaces are Hilbert spaces with the scalar products and the norms

respectively. Obviously,

Rewriting (1) in the operator equation and applying to the resulting expression, we get the Cauchy problem equivalent to problem (1), (2), and (3):

For each , , we denote the Banach space

which is equipped with the usual graph norm,

Theorem 1. Assume that   ,

where

where , , , , and .
Consider

where ,
  , .
Then problem (14) and (15) admits a unique weak solution , with . More precisely, the solution possesses the following properties:

(i)There exists a small positive constant such that

(ii)When , the solution is Lipschitz continuous in the weaker space as ; that is,

for some , where and are, respectively, the weak solutions of (14) corresponding to initial data and .

Remark 2. The formula (20) implies that every ; there exists , such that

where

Lemma 3 (see [29]). Let , and be the Banach spaces, ,

Then,

Proof of Theorem 1. We first obtain a priori estimate to the solutions of problem (14) and (15).
Let and rewrite (1); we have

Using the multiplier in (26), we get

where

Hence,

Applying the Gronwall lemma to (29),

where , , .
Using the multiplier , in (26),

(31) + (32); we have

Obviously,

Indeed,

where , ; , , and

Thus

That is,

It follows from (14) and (38) that

Now, we look for the approximate solutions of problem (14) and (15) of the form

where , is an orthonormal basis in , and at the same time an orthogonal one in , and with

Obviously, the estimate (38) is valid for . So we can extract a subsequence, still denoted by , such that

Applying Lemma 3 to (33), we have

Indeed, when , we have ; hence

when , and by virtue of the interpolation theorem,

Letting in (41) we see that is a weak solution of problem (14) and (15), with .
Integrating (31) over ,

Obviously,

we prove that , and (22).
(ii) Now, we show that is Lipschitz continuous in the weak space .
In fact, let be two solutions of problem (14) and (15) as shown above corresponding to initial data and , respectively. Then solves

Using the multiplier in (48),

We get

Theorem 1 is proved. Under the assumptions of Theorem 1, with , , we can define the solution operator ,

where is the weak solution of problem (14) and (15). Theorem 1 shows that constitutes a semigroup on , which is Lipschitz continuous in .

Theorem 4 (existence of the global attractor). Under the same assumptions of Theorem 1, with , has a global attractor in and , where is bounded set in , .

Proof of Theorem 4. Estimate (38) implies that the ball

is an absorbing set of the semigroup in . For every bound in ,

Let , where , and

It is easy to get

, , solves

where .
Let ,

because , , , .
Therefore

We know , so

Thus is bounded in . ,

is compact in . For every bounded set ,

Therefore has the global attractor and

that is . This completes the proof.

3. Exponential Attractor

Definition 5. The set is called an exponential attractor for the solution semigroup of acting on the energy space if

(i)the set is a compact set in ;(ii) is forward invariant set; that is, , ;(iii) attracts exponentially the images of all bounded set in ; that is,

for all bounded set ;(iv)it has finite fractal dimension in ; that is, .

From Theorem 1, estimate (38) implies that the ball

is an absorbing set of the semigroup in for . Without loss of generality we assume that is a forward invariant set. Let

where is chosen such that for and stands for the closure in space . Obviously, the set is bounded closed set in , , and it is also an absorbing set of . constitutes a complete metric space (with the norm) and one sees from (22) that the solution semigroup is continuous on , and the system constitutes a dissipative dynamical system.

Lemma 6 (see [19]). Let be a Banach space and a bounded closed set in . Assume that the mapping possesses the following properties:

(i) is Lipschitz on ; that is, there exists such that

(ii)there exist compact seminorms and on such that

for any , where and are constants. Then for any and , there exists a positively invariant compact set of finite fractal dimension such that

where , and

where is the maximal number of pairs in possessing the properties

Lemma 7. Let be the metric spaces and let the mapping be -Hölder continuous on the set . Then

Theorem 8. Let the assumptions of Theorem 1 be in force, with . Then the solution semigroup has an exponential attractor in .

Proof. Define the operator

We show that the discrete system has an exponential attractor.

Definition 9. We introduce the functional space

equipped with the norm

and the functional space

equipped with the usual graph norm; that is

Obviously, the spaces and are Banach spaces. Let the set

Define the operator

where and in the following means , .

Lemma 10. The set is a bounded closed set in .

Proof. Obviously, is bounded in . For any sequence ,

Since and is closed in , . By the Lipschitz continuity of in ,

so by the uniqueness of the limit; , that is, , where is closed in .

