Research Article | Open Access
The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations
We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domain . Assuming , we establish the existence of the uniform attractor in and . The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.
In this paper, we study the behavior of solutions of the nonautonomous 2D g-Navier-Stokes equations. 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 . This property implies that
We consider this equation in an appropriate Hilbert space and show that there is an attractor which all solutions approach as . The basic idea of our construction, which is motivated by the works of [1, 2].
In [1, 2] the author established the global regularity of solutions of the g-Navier-Stokes equations. For the boundary conditions, we will consider the periodic boundary conditions, while same results can be got for the Dirichlet boundary conditions on the smooth bounded domain. For many years, the Navier-Stokes equations were investigated by many authors and the existence of the attractors for 2D Navier-Stokes equations was first proved by Ladyzhenskaya [3, 4] and independently by Foias and Temam . The finite-dimensional property of the global attractor for general dissipative equations was first proved by Mallet-Paret . For the analysis on the Navier-Stokes equations, one can refer to  and specially  for the periodic boundary conditions.
The book in  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,  presents 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 the two dimensional g-Navier-Stokes equations in the periodic boundary conditions . We apply measure of noncompactness method to nonautonomous g-Navier-Stokes equations equation 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
Let and we assume that the function satisfies the following properties:
(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 byas well as with the normwhere .
Then for the functional setting of (1.1), we use the following functional spaceswhere is endowed with the scalar product and the norm in , and is the spaces with the scalar product and the norm given byAlso, we define the orthogonal projection asand we have that , whereThen, we define the -Laplacian operatorto have the linear operatorFor the linear operator , the following hold (see [1, 2]):
(1) is a positive, self-adjoint operator with compact inverse, where the domain of .
(2) There exist countable eigenvalues of satisfyingwhere 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 formwhere lie in appropriate subspaces of . Then, the form satisfies
We denote a linear operator on byand have as a continuous linear operator from into such that
We now rewrite (1.1) as abstract evolution equations,
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
By we denote the collection of the bounded sets of . We consider a family of processes depending on a parameter . The parameter is said to be the symbol of the process and the set is said to be the symbol space. In the sequel is assumed to be a complete metric space.
A family of processes , is said to be uniformly (with respect to ) bounded if for any the set
A set is said to be uniformly () absorbing for the family of processes , if for any and every there exists such that for all .
A set is said to be uniformly () attracting for the family of processes if for an arbitrary fixed ,
A family of processes possessing a compact uniformly absorbing set is called uniformly compact and a family of processes possessing a compact uniformly attracting set is called uniformly asymptotically compact.Definition 3.2. A closed set is said to be the uniform attractor of the family of processes if it is uniformly attracting and it is contained in any closed uniformly attracting set of the family of processes .
A family of processes acting in is said to be -continuous, 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 setis 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 (3.8) in term of the concept of measure of noncompactness that was put forward first by Kuratowski (see [11, 12]).
Let . Its Kuratowski measure of noncompactness is defined byDefinition 3.3. 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 [13, 14]).Definition 3.4. 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 3.5. Let be a metric space and let be a continuous invariant semigroup on .
A family of processes acting in is -(weakly) continuous and possesses the compact
uniform attractor satisfyingif 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 g-Navier-Stokes Equations
It is similar to autonomous case that we can establist the existence of solution of (2.14) by the standard Faedo-Galerkin method.
In [1, 2], the authors have shown that the semigroup associated with the autonomous systems (2.14) possesses a global attractor. The main objective of this section is to prove that the nonautonomous system (2.14) has uniform attractors in and .
To this end, we first state some the following results of existence and uniqueness of solutions of (2.14).Proposition 4.1. Let be given. Then for every there exists a unique solution on of (2.14), satisfying . Moreover,one hasFinally, if , then
Now recall the following facts that can be found in .Definition 4.2. A function is said to be normal if for any , there exists such thatRemark 4.3. Obviously, . Denote by the class of translation compact functions , whose family of is precompact in . It is proved in  that and are closed subspaces of , but the latter is a proper subset of the former (for further details see ).
We now define the symbol space for (4.3). 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
Letbe the so-called kernel of the process .Proposition 4.4. The process associated with the (4.3) possesses absorbing setswhich 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 4.5. If is normal function in , then the processes corresponding to problem (2.14) 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 .
Now, we only have to verify Condition Namely, we need to estimate , where is a solution of (2.14) given in Proposition 4.1.
Multiplying (2.14) by , we haveIt follows thatSince satisfies the following inequality (see ):thus,Next, the Cauchy inequality,Finally, we havePutting (4.13)–(4.15) together, there exist constant such thatTherefore, we deduce thatHere depends on , is not increasing as increasing.
By the Gronwall inequality, the above inequality impliesApplying (4.4) for any Using (2.9) and let , then impliesTherefore, we deduce from (4.18) thatwhich indicates satisfying uniform () Condition in . Applying Theorem 3.5 the proof is complete.
Theorem 4.6. If is normal function in , then the processes corresponding to problem (2.14) 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 .
we have the family of processes , corresponding to (4.3) possesses the uniformly absorbing set in .
Now we prove the existence of compact uniform attractor in by applying the method established in Section 3, that is, we testify that the family of processes corresponding to (4.3) satisfies uniform Condition .
Multiplying (2.14) by , we haveIt follows thatTo estimate , we recall some inequalities : for every :(see )from which we deduce thatand using, (4.26)Expanding and using Young's inequality, together with the first one of (4.28) and the second one of (4.25), we havewhere we useand setNext, using the Cauchy inequality,Finally, we estimate byPutting (4.29)–(4.33) together, there exists a constant such thatHere depends on , is not increasing as increasing. Therefore, by the Gronwall inequality, the above inequality implies Applying (4.4) for any Using (2.9) and let , then impliesTherefore, we deduce from (4.35) thatwhich indicates satisfying uniform () Condition in .
5. Dimension of the Uniform Attractor
Process is said to be uniformly quasidifferentiable on , if there is a family of bounded linear operators such that
We want to estimate the fractal dimension of the kernel sections of the process generated by the abstract evolutionary (2.14). Assume that is generated by the variational equation corresponding to (2.14)that is, is the solution of (5.2), and is the solution of (2.14) with initial value . For natural number , we setwhere is trace of the operator.
We now consider (2.14) with . The equations possess a compact uniform () attractor and . By [2, 10, 15], we know that the associated process is uniformly quasidifferentiable on and the quasidifferential is Hölder-continuous with respect to . The corresponding variational equation is
We have the main results in this section.Theorem 5.2. Suppose that satisfies the assumptions of Theorem 4.5. Then, if , the Uniform attractor defined by (4.8) satisfieswherethe constant of (3.29) and (3.32) of Chapter in  and , is the first eigenvalue of the Stokes operator and .
Theorem 4.5 at our disposal we may apply the abstract framework in [2, 10, 15, 17].
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 4.5 and 4.6, we may obtainalmost everywhere . From this equality, and in particular using the Schwarz and Lieb-Thirring inequality (see [2, 10, 15, 17]), one obtainson the other hand, we can deduce (2.14) thatfor , and then
Now we defineUsing Theorem 5.1, we haveHence
The author would like to thank the reviewers and the editor for their valuable suggestions and comments.
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