On the Dimension of the Pullback Attractors for g-Navier-Stokes Equations

We consider the asymptotic behaviour of nonautonomous 2D g-Navier-Stokes equations in bounded domain Ω . Assuming that 𝑓 ∈ 𝐿 2 l o c , which is translation bounded, the existence of the pullback attractor is 
proved in 𝐿 2 ( Ω ) and 𝐻 1 ( Ω ) . It is proved that the fractal dimension of the 
pullback attractor is finite.


Introduction
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 g g x 1 , x 2 is a suitable smooth real-valued function defined on x 1 , x 2 ∈ Ω and Ω is a suitable bounded domain in Ê 2 .Notice that if g x 1 , x 2 1, then 1.1 reduce to the standard Navier-Stokes equations.
In addition, we assume that the function f •, t : f t ∈ L 2 loc Ê; E is translation bounded, where E L 2 Ω or H −1 Ω .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 1 .
Let Ω 0, 1 × 0, 1 .We assume that the function g x g x 1 , x 2 satisfies the following properties: per Ω , 2 there exist constants m 0 m 0 g and M 0 M 0 g such that, for all x ∈ Ω, 0 < m 0 ≤ g x ≤ M 0 .Note that the constant function g ≡ 1 satisfies these conditions.
We denote by L 2 Ω, g the space with the scalar product and the norm given by as well as H 1 Ω, g with the norm where ∂u/∂x i D i u.
Then for the functional setting of the problems 1.1 , we use the following functional spaces: where H g is endowed with the scalar product and the norm in L 2 Ω, g and V g is the spaces with the scalar product and the norm given by Also, we define the orthogonal projection P g as and we have that Q ⊆ H ⊥ g , where per Ω,g ∇φ : φ ∈ C 1 Ω, Ê .

1.8
Then, we define the g-Laplacian operator to have the linear operator For the linear operator A g , the following hold see 1 .
1 A g is a positive, self-adjoint operator with compact inverse, where the domain of where λ g 4π 2 m 0 /M 0 and λ 1 is the smallest eigenvalue of A g .In addition, there exists the corresponding collection of eigenfunctions {e 1 , e 2 , e 3 , . ..} which forms an orthonormal basis for H g .Next, we denote the bilinear operator B g u, v P g u • ∇ v and the trilinear form 12 where u, v, and w lie in appropriate subspaces of L 2 Ω, g .Then, the form b g satisfies We denote a linear operator R on V g by and have R as a continuous linear operator from V g into H g such that 15 We now rewrite 1.1 as abstract evolution equations: du dt νA g u B g u νRu P g f, u τ u τ .

1.16
In 1 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 2 and independently in 3 .The finite-dimensional property of the global attractor for general dissipative equations was first proved in 4 .For the analysis of the Navier-Stokes equations, one can refer to 5 , specially 6 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 18 , 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 Ω with periodic boundary condition.It is proved that the fractal dimension of the pullback attractor is finite.
Hereafter c 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.

Abstract Results
We now recall the preliminary results of pullback attractors, as developed in 8-10, 13 .
The semigroup S t property is replaced by the process U t, τ composition property and, obviously, the initial condition implies that U τ, τ Id.As with the semigroup composition S t S τ S t τ , 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 U is viewed as an element of some parameter space J, so that the solution can be written as U t, p , and a shift map θ t : J → J 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 U t, τ to 1.16 by using the concept of measure of noncompactness to obtain pullback attractors.
By B E we denote the collection of the bounded sets of E.
Definition 2.1.Let U be a process on a complete metric space E. A family of compact sets {A t } t∈Ê is said to be a pullback attractor for U if, for all τ ∈ Ê, it satisfies 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 {A t } t∈Ê is said to be a forward attractor for U if, for The forward attractor is said to be uniform if the attraction property is uniform in time, that is, In the definition, dist A, B is the Hausdorff semidistance between A and B, 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 t when the initial time t − s goes to −∞.
The notion of an attractor is closely related to that of an absorbing set.Moreover if E is a uniformly convex Banach space, then the converse is true.

