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

The paper studies the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary condition u tt −Δu+Δ 2 u−Δu t −Δg(u) = f(x). The main result is concerned with nonlinearities g(u)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 V 2 × H.


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 Ω is a bounded domain in R  with the smooth boundary Ω, on which we consider the clamped boundary condition where ] is the unit outward normal on Ω, and the initial condition  (, 0) =  0 () ,   (, 0) =  1 () , and the assumptions on () and  will be specified later.
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  = p is solved in  1 .In the concrete, when 1 ≤  ≤ p, the global and exponential attractor in  1 is established, and the higher regularity of the global attractor is obtained.See Theorem 15; (3) the restriction  ≤ 5 is removed in subcritical case.
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.

Global Existence of Weak Solutions
For brevity, we use the following abbreviations: with  ≥ 1, where   are the  2 -based Sobolev spaces and   0 are the completion of  ∞ 0 (Ω) in   for  > 0. 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.
(ii) Now, we show that (,   ) is Lipschitz continuous in the weak space .
In fact, let , V be two solutions of problem ( 14) and (15) as shown above corresponding to initial data  0 ,  1 and V 0 , V 1 , respectively.Then  =  − V solves Using the multiplier   for every ( 0 ,  1 ) ∈ ,  ≥ 0, where  is the weak solution of problem ( 14) and ( 15).Theorem 1 shows that {()} constitutes a semigroup on , which is Lipschitz continuous in .
Let  = û , because We know that is A ⊂ .This completes the proof.

Exponential Attractor
Definition 5.The set A exp ⊂  is called an exponential attractor for the solution semigroup () of acting on the energy space  if (i) the set A exp is a compact set in ; (ii) A exp is forward invariant set; that is, ()A ⊂ A,  ≥ 0; (iii) A exp attracts exponentially the images of all bounded set in ; that is, for all bounded set  ⊂ ; (iv) it has finite fractal dimension in ; that is, dim  {A exp , } < +∞.
From Theorem 1, estimate (38) implies that the ball is an absorbing set of the semigroup () in  for  >  0 .Without loss of generality we assume that   is a forward invariant set.Let where  0 > 0 is chosen such that ()  ⊂   for  ≥  0 and [ ]  stands for the closure in space .Obviously, the set B  is bounded closed set in , ()B  ⊂ B  ,  ≥ 0, and it is also an absorbing set of ().B  constitutes a complete metric space (with the  norm) and one sees from ( 22) that the solution semigroup () is continuous on B  , and the system ((), B  ) constitutes a dissipative dynamical system.
Lemma 10 implies that   is complete with respect to the topology of   , and the dynamical system (V  ,   ) constitutes a discrete dissipative dynamical system.Lemma 11.Under the same assumptions of Theorem 1, then discrete dissipative dynamical system (V  ,   ) has an exponential attractor A.
Proof.(1)  is compact because  is the image of the compact set A under the continuous mapping Π.
(3) Obviously, for some 0 <  < 1 (see Lemma 11). ( Hence,  is a desired exponential attractor.Lemma 12 is proved. Let By the method used in [11], one easily knows that A exp is an exponential attractor of ((), B  ), with  topology.So by the definition of the exponential attractor, there exists a constant  > 0, such that Since the set A exp ⊂ B 0 is bounded in , we claim that A exp is an exponential attractor of the system ((), ).Indeed, (i) obviously, A exp is forward invariant; (ii) define the project operator for any   ,   1 ,   2 ∈ B  , ,  1 ,  2 ∈ [0, ], which imply that the mapping  is 1/2-Hölder continuous.Therefore, (iii) Consider the following: (iv) For any  ∈ R + , there exists a  ∈ N + , such that ()B  ⊂ ()B  as  ∈ (, ( + 1)].On account of  = (0 Therefore, Theorem 8 is proved.
Proof.The existence of the weak solutions can be easily proved by the same way of Theorem where N is the set of all fixed points of (); that is, (103) Lemma 17 (quasi-stability).Let the assumptions of Theorem 13 be valid and let , V be the solutions of problem ( 14)- (15) where where  =    2 /.Lemma 17 is proved.
Proof of Theorem 15.The estimates (93) and (104) show that the dissipative system ((),  1 ) is quasi-stable on the absorbing set B  , so the conclusions (i) and (ii) follow directly from the standard theory on global attractor (cf.Theorems 7.9.4-7.9.6 and 7.9.8 in [30]).
The energy equality holds and shows that (,   ) is a strictly Lyapunov function on  1 , so the dynamical system ((),  1 ) is gradient, and by conclusion (ii), it has a compact global attractor.Therefore, the conclusion (iii) of Theorem 4 holds (cf.Theorems 2.28 and 2.31 in [20]).
We see from the conclusion (ii) that the global attractor A is included and bounded in  2 =  2 ×  1 .Let D be the closure of the 1-neighborhood of A in  2 ; that is, Then D is bounded in So () has in  1 an exponential attractor (cf.Theorem 7.9.9 in [30]).Theorem 15 is proved.
Remark 18. Comparing Theorem 8 with Theorem 2.2 in [17] one finds that the critical case  = p is solved in natural energy space , the restriction  ≤ 5 is removed in the subcritical case  < p, the higher regularity of the global attractor is obtained, and the exponential attractor is established in  1 .
Let the assumptions of Theorem 1 be in force, with 1 ≤  <  * .Then the solution semigroup () has an exponential attractor A  in .