Abstract

We consider the uniform attractors for the three-dimensional nonautonomous Camassa-Holm equations in the periodic box . Assuming , we establish the existence of the uniform attractors in and . The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

1. Introduction

We consider the following viscous version of the three-dimensional Camassa-Holm equations in the periodic box : where is the modified pressure, while is the pressure, is the constant viscosity, and is a constant density. The function is a given body forcing and are scale parameters. Notice is dimensionless while has units of length. Also observe that at the limit , we obtain the three-dimensional Navier-Stokes equations with periodic boundary conditions.

We consider this equaton in an appropriate space and show that there is an attractor which all solutions approach as . The basic idea of our construction, is motivated by the works of [1].

In addition, we assume that the function is translation bounded, where . This property implies that

In [1] the authors established the global regularity of solutions of the autonomous Camassa-Holm, or Navier-Stokes-alpha (NS-) equations, subject to periodic boundary conditions. The inviscid NS- equations (Euler-) were introduced in [2] as a natural mathematical generalization of the integrable inviscid 1D Camassa-Holm equation discovered in [3] through a variational formulation. An alternative more physical derivation for the inviscid NS- equations (Euler-) was introduced in [4–8].

In the book [9], Haraux considers some special classes of such systems and studies systematically the notion of uniform attractor parallelling to that of global attractor for autonomous systems. Later on, [10] present a general approach, that is, well suited to study equations arising in mathematical physics. In this approach, to construct the uniform (or trajectory) attractors, instead of the associated process , one should consider a family of processes , , in some Banach space , where the functional parameter is called the symbol and is the symbol space including . The approach preserves the leading concept of invariance which implies the structure of uniform attractor described by the representation as a union of sections of all kernels of the family of processes. The kernel is the set of all complete trajectories of a process.

In the paper, we study the existence of compact uniform attractor for the nonautonomous three-dimensional Camassa-Holm equations in the periodic box . We apply measure of noncompactness method to nonautonomous Camassa-Holm equations with external forces in which is normal function (see Definition 4.2). Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

2. Functional Setting

From (1.1) one can easily see, after integration by parts, that On the other hand, because of the spatial periodicity of the solution, we have . As a result, we have , that is, the mean of the solution is invariant provided that the mean of the forcing term is zero. In this paper, we will consider forcing terms and initial values with spatial means that are zero, that is, we will assume and hence .

Next, let us introduce some notation and background.

(i)We denote is a vector-valued trigonometric polynomial defined on , such that and , and let and be the closures of in and in , respectively, observe that , the orthogonal complement of in , is (cf. [11] or [12]).

(ii)We denote the orthogonal projection, usually referred as Helmholtz-Leray projector, and by the Stokes operator with domain . Notice that in the case of periodic boundary condition, is a self-adjoint positive operator with compact inverse. Hence the space has an orthonormal basis of eigenfunctions of , that is, , with in fact, these eigenvalues have the form with .

(iii)We denote the -inner product and by the corresponding -norm. By virtue of Poincaré inequality, one can show that there is a constant such that and that

Moreover, one can show that (cf. [11, 12]). We denote and the inner product and norm on , respectively. Notice that, based on the above, the inner product , restricted to , is equivalent to the inner product provided . We denote is the dual of .

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.

3. Abstract Results

Let be a Banach space, and let a two-parameter family of mappings act on :

Definition 3.1. A two-parameter family of mappings is said to be a process in if

A family of processes , acting in is said to be -, if for all fixed and , the mapping is continuous from into .

A curve is said to be a complete trajectory of the process if

The kernel of the process consists of all bounded complete trajectories of the process :

The set is said to be the kernel section at time .

For convenience, let , the closure of the set and . Define the uniform () -limit set of by which can be characterized, analogously to that for semigroups, the following:

We recall characterize the existence of the uniform attractor for a family of processes satisfying (3.6) in term of the concept of measure of noncompactness that was put forward first by Kuratowski (see [13, 14]).

Let its Kuratowski measure of noncompactness is defined by

Definition 3.2. A family of processes , is said to be uniformly -limit compact if for any and the set is bounded for every and .

We present now a method to verify the uniform -limit compactness (see [15, 16]).

Definition 3.3. A family of processes , is said to satisfy uniformly Condition if for any fixed , and , there exist and a finite dimensional subspace of such that(i) is bounded; and(ii),where is a bounded projector.

Therefore, we have the following results.

Theorem 3.4. Let be a metric space and let be a continuous invariant semigroup on . A family of processes , acting in is -continuous (weakly) and possesses the compact uniform attractor satisfying if it(i)has a bounded uniformly absorbing set ; and(ii)satisfies uniformly Condition
Moreover, if is a uniformly convex Banach space, then the converse is true.

4. Uniform Attractor of Nonautonomous Camassa-Holm Equations

This section deals with the existence of the attractor for the three-dimensional nonautonomous Camassa-Holm equations with periodic boundary condition. To this end, we first state some the following results.

Proposition 4.1. Let and let . Then problem (1.1) has a unique solution such that for any , and such that for almost all and for any , here for every .

Proof. We use the Galerkin procedure to prove global existence. The proof of Proposition 4.1 is similar to autonomous Camassa-Holm in [1].

If we denote , the system (1.1) can be written as In [1] the authors have shown that the semigroup associated with the autonomous system (4.4) possesses a global attractor in and . The main objective of this section is to prove that the nonautonomous system (4.4) have uniform attractors in and .

Now recall the following facts that can be found in [15].

Definition 4.2. A function is said to be if for any , there exists such that

We denote by the set of all normal functions in .

Remark 4.3. Obviously, . Denote by the class of translation compact functions , whose family of is precompact in . It is proved in [15] that and are closed subspaces of , but the latter is a proper subset of the former (for further details see [15]).

We now define the symbol space   for (4.4). Let a fixed symbol be normal functions in , that is, the family of translation forms a normal function set in , where is an arbitrary interval of the time axis . Therefore,

Now, for any , problem (4.4) with instead of possesses a corresponding process acting on . As is proved in [10], the family of processes is -continuous.

Let be the so-called kernel of the process .

Proposition 4.4. The process associated with (4.4) possesses absorbing sets

Proof. The proof of Proposition 4.4 is similar to autonomous Camassa-Holm equation.

The main results in this section are as follows.

Theorem 4.5. If is a normal function in , then the processes corresponding to problem (1.1) possess compact uniform attractor in which coincides with the uniform attractor of the family of processes : where is the uniformly absorbing set in , and is the kernel of the process . Furthermore, the kernel is nonempty for all .

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 (4.4) given in Proposition 4.1.
Letting in (4.2), we have Notice that From the above inequalities we get Since satisfies the following inequality (see [1, 12]): then by Young's inequality, Thus, we obtain Therefore, we deduce that Here, depends on , and is not increasing as increasing.
By the Gronwall inequality, the above inequality implies Applying [10, Definition 4.1 and Lemma II 1.3] for any , Using (2.2) and letting , then implies Therefore, we deduce from (4.18) that which indicates satisfying uniform () Condition (C) in . Applying Theorem 3.4, the proof is complete.

According to Propositions 4.1 and 4.4, we can now regard that the families of processes for (1.1) are defined in and is a uniformly () absorbing set in .

Theorem 4.6. If is normal function in , then the processes corresponding to problem (1.1) possesses compact uniform attractor in which coincides with the uniform attractor of the family of processes : where is the uniformly absorbing set in and is the kernel of the process . Furthermore, the kernel is nonempty for all .