Abstract
In this paper, we discuss the long-time behavior of -Navier-Stokes equations with weak dampnesss and time delay. The uniformly attracting sets of processes are obtained. On the basis of the method with asymptotic compactness, the existence of the uniform attractor for the equation is proved with the restriction of the forcing term belonging to translational compacted function space.
1. Introduction
The understanding of the behavior with dynamical systems was one of the most important problems of modern mathematical physics (see [1–17]). In the last decades, -Navier-Stokes equations have received increasing attention due to their importance in the fluid motion. In [2–4], the existence of weak solution and strong solution for the 2D -Navier-Stokes equation on some bounded domain was studied. The Hausdorff and fractal dimension of the global attractor about the 2D -Navier-Stokes equation for the periodic and Dirichlet boundary conditions and the global attractor of the 2D -Navier-Stokes equation on some unbounded domains were researched in [5]. In [6–10], the finite dimensional global attractor and the pullback attractor for -Navier-Stokes equation were studied. Moreover, Anh et al. studied long-time behavior for 2D nonautonomous -Navier-Stokes equations and the stability of solutions to stochastic 2D -Navier-Stokes equation with finite delays in [11, 12]; Quyee researched the stationary solutions to 2D -Navier-Stokes equation and pullback attractor for 2D -Navier-Stokes equation with infinite delays in [13]. Recently, the random attractors for the 2D stochastic -Navier-Stokes equation were researched in [14]. From these researches, we can see that the attractor of 2D -Navier-Stokes equation is still important. We would like to use the theory of uniform attractors to study it. So, the present research is necessary and has a theoretical basis.
In this paper, we study the existence of the uniform attractor of the -Navier-Stokes equation with weak dampness and time delay which have the following form: where and denote the velocity and pressure, respectively. is the viscosity coefficient, denotes linear dampness, and is positive constant. is the time-dependent external force term, is another external force term with time delay. and are suitable real-valued smooth functions; when , Equation (1) becomes the usual 2D Navier-Stokes equations.
This paper is organized as follows. In Section 2, we first introduce some notations and preliminary results for the -Navier-Stokes equation. In Section 3, we prove existence of the uniform attractor of 2D -Navier-Stokes equation with weak dampness and time delay on the bounded domains.
2. Preliminaries
We assume that the Poincare inequality holds on , i.e., there exists , such that
Let with the inner products and the norms Let , which is endowed with the inner products and the norms , where
Let be the space of function with the compact support contained in , and let ; the closure of in is ; the closure of in is . has the inner product and norm of , And has the inner product and norm of .
It follows from (2) that
We define a -Laplacian operator as follows: .
Using the -Laplacian operator, we rewrite the first Equation (1) as follows:
From [2], we can define a -orthogonal projection and a -Stokes operator .
Applying the projection into (4), we can obtain the following weak formulation of (1): let and , we find that
such that . where is given by and , such that . Then, the weak formulation of (6) and (7) is equivalent to the functional equations where is the -Stokes operator defined by is bilinear operator and , where and satisfy the following inequalities [2, 4]:
Let , . For every , we define . For convenience, we denote .
Let satisfy the following assumptions: (I) is measureable,(II)(III), such that , there is (IV), such that
, , from (IV), we have
Definition 1. Let and satisfy the hypotheses (I)-(IV). For every , a function is called a weak solution of problem (1) if it fulfils
We can obtain the following theorem by the standard Faedo-Galerkin methods, where we let . Other cases can be similarly proved.
Theorem 2. Let satisfies the assumptions (I)-(IV), there exists a unique solution such that (6) and (7) holds.
Proof. We apply the Faedo-Galerkin methods. Since is separable and is dense in , there exists a sequence , which forms a complete orthonormal system in and a basic for . Let be a positive integer, for each , we define an approximate solution of (6) as , which satisfies
for and , where is the orthegonal projection in of onto the space spanned by . Then, we can obtain
We can write the differential equations in the usual form
where .
Let be the ith component of . The nonlinear ordinary differential system (18) has a maximal solution defined on some interval . If , then as . The following we will prove . We need several estimates to do.
We multiply (16) by and add these equations for to obtain
Then, we have
so that
Let , then
By the Gronwall inequality, we have
Hence,
which implies that the sequence remains in bounded set of . From (22), we have
Then,
We intergrate (26) from 0 to ; we have
So, the remains in a bounded set of .
