Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 28, Number 4, August 2023 282 - 290 https://doi.org/10.1051/wujns/2023284282 06 September 2023

This 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 a class of three-dimensional Brinkman-Forchheimer equation as follows:

(1)

where is an open set, not necessarily bounded, which is sufficiently regular and satisfies Poincaré's inequality. Here is the fluid velocity vector, is the Brinkman coefficient, is the Darcy coefficient, are the Forchheimer coefficients, is the pressure and is the initial data.

Brinkman-Forchheimer equation describes the motion of fluid flow in a saturated porous medium[1,2], and has been studied by many researchers[3-6]. From the physical viewpoint, Gilver and Altobelli[7] obtained a determination of effective viscosity for the Brinkman-Forchheimer flows model. Nield[8] dealt with the momentum equation in a porous medium, involving the fluid mechanics of the interface region between a porous medium and a fluid layer. Vafai and Kim[9,10] obtained an exact solution to Brinkman-Forchheimer equation by using a generalized momentum equation. From the mathematical viewpoint, the discussion of three-dimensional Brinkman-Forchheimer equations is mainly concerned with the well-posedness, regularity and long-time behavior of solutions. The global attractor is the core concept of infinite dimensional dynamical systems. If a system has a global attractor, the attractor will contain all possible limit states of the solution of the system. For the autonomous Brinkman-Forchheimer equation, Uğurlu[11] showed the existence of global attractor of the system (1) in . If the term "" in system (1) is replaced by "", Ouyang and Yang[12] proved the existence of global attractor in when by condition-(C) method. In Ref.[13], the existence of -pullback attractors for the 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], Qiao et al proved the existence of the global attractor for the strong solution of the Brinkman-Forchheimer equation in a three-dimensional bounded domain. In Ref.[16], Song and Wu discussed a non-autonomous Brinkman-Forchheimer equation with singularly oscillating external force in 3D bounded domains. To the best of our knowledge, there is no discussion of the existence of global attractors in three-dimensional unbounded domains for Brinkman-Forchheimer equation. In this paper, we will discuss the existence of global attractor of system (1) in three-dimensional unbounded domains that satisfy the Poincaré's inequality. We will use the method of uniformly estimating the tail of the solution to obtain the asymptotic compactness of the corresponding solution operator of the equation. This method was first proposed by Wang in Ref.[17].

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. Meanwhile, we provide some uniform estimates of the solution of equation (1), which will be used in the following two sections. In Section 2, we estimate the boundedness of the tail of the solution in . In Section 3 we prove the asymptotic compactness of the solution in and then obtain the existence of global attractor.

1 Preliminaries

Let Throughout this paper, we use to denote the norm in stands for a generic positive constant, depending on and some constants, but independent of time .

The Hausdorff semidistance in from set to set is defined as

We set is the closure of the set in topology, and is the closure of the set in topology. and are the dual spaces of and , clearly, , where the injection is dense and continuous. We denote by and the inner product and norm in . That is,

and denote the inner product and norm in that is,

and

We call 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 defined as is the orthogonal projection from to .

Then we introduce some useful inequalities and lemmas.

(4)

(5)

Sobolev's inequality[19]:

(6)

Lemma 1   (Grownwall's inequality) If , and is an absolutely continuous function on , and the following inequality holds:

where , then

Especially if , then

Now we give the existence and uniqueness theorem of the strong solution of equation (1).

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

Proof   The proof of this theorem is similar to the proof of Theorem 3.2 in Ref.[15], so here we omit the proof process.

Then we give some uniform estimates of the solution below.

Proposition 1   Suppose and . Then there exists a time , a positive constant such that

Proposition 2   Suppose and . Then there exists a time , a positive constant such that

Proposition 3   Suppose and . Then there exists a time , a positive constant such that

The propositions 1, 2, and 3 are proved in Ref.[15], and it is easy to verify that they are still valid in the three-dimensional unbounded domain that satisfies the Poincaré's inequality.

