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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 893240, 16 pages
On the Dimension of the Pullback Attractors for g-Navier-Stokes Equations
College of Science, China Jiliang University, Hangzhou 310018, China
Received 3 June 2010; Accepted 14 November 2010
Academic Editor: Binggen Zhang
Copyright © 2010 Delin Wu. 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.
We consider the asymptotic behaviour of nonautonomous 2D g-Navier-Stokes equations in bounded domain . Assuming that , which is translation bounded, the existence of the pullback attractor is proved in and . It is proved that the fractal dimension of the pullback attractor is finite.
In this paper, we study the behavior of solutions of the nonautonomous g-Navier-Stokes equations in spatial dimension 2. These equations are a variation of the standard Navier-Stokes equations, and they assume the form where is a suitable smooth real-valued function defined on and is a suitable bounded domain in . Notice that if , then (1.1) reduce to the standard Navier-Stokes equations.
In addition, we assume that the function is translation bounded, where or . This property implies that
We consider this equation in an appropriate Hilbert space and show that there is a pullback attractor . This is the basic idea of our construction, which is motivated by the works of .
Let . We assume that the function satisfies the following properties:(1),(2)there exist constants and such that, for all , . Note that the constant function satisfies these conditions.
We denote by the space with the scalar product and the norm given by as well as with the norm where .
Then for the functional setting of the problems (1.1), we use the following functional spaces: where is endowed with the scalar product and the norm in and is the spaces with the scalar product and the norm given by Also, we define the orthogonal projection as and we have that , where Then, we define the -Laplacian operator to have the linear operator For the linear operator , the following hold (see ).
(1) is a positive, self-adjoint operator with compact inverse, where the domain of is .
(2) There exist countable eigenvalues of satisfying where and is the smallest eigenvalue of . In addition, there exists the corresponding collection of eigenfunctions which forms an orthonormal basis for .
Next, we denote the bilinear operator and the trilinear form where , , and lie in appropriate subspaces of . Then, the form satisfies We denote a linear operator on by and have as a continuous linear operator from into such that
We now rewrite (1.1) as abstract evolution equations: In  the author established the global regularity of solutions of the g-Navier-Stokes equations. The Navier-Stokes equations were investigated by many authors, and the existence of the attractors for 2D Navier-Stokes equations was first proved in  and independently in . The finite-dimensional property of the global attractor for general dissipative equations was first proved in . For the analysis of the Navier-Stokes equations, one can refer to , specially  for the periodic boundary conditions.
The theory of pullback (or cocycle) attractors has been developed for both the nonautonomous and random dynamical systems (see [7–13]) and has shown to be very useful in the understanding of the dynamics of nonautonomous dynamical systems.
The understanding of the asymptotic behaviour of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem for a system having some dissipativity properties is to analyse the existence and structure of its global attractor, which, in the autonomous case, is an invariant compact set which attracts all the trajectories of the system, uniformly on bounded sets. This set has, in general, a very complicated geometry which reflects the complexity of the long-time behaviour of the system (see [14–17] and the references therein). However, nonautonomous systems are also of great importance and interest as they appear in many applications to natural sciences. In this situation, there are various options to deal with the problem of attractors for nonautonomous systems (kernel sections , skew-product formalism [16, 19], etc.); for our particular situation we have preferred to choose that of pullback attractor (see [9, 10, 13, 20]) which has also proved extremely fruitful, particularly in the case of random dynamical systems (see [11, 13]).
In this paper, we study the existence of compact pullback attractor for the nonautonomous g-Navier-Stokes equations in bounded domain with periodic boundary condition. It is proved that the fractal dimension of the pullback attractor is finite.
Hereafter will denote a generic scale invariant positive constant, which is independent of the physical parameters in the equation and may be different from line to line and even in the same line.
2. Abstract Results
The semigroup property is replaced by the process composition property and, obviously, the initial condition implies that . As with the semigroup composition , this just expresses the uniqueness of solutions.
It is also possible to present the theory within the more general framework of cocycle dynamical systems. In this case the second component of is viewed as an element of some parameter space , so that the solution can be written as , and a shift map is defined so that the process composition becomes the cocycle property However, when one tries to develop a theory under a unified abstract formulation, the context of cocycle (or skew-product flows) may not be the most appropriate to deal with the problem. In this paper, we apply a process to (1.16) by using the concept of measure of noncompactness to obtain pullback attractors.
By we denote the collection of the bounded sets of .
Definition 2.1. Let be a process on a complete metric space . A family of compact sets is said to be a pullback attractor for if, for all , it satisfies(i) for all ,(ii), for .The pullback attractor is said to be uniform if the attraction property is uniform in time, that is,
Definition 2.2. A family of compact sets is said to be a forward attractor for if, for all , it satisfies(i) for all ,(ii), for .The forward attractor is said to be uniform if the attraction property is uniform in time, that is, In the definition, is the Hausdorff semidistance between and , defined as Property (i) is a generalization of the invariance property for autonomous dynamical systems. The pullback attracting property (ii) considers the state of the system at time when the initial time goes to .
