Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 1, February 2023
Page(s) 1 - 10
DOI https://doi.org/10.1051/wujns/2023281001
Published online 17 March 2023

© Wuhan University 2023

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

In this paper, we study the following three-dimensional Brinkman-Forchheimer equation:

(1)

where is the Darcy coefficient, are the Forchheimer coefficients, is a constant, and is an open and bounded set, which is sufficiently regular. Here is the fluid velocity vector, denotes the pressure field, is the external force, is the Brinkman coefficient and is the initial velocity.

As a mathematical model, Brinkman-Forchheimer equation describes the motion of a fluid flowing in saturated porous media [1-3], which has received much attention on several issues over the last decades. From a mathematical point of view, the research on the three-dimensional Brinkman-Forchheimer equation is mainly divided into two categories. One is the structural stability of the equation with respect to the coefficients and [4-9], and the other is the long-term behavior of the solution of the equation [10-18].

If a system has a global attractor, then the attractor will contain all possible limit states of the solutions of the system. Therefore, studying the dynamic system restricted to the global attractor will be able to reveal a lot of information about the original system. So proving the existence of global attractors is a basic and important problem in infinite-dimensional dynamical systems. For the 3D Brinkman-Forchheimer equation, Ugurlu [10], Ouyang and Yang [11] showed the existence of global attractor in when and by condition-(C) method, respectively. Wang and Lin[12] showed the existence of global attractor in when . In Ref.[13], the existence of -pullback attractors for three-dimensional non-autonomous Brinkman-Forchheimer equation is deduced by establishing the -pullback asymptotical compactness of -cocycle. In Ref.[14], Song et al discussed the decay of the weak solution of the Brinkman-Forchheimer equation in three-dimensional full space. In Ref.[15], Song and Wu investigated the uniform boundedness of uniform attractor of equation (1) with singularly oscillating external force. They established the convergence of the attractor to the attractor of the averaged equation as . In Ref.[16], the pullback dynamics and asymptotic stability for a 3D Brinkman-Forchheimer equation with finite delay was concerned. In Ref.[17], by some estimates and the variable index to deal with the delay term, Yang et al got the sufficient conditions for asymptotic stability of trajectories inside the pullback attractors for a fluid flow model in porous medium by generalized Grashof numbers. In Ref.[18], Qiao et al proved the existence of a global attractor for the strong solution of the Brinkman-Forchheimer equation in a three-dimensional bounded domain.

In the geometric structure of the global attractor, the dimension is a very important property. This is because if the fractal dimension of the global attractor is finite, the original infinite-dimensional dynamical system can be reduced to a finite-dimensional ordinary differential equation system, so that the relatively complete theory of the finite-dimensional dynamical system can be used to study infinite dimensional dynamical system. As far as we know, there is no results on dimension estimate of global attractor of 3D Brinkman-Forchheimer. In this paper, inspired by Refs.[19-21], based on Ref.[18], we will discuss the Hausdorff dimension and fractal dimension of global attractors for strong solutions of the equation.

The structure of this paper is arranged as follows: In Section 1, we give some function space symbols and some inequalities that will be used later. In Section 2, we discuss the Hausdorff dimension and fractal dimension of global attractors for strong solution of the equation.

1 Preliminaries

In this section, we introduce some notations and preliminaries, which will be used throughout this paper.

First, let us introduce the following function spaces:

where denotes the closure in space . is the closure of the set in topology, and is the closure of the set in topology. and are the dual spaces of and , respectively. and are equipped with the following inner products:

and norms Let Throughout this paper, we use to denote the norm in . or will stand for some generic positive constants, depending on and some constants, but independent of time .

We call is a weak solution of problem (1) on , if

(2)

The weak form (2) is equivalent to the following functional equation:

(3)

Here is the Stokes operator subject to the no-slip homogeneous Dirichlet boundary condition with the domain defined as . is the orthogonal projection from onto . Obviously, the operator is a non-negative self-adjoint operator in with and for all .

For a bounded domain , the operator is invertible, and its inverse operator is bounded, self-adjoint and compact in . Thus, the spectrum of consists of an infinite sequence with as its eigenvalue (Theorem 2.2, Corollary 2.2, Ref.[22]). For all and we have where and is the n-dimensional Lebesgue measure of . For we find

(4)

Next we formulate some well-known inequalities and a Gronwall type lemma that we will use in what follows.

Poincaré's inequality [23]:

(5)

where is the first eigenvalue of operator under the homogeneous Dirichlet boundary condition.

Agmon's inequality [23]:

(6)

-inequality [24]:

(7)

Series inequlity[25]:

(8)

Lemma 1   (Gagliardo-Nirenberg's inequality) [24] Assuming that or be a bounded domain, which has a sufficiently smooth boundary , . Then there is a constant such that where depends on n,m,j,a,q,r.

Lemma 2   ( Gronwall's inequality) [23] Let be non-negative integrable functions on . If there is , such that then

Now we recall the existence and uniqueness theorem of the strong solutions of equation (1).

Theorem 1[18]Suppose and . Then there exists a strong solution of equation (1) satisfying

Moreover when the strong solution is unique.

Now we will review the uniform estimates of strong solution to the problem (1) when .

Lemma 3[18] Suppose Then there exists a time , constants such that when we have for .

Lemma 4[18]Suppose Then there exists a time ,a constant such that

Lemma 5[18] Suppose Then there exists a time ,a constant such that

Lemma 6[18]Suppose Then there exists a constant such that

Finally, we give the result of existence of global attractor in for the 3D Brinkman-Forchheimer equation.