Proposition 4   Suppose and . Then there exists a positive constant such that

Proof   Applying Minkowski's inequality, from (1) we have

(7)

According to Ladyzhenskaya's inequality we have

(8)

And by Sobolev's inequality, we get

(9)

Substituting (8) and (9) into (7), we get

So there must exist a positive constant such that

Proposition 5   When , the map is a Lipschitz continuous map on

Proof   Assuming that are two solutions of equation (1), with initial values and , respectively. Let , then we have

(10)

Taking the inner product of (10) with in , we get

(11)

Since

and in the last step of the above inequality, we used the following simple inequality

So we have

(13)

Substituting (13) into (11), we get

(14)

Applying Gronwall's inequality to (14), we have

(15)

According to Sobolev's inequality: and because of , it yields

The proof is completed.

2 Uniform Estimate on the Tail of the Solution

In this section, we will employ the technique of uniform estimate on the tail of solution to establish the -asymptotic compactness of the Brinkman-Forchheimer equation in a three-dimensional unbounded domain satisfying Poincaré's inequality.

Given , we denote by the set and the complement of . For our purpose, we choose a cut-off function with two order continuous derivative such that and

(16)

We have the following lemma.

Lemma 2   Suppose , and , which is a bounded set in . Then for every , there exist and such that where and depend on .

Proof   We will use the tail estimation method, which has been used by Wang in Ref.[17] to establish the existence of global attractor for reaction-diffusion equation in unbounded domain. The method has also been used in Ref.[20] to discuss the existence of global attractor for Newton Boussinesq equation in two-dimensional channel.

Multiplying the first equation of (1) by , we have

For the first term on the left-hand side of (17), applying Green's formula, we have

For the second term on the left-hand side of (17), we obtain

(19)

For the third term on the left-hand side of (17), we get

(20)

Taking (17)-(20) into account, we get

(21)

We now estimate the right-hand side of (21) term by term. Applying Young's inequality and Holder's inequality, we find

(22)

(23)

Since so we have

(24)

Similar with (24), we have

(25)

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

(26)

And because

So we have

(27)

By (22)-(27) and (21) we get

(28)

Now, from Proposition 1, Proposition 2, Proposition 3 and Proposition 4, we have known that there exists some positive constant and time , such that for all . Hence, given there exists such that

(29)

For the given , there also exists , such that

(30)

Therefore, for the given , there exists , such that

(31)

Furthermore, according to Sobolev's inequality, we have

So for the given , there also exists such that

(32)

Since , so there exists such that

(33)

Let , by (28)-(33), we have

(34)

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

for all

Therefore,

The proof is completed.

3 Existence of Global Attractor

In this section, we prove the existence of global attractor for problem (1) in . To this end, we need to establish the asymptotic compactness of the solution operator which is stated as follows.

Lemma 3   Suppose and . Then the dynamical system is asymptotically compact in , i.e., if and is bounded in , then the sequence has a convergent subsequence in .

Proof   Since is bounded in , there exists a positive constant such that

(35)

It is well known that is an unbounded self-adjoint operator with domain:

And defines a norm in which is equivalent to the norm in , in other words, there exists a constant depending only on such that

By Proposition 4, there is a positive constant and a time , such that for every , the following holds

(36)

Since , there is such that for all . Therefore we have, for ,

(37)

By (37) we find that there is a such that, up to a subsequence,

(38)

Given , by Lemma 2, there are positive constants and such that for any and with satisfies

(39)

Let be large enough such that for all . Then by (39) we obtain, for ,

(40)

Notice that (37) implies that the sequence is bounded in and hence precompact in . Therefore, there is such that, up to a subsequence,

(41)

By (38) and (41), we find which means that for every

(42)

In other words, for the given , there is such that for all and ,

(43)

Since , there is such that for all ,

(44)

Let and . Then for all , we have

(45)

where the last inequality is obtained by (40), (43) and (44). Notice that (45) shows that

and hence is asymptotically compact in . The proof is completed.

Theorem 2   Suppose and . Then the problem (1) has a global attractor in , which is compact, invariant and attracts every bounded set with respect to the norm of .

Proof   By Proposition 2, the dynamical system has a bounded absorbing set in , and by Lemma 3, is asymptotically compact in . Then the existence of a global attractor follows immediately from the standard attractor theory (see Refs.[21]-[25]).

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