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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 541426, 18 pages
Dimension Estimate for the Global Attractor of an Evolution Equation
Departamento de Matemática, Pontificia Universidad Javeriana, Carrera 7 No. 43-82, Bogotá, Colombia
Received 29 September 2011; Accepted 13 December 2011
Academic Editor: Juan J. Nieto
Copyright © 2012 Renato Colucci and Gerardo R. Chacón. 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 estimate the dimension of the global attractor of an evolution equation by the study of the evolution of the n-dimensional volumes under the flow. We compare these results with the estimate of the dimension of the inertial manifold.
One of the most interesting problems in the analysis of partial differential equations is the study of the asymptotic behavior of solutions. Guided by the finite-dimensional case, in the last decades, the concept of global attractor was introduced in order to study the long-term dynamics of dissipative equations (see, e.g., [1, 2] and the references therein). This theory has been generalized to the case on non-autonomous systems by introducing the definition of uniform attractor; see, for example, the interesting article  in which a reaction diffusion equation is studied with nonlinear boundary conditions in competitions with the dissipative terms. However, the existence of uniform attractors fails in many cases and that is the reason why the new definition of Pullback Attractor is introduced (see ). This new concept works for both the non-autonomous (see ) and the random systems, for parabolic (see ), and hyperbolic stochastic equations (see ) in bounded and in unbounded domains. A further generalization of the concept of attractor to the setting of multivalued processes can be seen in , and an application to the case of 3D Navier Stokes systems can be seen in .
Once the existence of a global attractor has been established, in order to analyze its structure, it is a common question to investigate the (fractal) dimension of it. The classical method to do that consists in the study of the contraction of an -dimensional volume element under the action of the flow (see [1, 2]). This is a very sharp method that works also in the case of non-autonomous systems; see, for instance,  in which also is pointed out the influence of the geometry of the domain in determining the dimension of attractors.
This method also works in numerical context; in fact there are also numerical schemes (see, e.g., ) that preserve important properties of the solution and that preserve the dimension of the attractor (see, for instance, ).
The disadvantage of the study of volume contraction method is that it requires higher regularity of the data and the solutions this can be avoided by using the theory of exponential attractors (see ) or the so-called method of l-trajectories (see, e.g., ). Both methods require less regularity and prove finite dimension of the attractor.
In this paper, we will study the dimension of the global attractor of an autonomous evolution equation by using different methods. The autonomous evolution equation considered here is a simple model of a dynamical system which presents different time scales in which the dynamics presents different features. We consider the following fourth-order evolution equation: where the function is the so-called double well potential and is an open interval such that its length . In particular, (1.1) is the -gradient flow associated to the functional: In  it is studied the global dynamics of an evolution equation like (1.1) with a more general nonconvex function . They use numerical experiments to discuss the dynamical behavior of the solutions for small values of . In particular it was pointed out the existence of three different time scales with peculiar dynamic behavior.
In a first time scale of order , there is a drastic reduction of the energy of the initial data and a formation of microstructure in the region where is not convex.
In the second time scale of order , we have that the equation exhibits a heat equation-like behavior in the convex regions while slow motion in the nonconvex ones.
In the third time scale of order , the infinite dynamical system is reduced to a finite one. The solution is approximately the union of consecutive segments. The dynamic is slow and the number of segments decreases letting decrease the number of freedom degrees of the problem.
We use the following notation throughout the paper. We develop the analysis in the following spaces:
In  the asymptotic behavior of (1.1) was studied and it was proven the existence of a global attractor for the semigroup associated to (1.1), that is, the semigroup of operators defined as The main tool in proving the existence of a global attractor is to show the existence of an absorbing set in (see Theorem 1.1 in ).
Theorem 1.1 (see ). There exist positive constants , with such that (i)the semigroup posses an absorbing set in : where (ii)the semigroup posses an absorbing set in : where (iii)the semigroup posses an absorbing set in : where
Theorem 1.1 let us conclude that the set is absorbing for all the bounded sets of . In particular the global attractor is given as the omega limit set of the absorbing set, . The authors also obtained in  a bound for the dimension of by proving the existence of an exponential attractor whose dimension is of order . Moreover, in we have the following estimates for : In accordance with the numerical experiments, the time in which the bounded solutions enter the absorbing set is of order (see [14, 15]).
