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 
Mathematics
CLC number: O175.29
Dimension Estimate of the Global Attractor for a 3D Brinkman Forchheimer Equation
College of Science, Xi'an University of Science and Technology, Xi'an 710054, Shaanxi, China
Received:
22
June
2022
In this paper, we study the dimension estimate of global attractor for a 3D BrinkmanForchheimer equation. Based on the differentiability of the semigroup with respect to the initial data, we show that the global attractor of strong solution of the 3D BrinkmanForchheimer equation has finite Hausdorff and fractal dimensions.
Key words: BrinkmanForchheimer equation / global attractor / Hausdorff dimension / fractal dimension
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 the following threedimensional BrinkmanForchheimer equation:
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, BrinkmanForchheimer equation describes the motion of a fluid flowing in saturated porous media^{ [13]}, which has received much attention on several issues over the last decades. From a mathematical point of view, the research on the threedimensional BrinkmanForchheimer equation is mainly divided into two categories. One is the structural stability of the equation with respect to the coefficients and ^{[49]}, and the other is the longterm behavior of the solution of the equation^{ [1018]}.
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 infinitedimensional dynamical systems. For the 3D BrinkmanForchheimer 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 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], 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 BrinkmanForchheimer 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 BrinkmanForchheimer equation in a threedimensional 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 infinitedimensional dynamical system can be reduced to a finitedimensional ordinary differential equation system, so that the relatively complete theory of the finitedimensional 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 BrinkmanForchheimer. In this paper, inspired by Refs.[1921], 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
The weak form (2) is equivalent to the following functional equation:
Here is the Stokes operator subject to the noslip homogeneous Dirichlet boundary condition with the domain defined as . is the orthogonal projection from onto . Obviously, the operator is a nonnegative selfadjoint operator in with and for all .
For a bounded domain , the operator is invertible, and its inverse operator is bounded, selfadjoint 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 ndimensional Lebesgue measure of . For we find
Next we formulate some wellknown inequalities and a Gronwall type lemma that we will use in what follows.
Poincaré's inequality ^{[23]}:
where is the first eigenvalue of operator under the homogeneous Dirichlet boundary condition.
Agmon's inequality ^{[23]}:
inequality^{ [24]}:
Series inequlity^{[25]}:
Lemma 1 (GagliardoNirenberg'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 nonnegative 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 BrinkmanForchheimer 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 BrinkmanForchheimer system has finite Hausdorff and fractal dimensions. We will use the similar techniques as in Refs.[1921], 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
So we have
Applying Gronwall's inequality, we find
So there is a time which we can take as , such that for all , we have
Integrating the inequality (10) from 0 to , we obtain
so we get
Theorem 3 Let and be two members of . Then there exists a constant such that
where the linear operator for is the solution operator of the problem:
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
and
Let . Combining (17) with (18), we obtain
Let us define . Then satisfies:
Let us take inner product with in to the first equation in (20) to obtain
Let us consider the first term on the righthand side of (21). We have
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 righthand side of (21), we have
For the third term on the righthand side of (21), we get
Similar with (24), for the fourth term on the righthand side of (21), we have
For the fifth term on the righthand side of (21), we have
In inequality (26), we used inequality and the following GagliardoNirenberg's inequality:
Combining (21)(26), we find
Taking inner product with in to the first equation of (19), we have
where
and
For the second term on the righthand side of (29), from inequality (26), we have
For the second term on the righthand side of (30), we have
Combining (29)(32) with (28), we obtain
Integrating (33 ) from 0 to , we have
Applying Gronwall's inequality to (34), we get
Integrating (27) from 0 to , due to (35), we infer that
An application of Gronwall's inequality in (36) yields
Thus
which completes the proof.
Now we rewrite the system (3) as
Let us now set and using it in (39) to obtain
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
Thus, the semigroup has the global attractor , where
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
Let us first consider the linear variations of the system (40). The linear variational equation corresponding to (40) has this form
where
The adjoint of is given by
Hence, can be computed as
Then, we derive the following results.
Proposition 1 Let . Then, we have
where is given in (6) which only depends on .
Proof Let us take the inner product with in to equation (47), we obtain
And because
and
Combining (50) and (51) with (49), we deduce that
Proposition 2 Suppose Then the global attractor has the finite fractal dimension in , with
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:
for all . Using Lemma 3.5 in Ref.[26], we know that the volume elements satisfy
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
From (56), we have
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
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 BrinkmanForchheimer 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 .
