Issue |
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 1, February 2023
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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 Brinkman-Forchheimer 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 Brinkman-Forchheimer equation has finite Hausdorff and fractal dimensions.
Key words: Brinkman-Forchheimer equation / global attractor / Hausdorff dimension / fractal dimension
Biography: SONG Xueli, female, Associate professor, research direction: infinite dimensional dynamical systems. E-mail: 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 three-dimensional Brinkman-Forchheimer 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, 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
The weak form (2) is equivalent to the following functional equation:
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
Next we formulate some well-known 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 (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
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 right-hand 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 right-hand side of (21), we have
For the third term on the right-hand side of (21), we get
Similar with (24), for the fourth term on the right-hand side of (21), we have
For the fifth term on the right-hand side of (21), we have
In inequality (26), we used -inequality and the following Gagliardo-Nirenberg'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 right-hand side of (29), from inequality (26), we have
For the second term on the right-hand 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 n
N, with N
0 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 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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