We are interested in finding estimates for the dimension of . In Section 2 we find an estimate of the fractal and Hausdorff dimension of by studying the evolution of -dimension volume elements, this following a classical theory by Temam (see ). In Section 3 we prove the existence of an inertial manifold (see ) for the system (1.1), that is, a smooth manifold that attracts all the orbit at an exponential rate and that contains the global attractor. By estimating the dimension of , we obtain, in another way, a different estimate of . In Section 4 we study the regularity of the attractor while in the last section we prove the existence of an absorbing set in . This is fundamental information in order to study the approximation of the attractor (see ).
We define the linear operator . The set is its domain. Also, then where and the operator is self-adjoint in . From classical results on solutions of evolution equations (see  or see Theorems 3.1 and 3.3 on ), we have the following:
Theorem 1.2. Problem (1.1) admits a unique solution such that
2. Dimension of the Attractor
Theorem 2.1. The global attractor of the semigroup has fractal dimension where .
Proof. In general if we consider an evolution equation of the type then the idea is to investigate the evolution of -dimensional infinitesimal volumes and find the smallest such that all the -infinitesimal volumes contract exponentially. Then one may guess that the attractor does not contain these elements and so that the dimension of is less than or equal to . We study the evolution of a set of infinitesimal displacement about a trajectory . In order to do that, we linearize the equation around the solution : where is the Fráchet derivative of . Each displacement evolves The volume is given by where then (see, e.g.,  for details) Then if is the projector on the space spanned by the displacements with base , we have and consequently the asymptotic growth rate is given by As we need the maximal growth rate, we compute the over the and all the -dimensional projectors: In order to have exponential decay, we look for the smallest such that the number is negative. Thus, using theorem 13.16 page 341 in  let us conclude that In order to estimate , we look for an uniform bound of the following expression: from which we conclude that . In our case the linearized equation is In details let be an orthonormal base for then with , We estimate the terms separately. The first term The second term from which by interpolation we get The last term is negative and consequently it can be neglected in the computation. Then we have the following estimate: where we have used that and where are the eigenvalues of the operator . Since from (2.18) we have that is uniformly bounded in time, then we conclude that where we have used a Young inequality and the constants , are given by This (see ) also gives an estimate for the Lyapunov exponents : Moreover (see , Theorem 2.2 pag. 396), we obtain an estimate on the dimension of .
Theorem 2.2. Let such that
then the -dimensional volume element in the phase space is exponentially decaying as . Moreover, the global attractor has finite dimension: , .
Then we have that From the above expression we have that . This provides a much better estimate than the one obtained in  by the fractal dimension of the exponential attractor .
3. Estimate via the Existence of the Inertial Manifold
In this section we will show the existence of the inertial manifold for system (1.1). Then the dimension of will provide a further estimate of the dimension of the global attractor since . In order to carry on our analysis, we follow the strategy explained in .
Theorem 3.1. The system (1.1) possesses an inertial manifold having dimension of order .
One first restricts (1.1) to the absorbing set . Then in order to prove the existence of one has to check the fulfillment of two conditions. The first condition consists of proving that the nonlinear term of (1.1) is Lipschitz from to :In particular the constant depends on the radius of .
The second condition is the so-called strong squeezing property.
Definition 3.2. Let , two solutions of (1.1), the strong squeezing property holds if for some and implies that and furthermore that if then for and some and where .
Again following  we have that a sufficient condition to prove that the strong squeezing property holds is the so-called spectral gap condition: for sufficiently large and where are the eigenvalues of the operator . The spectral gap condition will be the second property we will show holds. We divide the proof into two steps.
