We consider the pullback attractors for the three-dimensional nonautonomous Camassa-Holm equations in the periodic box

We consider the following viscous version of the three-dimensional Camassa-Holm equations in the periodic box

We consider this equaton in an appropriate Hilbert space and show that there is an attractor

In addition, we assume that the function

In [

The understanding of the asymptotic behavior 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. However, nonautonomous systems are also of great importance and interest as they appear in many applications to natural sciences. On some occasions, some phenomena are modeled by nonlinear evolutionary equations which do not take into account all the relevant information of the real systems. Instead some neglected quantities can be modeled as an external force which in general becomes time dependent. In this situation, there are various options to deal with the problem of attractors for nonautonomous systems (kernel sections [

In this paper, we study the existence of compact pullback attractor for the nonautonomous three-dimensional-Camassa-Holm equations in bounded domain

From (

Next, let us introduce some notation and background.

Let

We denote

We denote

The scalar product on

We now discuss the theory of pullback attractors, as developed in [

Instead of a family of one time-dependent maps

The semigroup property is replaced by the process composition property

It is also possible to present the theory within the more general framework of cocycle dynamical systems. In this case the second component of

As in the standard theory of attractors, we seek an invariant attracting set. However, since the equation is nonautonomous this set also depends on time. By

Let

A family of compact sets

In the definition,

The notion of an attractor is closely related to that of an absorbing set.

The family

Indeed, just as in the autonomous case, the existence of compact absorbing sets is the crucial property in order to obtain pullback attractors. For the following result see [

Let

Now we recall the abstract results in [

The family of processes

Let the family of processes

has a bounded (pullback) absorbing set

satisfies pullback Condition

This section deals with the existence of the attractor for the three-dimensional nonautonomous Camassa-Holm equations in a bounded domain

We use the notation in [

It is similar to autonomous case that we can establish the existence of solution of (

In [

To this end, we first state some the following results.

Let

The Proof of Proposition

The process

The proof of Proposition

The main results in this section are as follows.

Now we prove the existence of compact pullback attractors in

If

As in the previous section, for fixed

Multiplying (

From the above inequalities we get
^{1}—Estimates in [

Therefore, we deduce that

Applying continuous integral and [

According to Propositions

If

Using Proposition

Similarly, we only have to verify pullback Condition

Multiplying the equation (^{1}—Estimates in [

By the Gronwall inequality, the above inequality implies

The author would like to thank the reviewers and the editor for their valuable suggestions and comments.