Lemma 10 implies that is complete with respect to the topology of , and the dynamical system constitutes a discrete dissipative dynamical system.

Lemma 11. Under the same assumptions of Theorem 1, then discrete dissipative dynamical system has an exponential attractor .

Proof. Obviously,

the inequality (93) holds; then integrating (93) over we get

Hence,

where ,

It follows from (83) that

where . Since , the seminorm is compact in . Taking and making use of Lemma 6, we get the conclusion of Lemma 11. That is, the discrete dynamical system possesses an exponential attractor . Define the project operator

Lemma 12. is an exponential attractor of the discrete dynamical system .

Proof. (1) is compact because is the image of the compact set under the continuous mapping .
(2) ; we have ; thus .
(3) Obviously,

for some (see Lemma 11).
(4)
Hence, is a desired exponential attractor. Lemma 12 is proved.
Let

By the method used in [11], one easily knows that is an exponential attractor of , with topology. So by the definition of the exponential attractor, there exists a constant , such that

Since the set is bounded in , we claim that is an exponential attractor of the system . Indeed, (i) obviously, is forward invariant; (ii) define the project operator

for any , , which imply that the mapping is -Hölder continuous. Therefore, (the image of ) is compact in .
(iii) Consider the following:

(iv) For any , there exists a , such that as . On account of ,

Therefore, Theorem 8 is proved.

4. Global and Exponential Attractor in Nonsupercritical Case

Theorem 13. Let the assumptions of Theorem 1 be in force, with . Then problem (14)-(15) admits a unique weak solution , with , and the solution is Lipschitz continuous in ; that is,

for some , where , and are, respectively, the weak solutions of (14) corresponding to initial data and .

Proof. The existence of the weak solutions can be easily proved by the same way of Theorem 1. So we only prove (93) here. Taking -inner product by in (36), we have

Applying the Gronwall inequality to (94) we obtain (93).

Remark 14. (i) When , by (93), define the continuous semigroup

where as shown in Theorem 13.
(ii) It follows from Theorem 8 and Remark 14 that the dynamical system is dissipative; that is, it has a bounded absorbing set . Without loss of generality we assume that is positive invariant; that is, for .

Theorem 15. Let the assumptions of Theorem 13 be in force, especially when , ,

Then the following conclusions are valid.
(i) The solution semigroup possesses in a compact global attractor , which has finite fractal dimension.
(ii) Any full trajectory possesses the property

and there exists constant such that

(iii) The global attractor consists of full trajectory such that

where is the set of all fixed points of ; that is,

Furthermore, for any ,

(iv) The semigroup has in an exponential attractor.

Lemma 16. Let be an absolutely continuous function satisfying

where , , for and some . Then

Lemma 17 (quasi-stability). Let the assumptions of Theorem 13 be valid and let be the solutions of problem (14)-(15) with initial data in . Then satisfies the relation

for some constants .

Proof. (i) When , taking -inner product by in (48), with , we get

where

for suitably small. On account of , for and the interpolation theorem we have the control

Therefore, there exists constant such that

where .
(ii) When , rewrite (105) in the form

where

Since , there exists constant such that

which means

We infer from (109) that

where

for suitably small. Inserting (114) into (113), we get

There exists such that

Applying Lemma 16 to (115), we get

where . Lemma 17 is proved.

Proof of Theorem 15. The estimates (93) and (104) show that the dissipative system is quasi-stable on the absorbing set , so the conclusions (i) and (ii) follow directly from the standard theory on global attractor (cf. Theorems and in [30]).
The energy equality holds and shows that is a strictly Lyapunov function on , so the dynamical system is gradient, and by conclusion (ii), it has a compact global attractor. Therefore, the conclusion (iii) of Theorem 4 holds (cf. Theorems and 2.31 in [20]).
We see from the conclusion (ii) that the global attractor is included and bounded in . Let be the closure of the -neighborhood of in ; that is,

Then is bounded in and closed in , and it is an absorbing set of ; without loss of generality we assume that , . By Lemma 17, is quasi-stable on . For every , and by (14), ,

So has in an exponential attractor (cf. Theorem   in [30]). Theorem 15 is proved.

Remark 18. Comparing Theorem 8 with Theorem   in [17] one finds that the critical case is solved in natural energy space , the restriction is removed in the subcritical case , the higher regularity of the global attractor is obtained, and the exponential attractor is established in .

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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