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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 691615, 12 pages
Pullback Attractor for Nonautonomous Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics
1Department of Basic, Henan Mechanical and Electrical Engineering College, Xinxiang 453003, China
2Department of Mathematics, Nanjing University, Nanjing 210093, China
Received 9 April 2013; Accepted 18 June 2013
Academic Editor: Grzegorz Lukaszewicz
Copyright © 2013 Kun Li and Fang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the existence of -pullback attractor for nonautonomous primitive equations of large-scale ocean and atmosphere dynamics in a three-dimensional bounded cylindrical domain by verifying pullback condition.
This paper is concerned with the existence of pullback attractor for the following nonautonomous primitive equations of large-scale ocean and atmosphere dynamics: with the following boundary conditions: and the initial data in the three-dimensional bounded cylindrical domain where is a bounded domain in with smooth boundary and is a positive constant, the unknown functions denote the horizontal velocity, the temperature, and the pressure at , respectively, is the Coriolis parameter, Ro is the Rossby number which measures the significant influence of the Earth’s rotation to the dynamical behaviour of the ocean, is a given heat source, denotes the lateral boundary of is the normal vector to , and is a positive constant related with the turbulent heating on the surface of the ocean. The viscosity and the heat diffusion operators and are given by where are positive constants representing the horizontal and vertical Reynolds numbers, respectively, and are positive constants which stand for the horizontal and vertical eddy diffusivity, respectively. For the sake of simplicity, let be the horizontal gradient operator, and let be the horizontal Laplacian.
Nonautonomous equations appear in many applications in the natural sciences, and so they are of great importance and interest. The long-time behavior of solutions of such equations has been studied extensively in recent years (see [1–12], etc.). The first attempt was to extend the notion of global attractor to the nonautonomous case, leading to the concept of the so-called uniform attractor (see ). It is remarkable that the conditions ensuring the existence of the uniform attractor is parallel those for the autonomous case. However, one disadvantage of the uniform attractor is that it does not need to be “invariant,” unlike the global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many nonautonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems. In order to overcome this drawback, the concept of pullback attractor has been introduced for the nonautonomous case. The theory of pullback attractors has been developed for both nonautonomous and random dynamical systems, and it has been shown to be very useful in the understanding of the dynamics of nonautonomous dynamical systems (see ).
Large-scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, due to the shallowness of the oceans and atmosphere, that is, the depth of the fluid layer is very small in comparison to the radius of the Earth, the vertical large-scale motion in the oceans and atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result, one can obtain the system (1)–(4), which is known as the primitive equations for large-scale oceans and atmosphere dynamics (see [14–18]). In the case of ocean dynamics, we know that one has to add the diffusion-transport equation of the salinity to the system (1)–(4), but we will omit it here in order to simplify our mathematical presentation. However, we emphasize that our results are equally valid when the salinity effects are taken into account.
In recent years, the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean have been extensively studied from the mathematical point of view (see [14–16, 19–26], etc.). The mathematical framework of the primitive equations of the ocean was formulated, and the existence of weak solutions was proved by Lions et al. in . In , the authors proved the existence and the uniqueness of local in time strong solutions of the primitive equations of the ocean. The global existence of strong solutions of the three-dimensional primitive equations of large-scale ocean, by assuming that the initial data are small enough, and the local existence of strong solutions of the three-dimensional primitive equations of large-scale ocean for all initial data were proved by Guillén-González et al. in . In , the authors proved the maximum principles for the primitive equations of the atmosphere. The existence and uniqueness of strong solutions, global in time, to the primitive equations in thin domains for a large set of initial data whose sizes depend inversely on the thickness were established in . In , Temam and Ziane considered the local existence of strong solutions for the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean. Asymptotic analysis of the primitive equations under the small depth assumption was established in . In , the authors proved the existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics. The existence and uniqueness of global strong solutions for the initial boundary value problem of the three-dimensional viscous primitive equations of large-scale ocean were established by the authors in . In , the authors considered the long-time dynamics of the primitive equations of large-scale atmosphere and obtained a weakly compact global attractor which captures all the trajectories with respect to -weak topology. Under the assumption of the initial data , the existence of compact global attractor in for the primitive equations of large-scale atmosphere was established by the elliptic regularity theory in . In , the authors proved the global existence and uniqueness of -weak solutions when the initial conditions satisfy some regularity. The existence of global attractors in and for three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinates was established in . In , the authors proved the existence of -global attractor for the primitive equations of large-scale atmosphere and ocean dynamics by use of the Aubin-Lions compactness theorem. Attractor is an important concept in the study of infinite-dimensional dynamical systems; however, the existence of pullback attractor for the three-dimensional nonautonomous viscous primitive equations of large-scale atmosphere and ocean dynamics remains unsolvable. In this paper, we prove the existence of pullback attractor for nonautonomous equations (1)–(5) by verifying pullback condition.
This paper is organized as follows. In Section 2, we introduce the mathematical framework of the system (1)–(5) and recall some auxiliary lemmas used to give some a priori estimates of strong solutions for (1)–(5) in Section 3, which are used to obtain the existence of -pullback absorbing set. Finally, in Section 4, we are devoted to proving the existence of -pullback attractor for the three dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics by verifying pullback condition.