The notion of an attractor is closely related to that of an absorbing set.
Definition 2.3. The family is said to be (pullback) absorbing with respect to the process if, for all and , there exists such that for all The absorption is said to be uniform if does not depend on the time variable .
Now we recall the abstract results in .
Definition 2.4. The family of processes is said to be satisfying pullback Condition (C) if, for any fixed and , there exist and a finite dimensional subspace of such that(i) is bounded,(ii), where is a bounded projector.
Theorem 2.5. Let the family of processes acting in be continuous and possess compact pullback attractor satisfying if it (i)has a bounded (pullback) absorbing set ,(ii)satisfies pullback Condition (C).Moreover if is a uniformly convex Banach space, then the converse is true.
3. Pullback Attractor of Nonautonomous g-Navier-Stokes Equations
This section deals with the existence of the attractor for the two-dimensional nonautonomous g-Navier-Stokes equations in a bounded domain with periodic boundary condition.
In , the author has shown that the semigroup associated with the autonomous systems (1.16) possesses a global attractor in and . The main objective of this section is to prove that the nonautonomous system (1.16) has uniform attractors in and .
To this end, we first state the following results of existence and uniqueness of solutions of (1.16).
Proposition 3.1. Let be given. Then for every there exists a unique solution on of (1.16), satisfying . Moreover, one has Finally, if , then
Proposition 3.2. The process associated with the system (1.16) possesses (pullback) absorbing sets, that is, there exists a family of bounded (pullback) absorbing sets in and for the process , which is given by which absorb all bounded sets of . Moreover and absorb all bounded sets of and in the norms of and , respectively.
The main results in this section are as follows.
Theorem 3.3. If and , then the processes corresponding to problem (1.16) possess compact pullback attractor in which coincides with the pullback attractor: where is the (pullback) absorbing set in .
Proof. As in the previous section, for fixed , let be the subspace spanned by , and the orthogonal complement of in . We write
Now, we only have to verify Condition (C). Namely, we need to estimate , where is a solution of (1.16) given in Proposition 3.1.
Multiplying (1.16) by , we have It follows that Since satisfies the following inequality (see ): thus, Next, using the Cauchy inequality, Finally, we have Putting (3.9)–(3.11) together, there exists constant such that Therefore, we deduce that Here, depends on , is not increasing as increasing.
By the Gronwall inequality, the above inequality implies that If we consider the time instead of (so that we can use more easily the definition of pullback attractors), we have Applying continuous integral and Lemma II 1.3 in  for any , there exists such that thus, we have Using (1.11) and letting , then implies that Therefore, we deduce from (3.14) that which indicates satisfying pullback Condition (C) in . Applying Theorem 2.5, the proof is complete.
Theorem 3.4. If , then the processes corresponding to problem (1.16) possess compact pullback attractor in : where is the absorbing set in .
Proof. Using Proposition 3.2, we have that the family of processes corresponding to (1.16) possess the pullback absorbing set in .
Now we testify that the family of processes corresponding to (1.16) satisfies pullback Condition (C).
Multiplying (1.16) by , we have It follows that To estimate , we recall some inequalities (see ), for every , from which we deduce that and using (3.26), Expanding and using Young's inequality, together with the first one of (3.28) and the second one of (3.25), we have where we use and set Next, using the Cauchy inequality, Finally, we estimate by Putting (3.29)–(3.33) together, there exists a constant such that Here, depends on , is not increasing as increasing. Therefore, by the Gronwall inequality, the above inequality implies that We consider the time instead of . The following result is similar to (3.17)–(3.19), for any : Using (1.11) and letting , then implies that Therefore, we deduce from (3.35) that which indicates satisfying pullback Condition (C) in .
4. The Dimension of the Pullback Attractor
To estimate the dimension of the pullback attractor , we will apply the abstract machinery in [18, 23]. Let be a given family of nonlinear operators such that, for all and any , there exists a unique function satisfying where .
Using the standard methods (see [17, 18]), we can show that is uniformly quasidifferentiable on . Then, for all and any , there exists a unique , which is a solution of For all , we define the linear operator by
Theorem 4.1. Suppose that satisfies the assumptions of Theorem 3.3. Then, if , the Pullback attractor (uniformly in the past) defined by (3.4) satisfies where with the constant , of (3.29) and (3.32) of Chapter VI in , is the first eigenvalue of the Stokes operator and .
Proof. With Theorem 3.3 at our disposal we may apply the abstract framework in [17, 18, 23, 24].
For , let , where . Let be an orthonormal basis for span . Since almost everywhere , we can also assume that almost everywhere . Then, similar to the proof process of Theorems 3.3 and 3.4, we may obtain almost everywhere . From this equality, and in particular using the Schwarz and Lieb-Thirring inequality (see [17, 18, 23, 24]), one obtains On the other hand, we can deduce that for , and then
Now we define Hence,
The author would like to thank the reviewers and the editor for their valuable suggestions and comments. This work was supported by the National Science Foundation of China (Grant no. 10901147).
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