Theorem 2[18]Suppose Then the problem (1) has a global attractor in , which is invariant and compact in and attracts every bounded subset of with the norm in .

2 Estimates of Dimensions of the Global Attractor

In this section, we will establish the differentiability of the semigroup with respect to the initial data. We show that the global attractor of the 3D Brinkman-Forchheimer system has finite Hausdorff and fractal dimensions. We will use the similar techniques as in Refs.[19-21], etc to obtain the desired results.

Let be the unique strong solution of the autonomous system (1) belonging to the global attractor .

Let us take inner product with in to the first equation in (1) to obtain

(9)

So we have

(10)

Applying Gronwall's inequality, we find

(11)

So there is a time which we can take as , such that for all , we have

(12)

Integrating the inequality (10) from 0 to , we obtain

(13)

so we get

(14)

Theorem 3   Let and be two members of . Then there exists a constant such that

(15)

where the linear operator for is the solution operator of the problem:

(16)

and , . In other words, for every , the solution as a map is Fréchet differentiable for the initial data, and its Fréchet derivative .

Proof   Let for any Then we have

(17)

and

(18)

Let . Combining (17) with (18), we obtain

(19)

Let us define . Then satisfies:

(20)

Let us take inner product with in to the first equation in (20) to obtain

(21)

Let us consider the first term on the right-hand side of (21). We have

(22)

In (22), Agmon's inequality is used. And in the last inequality of (22), because are members of , so we used the uniform estimates of solutions in Lemma 4 and Lemma 6 to obtain the desired result.

Similar with (22), for the second term on the right-hand side of (21), we have

(23)

For the third term on the right-hand side of (21), we get

(24)

Similar with (24), for the fourth term on the right-hand side of (21), we have

(25)

For the fifth term on the right-hand side of (21), we have

(26)

In inequality (26), we used -inequality and the following Gagliardo-Nirenberg's inequality:

Combining (21)-(26), we find

(27)

Taking inner product with in to the first equation of (19), we have

(28)

where

(29)

and

(30)

For the second term on the right-hand side of (29), from inequality (26), we have

(31)

For the second term on the right-hand side of (30), we have

(32)

Combining (29)-(32) with (28), we obtain

(33)

Integrating (33 ) from 0 to , we have

(34)

Applying Gronwall's inequality to (34), we get

(35)

Integrating (27) from 0 to , due to (35), we infer that

(36)

An application of Gronwall's inequality in (36) yields

(37)

Thus

(38)

which completes the proof.

Now we rewrite the system (3) as

(39)

Let us now set and using it in (39) to obtain

(40)

where . Note that the systems (40) and (39) are equivalent. Remember that the systems (39) is well posed in , while the system (40) is well posed in . Therefore, there exists a unique weak solution of (40) in . Moreover, the system (40) generates one family of strongly continuous semigroup of solution operators

Since and , the semigroup is connected to the original semigroup through the relation

(41)

Thus, the semigroup has the global attractor , where

(42)

and is the global attractor for .

Now we will show a bound for the fractal dimension of in . Besides, using the following argument, the fractal dimension in can easily yield the same bound. From Proposition 3.1, Chapter VI of Ref.[23], we know that under the Lipschitz maps, the fractal dimension estimates can be obtained. Furthermore, we infer that

(43)

Let us first consider the linear variations of the system (40). The linear variational equation corresponding to (40) has this form

(44)

where

(45)

The adjoint of is given by

(46)

Hence, can be computed as

(47)

Then, we derive the following results.

Proposition 1   Let . Then, we have

(48)

where is given in (6) which only depends on .

Proof   Let us take the inner product with in to equation (47), we obtain

(49)

And because

(50)

and

(51)

Combining (50) and (51) with (49), we deduce that

(52)

Proposition 2   Suppose Then the global attractor has the finite fractal dimension in , with

(53)

where is defined in (4), is given in Lemma 6 and is given in (6) which only dependents on .

Proof   Let for some , be an initial orthogonal set of infinitesimal displacements. The volume of the parallelopiped spanned by is given by where denotes the exterior product. The evolution of such displacements satisfies the following evolution equation:

(54)

for all . Using Lemma 3.5 in Ref.[26], we know that the volume elements satisfy

(55)

where is the orthogonal projection onto the linear span of in . And we also know that with and an orthonormal set spanning . Then, we define

(56)

From (56), we have

(57)

for all , where the supremum over is a supremum over all choices of initial orthogonal set of infinitesimal displacements that have taken around . Now let us prove that the volume element decays exponentially with time, whenever nN, with N0 to be determined later.

Let us use Proposition 1 to estimate

(58)

where . So we obtain

We need the right hand side of the above inequality must be negative, therefore we require , where and is defined in (4), which completes the proof.

Since has finite fractal dimension in with the bound (53), we can easily prove the following Theorem by using (43).

Theorem 4   Suppose . Then the global attractor obtained in Theorem 2 has finite Hausdorff and fractal dimensions, which can be estimated by

where is defined in (4), are given in Lemma 6 and is given in (6) which only depends on .

3 Conclusion

In this paper, we investigate the dimension of global attractor in of strong solution for a 3D Brinkman-Forchheimer equation. By setting , we rewrite system (3) as (40). And by proving (40) has a bound for the fractal dimension and Hausdorff dimension of in , we obtain the system (3) has a bound for the fractal dimension and Hausdorff dimension of in .

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