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 Ω with periodic boundary condition.
In 1 , the author has shown that the semigroup S t : H g → H g t ≥ 0 associated with the autonomous systems 1.16 possesses a global attractor in H g and V g .The main objective of this section is to prove that the nonautonomous system 1.16 has uniform attractors in H g and V g .
To this end, we first state the following results of existence and uniqueness of solutions of 1.16 .Proposition 3.1.Let f ∈ V be given.Then for every u τ ∈ H g there exists a unique solution u u t on 0, ∞ of 1.16 , satisfying u τ u τ .Moreover, one has Proof.The Proof of Proposition 3.1 is similar to autonomous case in 1, 17 .
Proposition 3.2.The process {U t, t − s } : V g → V g associated with the system 1.16 possesses (pullback) absorbing sets, that is, there exists a family {B t } t∈R of bounded (pullback) absorbing sets in H g and V g for the process U, which is given by

3.3
which absorb all bounded sets of H g .Moreover B 0 and B 1 absorb all bounded sets of H g and V g in the norms of H g and V g , respectively.
Proof.The proof of Proposition 3.2 is similar to autonomous g-Navier-Stokes equation.We can obtain absorbing sets in H g and V g from 1 .
The main results in this section are as follows. 3.9 Next, using the Cauchy inequality, .

3.10
Finally, we have

3.12
Therefore, we deduce that

3.13
Here, M 1 depends on λ m 1 , is not increasing as λ m 1 increasing.By the Gronwall inequality, the above inequality implies that

3.14
If we consider the time t − s instead of τ so that we can use more easily the definition of pullback attractors , we have Applying continuous integral and Lemma II 1.3 in 18 for any ε, there exists η η ε > 0 such that 3.16 thus, we have

3.20
Therefore, we deduce from 3.14 that which indicates {U t, τ } satisfying pullback Condition C in H g .Applying Theorem 2.5, the proof is complete.
According to Propositions 3.1-3.2,we can now regard that the families of processes {U t, τ } are defined in V g and B 1 is a pullback absorbing set in V g .
b R; H g , then the processes {U t, τ } corresponding to problem 1.16 possess compact pullback attractor A 1 t in V g : where B 1 is the absorbing set in V g .
Proof.Using Proposition 3.2, we have that the family of processes {U t, τ } corresponding to 1.16 possess the pullback absorbing set in V g .Now we testify that the family of processes {U t, τ } corresponding to 1. 16

3.24
To estimate B g u, u , Au 2 g , we recall some inequalities see 22 , for every u, v ∈ D A g , from which we deduce that 3.27 and using 3.26 ,

3.28
Expanding and using Young's inequality, together with the first one of 3.28 and the second one of 3.25 , we have

3.29
where we use

and set
L 1 log λ m 1 λ g .

3.31
Next, using the Cauchy inequality,

3.32
Finally, we estimate

The Dimension of the Pullback Attractor
To estimate the dimension of the pullback attractor A 0 t , we will apply the abstract machinery in 18, 23 .Let F : V g × Ê → V g be a given family of nonlinear operators such that, for all τ ∈ Ê and any u τ ∈ H g , there exists a unique function u t u t; τ, u 0 satisfying ∈ V g almost everywhere s ≥ τ, we can also assume that ϕ j s ∈ V g almost everywhere s ≥ τ.Then, similar to the proof process of Theorems 3.
Definition 2.3.The family {B t } t∈Ê is said to be pullback absorbing with respect to the process U if, for all t ∈ Ê and D ∈ B E , there exists S D, t > 0 such that for all s ≥ S D, tThe absorption is said to be uniform if S D, t does not depend on the time variable t.The family of processes {U t, t − s } is said to be satisfying pullback Condition C if, for any fixed B ∈ B E and ε > 0, there exist s 0 s B, t, ε ≥ 0 and a finite dimensional subspace E 1 of E such that i { P s≥s 0 U t, t − s B E } is bounded, ii I − P s≥s 0 U t, t − s B E ≤ ε, where P : E → E 1 is a bounded projector.Theorem 2.5.Let the family of processes {U t, τ } acting in E be continuous and possess compact pullback attractor A t satisfying A t B∈B ω B, t , for t ∈ Ê, 2.7 if it i has a bounded (pullback) absorbing set B, ii satisfies pullback Condition (C).