Upper Semicontinuous Property of Uniform Attractors for the 2D Nonautonomous Navier-Stokes Equations with Damping
Our aim is to investigate the long-time behavior in terms of upper semicontinuous property of uniform attractors for the 2D nonautonomous Navier-Stokes equations with linear damping and nonautonomous perturbation external force, that is, the convergence of corresponding attractors when the perturbation tends to zero.
In the present paper, we investigate the long-time behavior of uniform attractors for the nonautonomous 2D Navier-Stokes equations with damping and singular external force that governs the motion of incompressible fluid where is a bounded domain with smooth boundary , is the kinematic viscosity of the fluid, is the velocity vector field which is unknown, is the pressure, is positive constant, , and is a small positive parameter.
The functions and are taken from the space of translational bounded functions in , namely, for some constants .
We denote and note that is of the order as .
As a straightforward consequence of (9), we have
Since the last century, the global well-posedness and large-time behavior of solutions to the Navier-Stokes equations have attracted many mathematicians to study. For the well posedness of 3D incompressible Navier-Stokes equations, in 1934, Leray [1, 2] derived the existence of weak solution by weak convergence method; Hopf  improved Leray's result and obtained the familiar Leray-Hopf weak solution in 1951. Since the Navier-Stokes equations lack appropriate priori estimate and the strong nonlinear property, the existence of strong solution remains open. For the infinite-dimensional dynamical systems, Sell  constructed the semiflow generated by the weak solution which lacks the global regularity and obtained the existence of global attractor of the incompressible Navier-Stokes equations on any bounded smooth domain; Cheskidov and Foias  introduced a weak global attractor with respect to the weak topology of the natural phase space for 3D Navier-Stokes equation with periodic boundary; Flandoli and Schmalfuß  deduced the existence of weak solutions and attractors for 3D Navier-Stokes equations with nonregular force; Kloeden and Valero  investigated the weak connection of the attainability set of weak solutions of 3D Navier-Stokes equations; Cutland  obtained the existence of global solutions for the 3D Navier-Stokes equations with small samples and germs; Chepyzhov and Vishik [9–11] investigated the trajectory attractors for 3D nonautonomous incompressible Navier-Stokes system which is based on the works of Leray and Hopf. Using the weak convergence topology of the space (see below for the definition), Kapustyan and Valero  proved the existence of a weak attractor in both autonomous and nonautonomous cases and gave an existence result of strong attractors. Kapustyan et al.  considered a revised 3D incompressible Navier-Stokes equations generated by an optimal control problem and proved the existence of pullback attractors by constructing a dynamical multivalued process. For more results of the well-posedness and long-time behavior of the 2D autonomous incompressible Navier-Stokes equations, such as the existence of global solutions, the existence of global attractors, Hausdorff dimension, and inertial manifold approximation, we can refer to Ladyzhenskaya , Robinson , Sell and You , and Temam [17, 18]. Moreover, Caraballo and Real  derived the existence of global attractor for 2D autonomous incompressible Navier-Stokes equation with delays; Chepyzhov and Vishik [20, 21] investigated the long-time behavior and convergence of corresponding uniform (global) attractors for the 2D Navier-Stokes equation with singularly oscillating forces as the external force tend to be steady state by virtue of linearization method and estimate the corresponding difference equations; Foias and Temam [22, 23] gave a survey about the geometric properties of solutions and the connection between solutions, dynamical systems, and turbulence for Navier-Stokes equations, such as the existence of -limit sets; Rosa  and Hou and Li  obtained the existence of global (uniform) attractors for the 2D autonomous (nonautonomous) incompressible Navier-Stokes equations in some unbounded domain, respectively; Lu et al.  and Lu  proved the existence of uniform attractors for 2D nonautonomous incompressible Navier-Stokes equations with normal or less regular normal external force by establishing a new dynamical systems framework; Miranville and Wang  derived the attractors for nonautonomous nonhomogeneous Navier-Stokes equations.