3.1. Lipschitz Property for
Theorem 3.3. Given solutions and , then there exists a constant such that where denotes the nonlinear part of (1.1), that is,
Proof. First, we suppose that is a solution of (1.1), then, for any unitary vector in , we have that
and consequently the nonlinear operator restricted on is bounded from :
Now, if and are solutions of (1.1), let , then we proceed in a similar manner as before to get that if , then where the last inequality follows from (see ) Thus, if we take then we have that and it follows that the nonlinear functional is Lipschitz continuous. From the expression of the radii ,, it results that the Lipschitz constant is of order .
3.2. Spectral Gap Condition
Theorem 3.4. There exists a natural number sufficiently large so that
Proof. The eigenvalues of the operator are given by while the eigenvalues of are . Then the spectral gap condition is In details we have that is, Then if satisfies the previous inequality, we have that
Thus, the spectral gap condition is satisfied and the existence of the inertial manifold is assured. This proves Theorem 3.1.
We remark that the dimension of is Then comparing the estimates obtained, we have that we get by the method of -dimensional volume evolution and, respectively, and by the existence of the inertial set and the inertial manifold . It is not a surprise that the dimension of the inertial manifold is much bigger than that of the inertial set and global attractor since we are requiring the existence of a smooth structure.
4. Regularity of Attractor and Dimension
Theorem 4.1. If is a bounded domain and if and are functions, then the global attractor is a bounded subset of for every . In particular, if , then .
This regularity result let us conclude by theorem 15.1 in  that for , for almost every set of points in , the mapping is an embedding of into . This gives a parametrization of the attractor and gives an estimate of the degree of freedom of the system: . Consequently if we want to parameterize the attractor by equally spaced points in , it results that each subinterval of is of order (see  for a more detailed discussion). This is in full accordance with the numerical experiments proposed in  in which it was pointed out that the wave length of microstructure is of order .
This result has a connection with the definition of determining nodes (see ): a set of points in is said to be asymptotically determining if for two solutions and implies that From a theorem in , if attracts all the solutions in the norm of then almost every set of nodes in is asymptotically determining. In the last section we will show that this is true for (1.1); that is, there exists an absorbing set in .
We conclude this section by proving and giving explicit bounds for in . In order to do that, we follow a strategy suggested by Robinson in .
Theorem 4.2. The global attractor of (1.1) is bounded in .
Proof. Multiply the equation by and integrate over : By integrating the previous equation, with respect to time, over we get The integrand of the last term of the previous equality can be written as , and this yields to Since the attractor is bounded in and , we have We will prove that is uniformly bounded in . We derive (1.1) with respect to time, multiply it by , and integrate over : Then, using interpolation we get from which Integrating the previous inequality, with respect to time, in , we have Since the term is bounded in , we obtain the following inequality by setting : This gives an uniform bound for , and since , it is sufficient to conclude that . In particular, let and , we set for any . Then and so , from which we have that Now we consider the norm of the nonlinear term : where we have used (1.8), (1.10), and . Then, from the previous inequality, we get that is uniformly bounded in . Then since we have that is uniformly bounded in . Then from the following interpolating inequality we get that is bounded in .
5. Absorbing Set in
In this last section we show the existence of an absorbing set in .
Theorem 5.1. The semigroup defined in (1.4) posses an absorbing set in : where and is a positive constant such that
Proof. Let be a solution of (1.1) such that and .
Multiply (1.1) by and integrate on , we obtain from which and then By the boundary conditions, from the hypothesis that and from the interpolating inequality , we have that Then we can rewrite (5.6) only in terms of the -norm of : Integrating the previous inequality over , we get where the last inequality follows from . Then if we set we have that Now coming back to (5.5), we use a different estimate method from which Integrating the previous inequality with respect to time in , with , we get Moreover by an interpolating inequality, we have that We multiply (1.1) by and integrate over : from which by using an interpolating inequality and neglecting the positive term we get from which we conclude We apply the uniform Gronwall lemma (see ) with from which we obtain We fix , and then Then, using an interpolating inequality, we have that where This completes the proof, the set is absorbing for all the bounded sets of .
The first author would like to thank Professor Giorgio Fusco for suggesting the subject of research during his Ph.D. studies and for helpful hints and remarks.
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