Issue 
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 4, August 2023



Page(s)  282  290  
DOI  https://doi.org/10.1051/wujns/2023284282  
Published online  06 September 2023 
Mathematics
CLC number: O175.29
Existence of Global Attractor for a 3D BrinkmanForchheimer Equfation in Some Poincaré Unbounded Domains
College of Science, Xi'an University of Science and Technology, Xi'an 710054, Shaanxi, China
Received:
6
January
2023
In this paper, we study the existence of global attractor of a class of threedimensional BrinkmanForchheimer equation in some unbounded domains which satisfies Poincaré inequality. We use the tail estimation method to establish the asymptotic compactness of the solution operator and then prove the existence of the global attractor in .
Key words: BrinkmanForchheimer equation / global attractor / asymptotic compactness / tail estimation method
Biography: SONG Xueli, female, Associate professor, research direction: infinite dimensional dynamical systems. Email: songxlmath@163.com
Fundation item: Supported by the National Natural Science Foundation of China (12001420)
© Wuhan University 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 threedimensional BrinkmanForchheimer equation as follows:
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.
BrinkmanForchheimer equation describes the motion of fluid flow in a saturated porous medium^{[1,2]}, and has been studied by many researchers^{[36]}. From the physical viewpoint, Gilver and Altobelli^{[7]} obtained a determination of effective viscosity for the BrinkmanForchheimer 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 BrinkmanForchheimer equation by using a generalized momentum equation. From the mathematical viewpoint, the discussion of threedimensional BrinkmanForchheimer equations is mainly concerned with the wellposedness, regularity and longtime 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 BrinkmanForchheimer 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 threedimensional nonautonomous BrinkmanForchheimer 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 BrinkmanForchheimer equation in threedimensional full space. In Ref.[15], Qiao et al proved the existence of the global attractor for the strong solution of the BrinkmanForchheimer equation in a threedimensional bounded domain. In Ref.[16], Song and Wu discussed a nonautonomous BrinkmanForchheimer 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 threedimensional unbounded domains for BrinkmanForchheimer equation. In this paper, we will discuss the existence of global attractor of system (1) in threedimensional 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
The weak form (2) is equivalent to the following functional equation:
Here is the Stokes operator defined as is the orthogonal projection from to .
Then we introduce some useful inequalities and lemmas.
Ladyzhenskaya's inequality^{[18]}:
Sobolev's inequality^{[19]}:
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 threedimensional 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
According to Ladyzhenskaya's inequality we have
And by Sobolev's inequality, we get
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
Taking the inner product of (10) with in , we get
Since
and in the last step of the above inequality, we used the following simple inequality
So we have
Substituting (13) into (11), we get
Applying Gronwall's inequality to (14), we have
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 BrinkmanForchheimer equation in a threedimensional unbounded domain satisfying Poincaré's inequality.
Given , we denote by the set and the complement of . For our purpose, we choose a cutoff function with two order continuous derivative such that and
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 reactiondiffusion 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 twodimensional channel.
Multiplying the first equation of (1) by , we have
For the first term on the lefthand side of (17), applying Green's formula, we have
For the second term on the lefthand side of (17), we obtain
For the third term on the lefthand side of (17), we get
Taking (17)(20) into account, we get
We now estimate the righthand side of (21) term by term. Applying Young's inequality and Holder's inequality, we find
Since so we have
Similar with (24), we have
For the sixth term and the fifth term on the righthand side of (21) we have
And because
So we have
By (22)(27) and (21) we get
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
For the given , there also exists , such that
Applying Ladyzhenskaya's inequality, we have
Therefore, for the given , there exists , such that
Furthermore, according to Sobolev's inequality, we have
So for the given , there also exists such that
Since , so there exists such that
Let , by (28)(33), we have
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
It is well known that is an unbounded selfadjoint 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
Since , there is such that for all . Therefore we have, for ,
By (37) we find that there is a such that, up to a subsequence,
Given , by Lemma 2, there are positive constants and such that for any and with satisfies
Let be large enough such that for all . Then by (39) we obtain, for ,
Notice that (37) implies that the sequence is bounded in and hence precompact in . Therefore, there is such that, up to a subsequence,
By (38) and (41), we find which means that for every
In other words, for the given , there is such that for all and ,
Since , there is such that for all ,
Let and . Then for all , we have
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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