However, the infinite-dimensional systems for 3D incompressible Navier-Stokes equations have not been yet completely resolved, so many mathematicians pay attention to this challenging problem. In this regard, some mathematicians pay their attentions to the Navier-Stokes equation with damping. Let us recall some known results for the 3D incompressible Naver-Stokes equations with damping. For the 3D autonomous Navier-Stokes equation with damping, the authors of  showed that the initial boundary value problem of a 3D Navier-Stokes equation with damping has a unique weak solution and Song and Hou  derived the global attractors for the same autonomous system. Kalantarov and Titi  investigated the Navier-Stokes-Voight equations as an inviscid regularization of the 3D incompressible Navier-Stokes equations, and further obtained the existence of global attractors for Navier-Stokes-Voight equations. Recently, Qin et al.  showed the existence of uniform attractors by uniform condition-(C) and weak continuous method to obtain uniformly asymptotical compactness in and . However, there are fewer results for the upper semicontinuous and lower semicontinuous for the nonautonomous system with perturbation case. In this paper, we will show the long-time behavior in terms of upper semicontinuous property of uniform attractors for the problem (1)–(4), that is, the convergence of corresponding attractors when the perturbation tends to zero.
This paper is organized as follows: in Section 2, we will give some preliminaries of uniform attractors; in Section 3, the uniform boundedness of uniform attractors of 2D Navier-Stokes equation with damping for will be obtained; the main result will be stated in the last section.
2. Some Preliminaries of Uniform Attractors
The Hausdorff semidistance in from one set to another set is defined as () is the generic Lebesgue space and is the usual Sobolev space. We set , is the closure of the set in topology with norm or , is the closure of the set in topology, and is the closure of the set in topology.
The family of functions denote a local Bochner integration function class, and denotes all translation bounded functions which satisfies for all ; that is, is translation bounded in . is translation compact function in . Obviously, .
Operator is the Helmholtz-Leray orthogonal projection in onto the space , is the Stokes operator subject to the nonslip homogeneous Dirichlet boundary condition with the domain , is a self-adjoint positively defined operator on with domain , and is the first eigenvalue for the Stokes operator ; we define the Hilbert space as with its inner product and norm topology as .
The problems (1)–(4) and (5)–(8) can be written as a generalized abstract form where the pressure has disappeared by force of the application of the Leray-Helmholtz projection , and is the bilinear operator. The bilinear form can be extended as a continuous trilinear operator and satisfies
Lemma 1. For each , every nonnegative locally summable function on and every , one has for all .
Proof. See, for example, Chepyzhov et al. .
Lemma 2. Let fulfill the fact that for almost every , the differential inequality where, for every , the scalar functions and satisfy for some , , and . Then
Proof. See, for example, Chepyzhov et al. .
(1) Assume , ; then problem (17)–(20) possesses a unique global weak solution which satisfies
Moreover, one chooses an arbitrary nonautonomous external force and fixed, the global solution generates a process (, , ) which is continuous with respect to , where is a symbol which belongs to the symbol space , and means the closure in the topology .
(2) Assume that , ; then the family of processes , () generated by the global weak solution of problem (17)–(20) possesses a uniform (with respect to ) attractor in .
Theorem 4. Assume that ; the functions and are taken from the space of translational bounded functions in and (10)–(13) hold, and then the family of processes generated by the global solution of problem (1)–(4) possesses uniform (with respect to ) attractors for any fixed in .
Proof. As the similar argument in , we choose in , since and are translational bounded in , and then for any fixed , we can deduce that is translational bounded in and the existence of uniformly compact attractors for any fixed .
Proof. As the similar technique in , we can easily deduce the existence of a uniformly compact attractor if we choose since is translation bounded in .
The structure of the uniform attractor will be discussed as follows: since the functions and are translation bounded and satisfy (10)–(13), the global solution of problem (1)–(4) generates the family of processes acting on by the formula , , where is a solution to (1)–(4).
Similar to the procedure in  and by Theorem 4, the processes class has a uniformly (with respect to ) absorbing set which is bounded in for any fixed , which means that for any bounded set , there exists a time such that
Hence, are also uniformly absorbing with respect to as or which belongs to , is the integer part of .
The processes have a uniform global attractor as uniform -set where denotes the closure in and is an arbitrarily uniformly bounded absorbing set of the processes ; here, we can set .
On the other hand, for each fixed , is also bounded in , since (). Assuming , then . Besides, if and , then for some and .
Next, we consider the equation class as follows to describe the structure of the uniform attractor
For every external force , by the well-poseness of the abstract equation (17), we can derive that (38) generates a family of processes on , which shares similar properties to , corresponding to the original equation (1) with external force . Moreover, from Theorem 3 we know the map is -continuous.
Definition 6. The kernel of (17) is the family of all complete orbits which are uniformly bounded in . The set is called the kernel section of at time . For every , the following representation (complete orbit) of uniform attractors of (1) holds:
3. Uniform Boundedness of in
Firstly, we consider the auxiliary linear equation with nonautonomous external force and give some useful estimates and then prove the uniform boundedness of in .
Considering the linear equation we obtain the following lemmas.
