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

Geng, Fan; Li, Ruizhai; Zhang, Xiaojun; Ge, Xiangyu

doi:https://doi.org/10.1155/2016/5036048

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Volume 2016 | Article ID 5036048 | https://doi.org/10.1155/2016/5036048

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

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

Academic Editor: Taher S. Hassan

Received29 Nov 2015

Accepted07 Feb 2016

Published10 Mar 2016

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 conditionwhere is a bounded domain in with the smooth boundary , on which we consider the clamped boundary conditionwhere is the unit outward normal on , and the initial conditionand the assumptions on and will be specified later.

In 1872, Boussinesq [1] established the equationto 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 equationwith 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 equationwhich 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 dampingThey got global solution , where , , .

For the multidimensional case, Yang [17] proved the IVP of the Boussinesq equationThere 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 normsrespectively. 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 spacewhich is equipped with the usual graph norm,

Theorem 1. Assume that ,wherewhere , , , , and .
Considerwhere ,
, .
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 haveUsing the multiplier in (26), we getwhere Hence,Applying the Gronwall lemma to (29),where , , .
Using the multiplier , in (26),(31) + (32); we haveObviously, Indeed,where , ; , , and ThusThat is,It follows from (14) and (38) thatNow, we look for the approximate solutions of problem (14) and (15) of the formwhere , is an orthonormal basis in , and at the same time an orthogonal one in , and withObviously, the estimate (38) is valid for . So we can extract a subsequence, still denoted by , such thatApplying Lemma 3 to (33), we haveIndeed, 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 solvesUsing the multiplier in (48),We getTheorem 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 ballis an absorbing set of the semigroup in . For every bound in ,Let , where , andIt is easy to get, , solveswhere .
Let ,because , , , .
ThereforeWe know , soThus is bounded in . ,is compact in . For every bounded set ,Therefore has the global attractor andthat 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 ballis an absorbing set of the semigroup in for . Without loss of generality we assume that is a forward invariant set. Letwhere 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 spaceequipped 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 getHence,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 haveApplying the Gronwall inequality to (94) we obtain (93).

Remark 14. (i) When , by (93), define the continuous semigroupwhere 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 propertyand there exists constant such that(iii) The global attractor consists of full trajectory such thatwhere 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 satisfyingwhere , , 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 relationfor some constants .

Proof. (i) When , taking -inner product by in (48), with , we getwhere for suitably small. On account of , for and the interpolation theorem we have the control Therefore, there exists constant such thatwhere .
(ii) When , rewrite (105) in the formwhereSince , there exists constant such thatwhich meansWe infer from (109) thatwherefor suitably small. Inserting (114) into (113), we getThere 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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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.

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