References
 Nield D A, Bejan A. Convection in Porous Media [M]. New York: SpringerVerlag , 1992. [CrossRef] [Google Scholar]
 Payne L E, Straughan B. Stability in the initialtime geometry problem for the Brinkman and Darcy equations of flow in a porous media [J]. Journal de Mathématiques Pures et Appliquées, 1996, 75(3): 225271. [Google Scholar]
 Payne L E, Straughan B. Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions [J]. Journal de Mathématiques Pures et Appliquées, 1998, 77(4): 317354. [CrossRef] [MathSciNet] [Google Scholar]
 Celebi A O, Kalantarov V, Ugurlu D. Continuous dependence for the convective BrinkmanForchheimer equations [J]. Applicable Analysis, 2005, 84(9): 877888. [CrossRef] [MathSciNet] [Google Scholar]
 Celebi A O, Kalantarov V, Ugurlu D. On continuous dependence on coefficients of the Brinkman Forchheimer equations [J]. Applied Mathematics Letters, 2006, 19(8): 801807. [CrossRef] [MathSciNet] [Google Scholar]
 Liu Y. Convergence and continuous dependence for the BrinkmanForchheimer equations [J]. Mathematical and Computer Modelling, 2009, 49(78): 14011415. [Google Scholar]
 Payne L E, Straughan B. Convergence and continuous dependence for the BrinkmanForchheimer equations [J]. Studies in Applied Mathematics, 1999, 102(4): 419439. [CrossRef] [MathSciNet] [Google Scholar]
 Liu Y, Xiao S Z, Lin Y W. Continuous dependence for the BrinkmanForchheimer fluid inter facing with a Darcy fluid in a bounded domain [J]. Mathematics and Computers in Simulation, 2018, 150: 6682. [CrossRef] [MathSciNet] [Google Scholar]
 Li Y F, Lin C H. Continuous dependence for the nonhomogeneous BrinkmanForchheimer equations in a semiinfinitepipe [J]. Applied Mathematics and Computation , 2014, 244: 201208. [CrossRef] [MathSciNet] [Google Scholar]
 Ugurlu D. On the existence of a global attractor for the BrinkmanForchheimer equation [J]. Nonlinear Analysis: Theory, Methods & Applications, 2008, 68(7): 19861992. [CrossRef] [MathSciNet] [Google Scholar]
 Ouyang Y, Yang L E. A note on the existence of a global attractor for the Brinkman Forchheimer equations [J]. Nonlinear Analysis :Theory, Methods & Applications, 2009, 70(5): 20542059. [CrossRef] [MathSciNet] [Google Scholar]
 Wang B X, Lin S Y. Existence of global attractors for the threedimensional Brinkman Forchheimer equations [J]. Mathematical Methods in the Applied Sciences, 2010, 31(12): 14791495. [Google Scholar]
 Song X L. Pullback Dattractors for a nonautonomous BrinkmanForchheimer system [J]. Journal of Mathematical Research with Applications, 2013, 33(1): 90100. [Google Scholar]
 Song X L, Xu S, Qiao B M.  decay of solutions for the threedimensional Brinkman Forchheimer equations in [J] . Mathematics in Practice and Theory, 2020, 50(22): 307314(Ch). [Google Scholar]
 Song X L, Wu J H. Nonautonomous 3D BrinmanForchheimer equation with oscillating external force and its uniform attractor [J]. AIMS Mathematics, 2020, 5(2): 14841504. [CrossRef] [MathSciNet] [Google Scholar]
 Liu W J, Yang R, Yang X G. Dynamics of a 3D Brinman Forchheimer equation with infinite delay [J]. Communications on Pure and Applied Analysis, 2021, 20(5): 19071930. [CrossRef] [MathSciNet] [Google Scholar]
 Yang X G, Li L, Yan X J, et al. The structure and stability of pullback attractors for 3D BrinkmanForchheimer equation with delay [J]. Electronic Research Archive, 2020, 28(4): 13951418. [CrossRef] [MathSciNet] [Google Scholar]
 Qiao B M, Li X F, Song X L. The existence of global attractors for the strong solutions of threedimensional Brinkman Forchheimer equations [J]. Mathematics in Practice and Theory, 2020, 50(10): 238251(Ch). [Google Scholar]
 Kalantarov V K, Titi E S. Global attractors and determining modes for the 3D NavierStokesVoight equations [J]. Chinese Annals of Mathematics, Series B, 2009, 30(6): 697714. [Google Scholar]
 Zelati M C, Gal C G. Singular limits of Voigt models in fluid dynamics [J]. Journal of Mathematical Fluid Mechanics, 2015, 17(2): 233259. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Mohan M T. Global and exponential attractors for the 3D KelvinVoigtBrinkman Forchheimer equations [J]. Discrete and Continuous Dynamical Systems, Series B, 2020, 25(9): 33933436. [CrossRef] [MathSciNet] [Google Scholar]
 Ilyin A A. On the spectrum of the Stokes operator [J]. Functional Analysis and Its Applications, 2009, 43(4): 254263. [CrossRef] [MathSciNet] [Google Scholar]
 Temam R H. InfiniteDimensional Dynamical Systems in Mechanics and Physics[M]. New York: SpringerVerlag, 1997. [CrossRef] [Google Scholar]
 Cai X, Jiu Q. Weak and strong solutions for the incompressible NavierStokes equations with damping [J]. Journal of Mathematical Analysis and Applications, 2008, 343(2): 799809. [Google Scholar]
 Kuang J C. Applied Inequalities [M]. Jinan: Shandong Science and Technology Press, 2010(Ch). [Google Scholar]
 Constantin P, Foias C. Global Lyapunov exponents, KaplanYorke formulas and the dimension of the attractors for 2D NavierStokes equations [J]. Communications on Pure & Applied Mathematics, 1985, 38(1): 127. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
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