Lemma 8. Assume ; then problem (43) has a unique solution Moreover, the following inequalities hold for every and some constant , independent of the initial time .
Proof. Firstly, similar to the discussion in  or , by the Galerkin approximation method, we can obtain the existence of global solution; here we omit the details.
Then, multiplying (43) by , , and , respectively, using the Poincaré inequality, we get By the Gronwall inequality to (51), (53), integrating over for (50), (52), and (53), we can easily complete the proof.
Setting , , , we have the following lemma.
Lemma 9. Let . Assume that holds for some constant . Then the solution to the following Cauchy problem with satisfies the inequality where constant is independent of .
Proof. Noting that
and then using (54) and (57), we can deduce the following estimates of as
From Lemmas 2 and 8, we have
Similarly, we derive that
Hence, using the Poincaré inequality, by (45)–(47) and (58)–(60), we derive
Next, we set which implies that for any , since in (55).
Integrating (55) with respect to time from to , we see that is a solution to the problem such that we can deduce that from (62) to (63).
Using (46) and (61), we conclude
Noting that , , and , using (58), (68), and (69), we derive that
Hence, by (51), (58), and (62), we conclude
Combining (70) and (71), the proof for the lemma is finished.
Now, we will use the auxiliary linear equation and some estimates to prove the uniform boundedness of in . For convenience, we set and assume for some constants since and are translation bounded in .
Proof. Let be the solution to (1)–(4) with the initial data as . For , we consider the auxiliary linear equation
By Lemma 9, we have the estimate
Multiplying (75) with and integrating over , using the boundary value condition, we derive that
By the Gronwall inequality and similar to (60), noting that when tends to infinite, we can set such that since .
Setting the function as which satisfies the problem where is a solution for problem (1)–(4), and is a solution to (75), is the bilinear operator which is defined in Section 2.
Taking the scalar product of (80) with in , we obtain Here we use the property of trilinear operator (21)-(22); we observe that
Inserting (82)–(84) into (81) and then using the inequality and (78), we have which implies that where
Therefore, from Theorem 3, we derive from (88) that for any ,
Applying Lemma 2 with , , , , we get
Recalling that and using (85) and (90), we end up with for all .
Thus, for every , the processes have an absorbing set
On the other hand, if , the processes also possess an absorbing set
In conclusion, for every , the set is an absorbing set for the processes which is independent of . Since , (74) follows and hence the proof is finished.
4. Convergence of to
Next, we will study the difference of two solutions for (1) with and (4) with , which share the same initial data. Denote with belonging to the absorbing set which can be found in Section 3. In particular, for , since , we obtain for some , as the size of depends on .
Lemma 11. For every , , and , the difference where satisfies the estimate for some positive constants and , both independent of .
Proof. Since the difference solves
fulfills the Cauchy problem
where is the solution to (75).
Taking inner product in of (101) with , we obtain
Noting we derive
Next, we estimate each term on the right-hand side of (104).
Applying (22) to (27), we find
Hence, from (105) to (107), we obtain where and satisfy (76) and (96), respectively, and
Thus, it follows from (102) and (104) that
Noting that , by the Gronwall inequality, we get
Consequently, holds for some positive constants and .
Finally, since , using (76) to control , we may obtain where is a positive constant.
Define and we have
For any , we observe that is a solution to (38) with external force and . For , we investigate the property of the difference
Lemma 12. The inequality holds; here and are defined as in Lemma 11.
Proof. As the similar discussion to the proof of Lemma 11, replacing , , and by , , and , respectively, noting that (96) still holds for , and the family , (), is -continuous, and using (116) in place of (73), we can finally complete the proof of the lemma.
The main result of this paper reads as follows.
Proof. For , , from (110)-(111), we obtain that there exists a complete bounded trajectory of (38), with some external force
such that .
We choose such that
From the equality and applying Lemma 12 with , , we obtain
On the other hand, the set attracts all sets uniformly when . Then, for all , there exists some time which is independent on , such that
Choosing and using (123)-(124), we readily get
Since and is arbitrary, taking the limit , we can prove the theorem.
Xin-Guang Yang was in part supported by the Young Teacher Research Fund of Henan Normal University (qd12104) and the Innovational Scientists and Technicians Troop Construction Projects of Henan Province (no. 114200510011). Jun-Tao Li was in part supported by the Natural Science Foundation of China (61203293), Foundation of Henan Educational Committee (2011B120005), Key Scientific and Technological Project of Henan Province (122102210131), and College Young Teachers Program of Henan Province (2012GGJS-063).
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