Throughout this paper, let be a Banach space endowed with the norm , let be the -norm of , and let be positive constants, which may be different from line to line.
2.1. Functional Spaces and Some Lemmas
To study problem (1)–(5), we introduce some function spaces. Let Denote the closure of by with respect to the following norms, respectively, defined as follows: for any , and let the closure of with respect to the norm .
Now, we recall some lemmas used in the sequel.
Lemma 2 (see ) (Minkowski inequality). Let , be two measure spaces and be a measurable function about on . If for .. ,, and , then
2.2. New Formulation
In this subsection, we divide (1) into two systems with respect to and defined by
Taking the average of (1) in the direction over the interval and using the boundary conditions (3), we obtain with the boundary conditions Subtracting (14) from (1), we get with the boundary conditions
3. Some a Priori Estimates of Strong Solutions
3.1. The Well Posedness of Strong Solutions
We start with the following general existence and uniqueness of solutions which can be obtained by the standard Fatou-Galerkin methods  and the similar methods established in . Here, we only state the result as follows.
3.2. Some a Priori Estimates about Strong Solutions
In this subsection, assume that and ; we give some a priori estimates of strong solutions which imply the existence of pullback absorbing set.
3.2.1. Estimate of
Taking the inner product of (2) with in and combining the boundary conditions (4), we get Integrating by parts and using the boundary conditions (3), we obtain Since it is implied that It follows from (20) and (23) that Using (23) again, we get Therefore, we deduce from (24) and (25) that Integrating (26) from to , we obtain where we use the inequality That is, Let , for any ; integrating (24) from to and combining with (29), we obtain
3.2.2. Estimates of
Multiplying (1) by and integrating by parts over and using the boundary conditions (3), we deduce That is, It follows from that Thanks to (32) and (33), we get Integrating (34) over and using (23) and (27), we have where That is, Let , for any ; integrating (32) from to and combining with (23), (30), and (37), we obtain
3.2.3. Estimates of
Taking the inner product of (2) with in and using the boundary conditions (4), we have Using the Young inequality, we obtain Integrating (40) from to and combining with (27) and , we get Therefore, we have
3.2.4. Estimates of
Multiplying (16) by and integrating by parts over , we deduce It follows from interpolation inequality and Lemma 2 that which implies that We deduce from (44) and (46) that Repeating the similar process with the previously mentioned, we have where we use the inequality .
We deduce from (43) and (47)–(48) that which implies that Therefore, it follows from (35)–(38), (42), and Lemma 1 that there exists a positive constant independent of and a constant such that for any . For brevity, we omit writing out explicitly these bounds here, and we also omit writing out other similar bounds in our future discussion for all other uniform a priori estimates.
3.2.5. Estimates of
Taking the inner product of (14) with and combining the boundary conditions (15), we deduce We give the estimates of each term of the right-hand side of (53) as follows: It follows from (53)–(54) that In view of (27), (35)–(38), (52), and Lemma 1, we obtain for any .
3.2.6. Estimates of
Denote . It is clear that satisfies the following equation obtained by differentiating (1) with respect to : with the boundary conditions Multiplying (57) by , integrating over , and using the boundary conditions (58), we get We estimate the right-hand side of (59) term by term as follows:
It is shown in  that which implies that for any .
3.2.7. Estimates of
We estimate each term in the right-hand side of (67) as follows
3.2.8. Estimates of
We give the estimates of each term in the right-hand side of (73) as follows
4. The Existence of the Pullback Attractor
In this section, we recall some definitions and lemmas about pullback attractor and prove the existence of -pullback attractor.
Definition 5 (see [11, 13, 33]). Let be two Banach spaces. Then, the family is said to be a -pullback attractor for , if (i) is closed in and compact in for any ,(ii) for any (invariance property),(iii)it pullback attracts every bounded subsets of in the topology of ((,)-pullback attracting), that is, for any nonempty class of bounded subsets , for any .
Definition 6 (see [31, 33]). The process is said to be pullback asymptotically compact if for any and any nonempty class of bounded subsets , any sequence , and any sequence , the sequence is relatively compact in .
Lemma 7 (see ). Let be a continuous process such that is pullback asymptotically compact. If there exists a family of pullback absorbing sets , then has a unique pullback attractor and
Definition 8 (see ). A process is said to satisfy pullback condition (PDC), if for any fixed , any bounded subset of a Banach space and any , there exist a time and a finite-dimensional subspace of such that (i) is bounded, (ii) for any and any , where is a bounded projection and Id is the identity.
Lemma 9 (see ). A process satisfying pullback condition (PDC) is pullback asymptotically compact.
Thanks to the knowledge of functional analysis, we know that the operator with domain is a positive self-adjoint operator with compact inverse. Therefore, the space possesses an orthonormal basis of eigenfunctions of the operator such that where Let , and let be the orthogonal projection onto .
Proof. Let be a bounded subset of , where , and .
Taking the inner product of (1) with , we have From the Young inequality, we obtain By virtue of Lemma 1, we get where Therefore, we obtain for any , and any , if is sufficiently large.
Similarly, we get for any , and any , if is sufficiently large. Hence, for any , and any , if .
The authors of this paper would like to express their sincere thanks to the reviewer for valuable comments and suggestions.
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