Let denotes the function from into , which is equal to on and to 0 on the complement of this interval. The Fourier transform of is denoted by . Then, we will show that there exist a positive constant and such that
Since the remains in a bounded set of , the remains in a bounded set of . Since has two discontinuities at 0 and , the distribute derivative of is given by
where and are the dirac distributions at 0 and , and is the derivative of on . We obtain that
for , where and are distributions at 0 and , and on . By the Fourier transform, we have
where and denoting the Fourier transforms of and , respectively. We multiply (31) by and add the resulting equations for ; we get
We obtain
So, belongs to a bounded set in the space . For , we have Since and are bounded, from (31), we obtain
Let , we have
then
Since , by the Parseval equality and by the Schwarz inequality and the Parseval equality, we obtain
So, , and remains in a bounded set of and . There exists an element and a subsequence such that in weakly and in weak-star as . For any , we have strongly in .
For any support of , we have strongly in . Let be a continuously differentiable function on with , we multiply (16) by , then integrate by parts,
We have
where .
Finally, we prove that satisfies (7). We multiply (6) by and integrate
We compare (39) with (40) to obtain . Let , then we have . So, .
Now, we will prove the solution of (6) and (7) is unique. We let and be the solutions of (9) and . We have
We take the scalar product of (41) with , then
Therefore,
Then,
We have
Hence, . So, .
From [15], we can define a family of two parametric maps in ,
Here, is called the time symbol of the system. We have the following concepts and conclusions from [15].
Definition 3. For the given time symbol , a family of two-parametric maps with is called a process in , if Now, we define translation operator in . . We have Denote , where is the positive invariant semigroups acting on and satifying and Let be a constant, obviously Let be the Banach space; we use to denote the set of all bounded sets on and consider a family of processes with , the parameter is called the symbols of the process family , is called the symbol space, and we assume that is a complete metric space.
Definition 4. A family of processes is called uniformly bounded (), if any , both
Definition 5. A set is said to be uniformly absorbing for the family of processes , if for any and each , there exists , such that for all ,
Definition 6. A set is said uniformly atttracting set of , if for any , there is A family of processes is said to uniformly compact, if there exists a compacted uniformly absorbed set in . A family of processes is said to uniformly asymptotic compact, if there exists a compacted uniformly atttracting set in .
Definition 7. A closed set is said to be the uniform attractor of the family of processes , if (1) is uniformly attractive(2)is included in any uniformly attracting set of , that is
Theorem 8. Let be equicontinuous and for any is quasicompact in , then is relatively compact in .
Lemma 9 (Uniform Gronwall lemma). Let be local integrable function on , is also local integrable on , and , where is positive constant. Then,
3. The Existence of Uniform Attractor for 2D -Navier-Stokes Equations in Bounded Domain
First, we prove the existence of uniformly absorbing set in and ; we define , where is translation compact function. That is,
The following we use to represent the translation compact function class.
Lemma 10. Let for any , , assume that (I)-(IV) hold, then there exist bounded absorbing sets of process family in .
Proof. Since is bounded, then there exists , such that For any , we define , then taking the inner product of (9) with , we have Let , then Integrating both sides from to , then Then, for Let , then Taking , such that , then , so Let then Then, Let , we will prove the existence of the uniformly absorbing bounded set in . First, we must prove the boundedness of .
Lemma 11. Given that , then there exist and constant , such that where denotes any bounded set on the .
Proof. Taking the inner product of (9) with , Then, Integrating on both sides in , we have Then, When , that is , we have where
Lemma 12. For any , . Assume that (I)-(IV) hold, then there exists uniformly bounded absorbing set of process family in .
Proof. Let , taking the inner product of (9) with , we obtain for Then, Applying Lemma 9, where If taking , then Let , so . Then,
From [16], we have the following definition.
Definition 13. Let be Banach space, if , there exists , such that Then, is called normal function.
We will take the sets of all normal function classes in as . From [17], we can see that is the true subspace of . Therefore, the translation compact function must be a normal function.
Theorem 14. Suppose that nonlinear term satisfies (I)-(III), is translation compact function in , then process family exist uniform attractor , and .
Proof. Since is bounded set in and uniform absorbed set of . For each , we take a set
where denotes any compact self-adjoint operator, then , and is another uniform absorbing bounded set of in . Now, we will prove is relatively compact in . From Theorem 8, we only need to prove is equicontinuous and uniform bounded in . From the definition of , we can obtain it is uniformly bounded. Now, we will prove is equicontinuous. For any ,
Let , and denote as , then
We estimate the items on the right end of the above formula, let ,
Since
we let
When , and , then
Let
Then,
So,
And When , we have
Then, is equicontinuous, and is relatively compact in , so is compacted uniformly absorbing set of in , Let , since and the embedding mapping is continuous, so is compact in ; is also compacted uniformly absorbing set of in . Then, process family exists uniform attractor
Data Availability
The (data type) data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
The author would like to thank the referees for the helpful suggestions. This work is supported by the Projects of Natural Science Basic Research Plan in Shaanxi Province of China (no. 2018JM1042).