The full nonlinear dissipative quasigeostrophic model is shown to have a
unique temporally almost periodic solution when the wind forcing is temporally
almost periodic under suitable constraints on the spatial square-integral of
the wind forcing and the $\beta$ parameter, Ekman number, viscosity and the
domain size. The proof involves the pullback attractor for the associated
nonautonomous dynamical system

Duan, Jinqiao

Kloeden, Peter E.

English

arXiv

AbstractThe full nonlinear dissipative quasi-geostrophic model is shown to have a unique temporally almost periodic solution when the wind forcing is temporally almost periodic under suitable constraints on the spatial square-integral of the wind forcing and the β parameter, Ekman number, viscosity, and the domain size. The proof involves the pullback attractor for the associated nonautonomous dynamical system

Elsevier - Publisher Connector 

Dissipative Quasi-Geostrophic Motion under Temporally Almost Periodic Forcing 

Ž .Journal of Mathematical Analysis and Applications 236, 74]85 1999
Article ID jmaa.1999.6432, available online at http:rrwww.idealibrary.com on
Dissipative Quasi-Geostrophic Motion under
Temporally Almost Periodic ForcingU
Jinqiao Duan
Department of Mathematical Sciences, Clemson Uni¤ersity, Clemson,
South Carolina 29634
E-mail: duan@math.clemson.edu
and
Peter E. Kloeden
FB Mathematik, Johann Wolfgang Goethe Uni¤ersitat,È
D-60054 Frankfurt am Main, Germany
E-mail: kloeden@math.uni-franfurt.de
Submitted by L. Debnath
Received January 26, 1999
The full nonlinear dissipative quasi-geostrophic model is shown to have a unique
temporally almost periodic solution when the wind forcing is temporally almost
periodic under suitable constraints on the spatial square-integral of the wind
forcing and the b parameter, Ekman number, viscosity, and the domain size. The
proof involves the pullback attractor for the associated nonautonomous dynamical
system. Q 1999 Academic Press
Key Words: quasi-geostrophic fluid model; dissipative nonautonomous dynamics;
almost periodic motion; pullback attractor.
1. INTRODUCTION
Ž .The barotropic quasi-geostrophic QG flow model is derived as an
approximation of the rotating shallow water equations by an asymptotic
* This work was partly supported by the National Science Foundation Grant DMS-9704345
and the DFG Forschungschwerpunkt ``Ergodentheorie, Analysis und effiziente Simulation
dynamischer Systeme.'' This work began during a visit to the Oberwolfach Mathematical
Research Institute, Germany.
74
0022-247Xr99 $30.00
Copyright Q 1999 by Academic Press
All rights of reproduction in any form reserved.
ALMOST PERIODIC QUASI-GEOSTROPHIC MOTION 75
expansion in a small Rossby number. The lowest order approximation,
which is also the conservation law for the 0th order potential vorticity,
Žw x.gives the barotropic QG equation 16, p. 234 :
Dc q J c , Dc q bc s n D2c y r Dc q f x , y , t , 1Ž . Ž . Ž .t x
Ž .where c x, y, t is the stream function, b ) 0 the meridional gradient of
the Coriolis parameter, n ) 0 the viscous dissipation constant, r ) 0 the
Ž .Ekman dissipation constant, and f x, y, t the wind forcing. In addition,
Ž .D s › q › is the Laplacian operator in the plane and J f , g s f g yx x y y x y
f g is the Jacobian operator.y x
Ž . Ž .Equation 1 can be rewritten in terms of the relative vorticity v x, y, t
Ž .s Dc x, y, t as
v q J c , v q bc s n Dv y rv q f x , y , t . 2Ž . Ž . Ž .t x
< <For an arbitrary bounded planar domain D with area D and piecewise
smooth boundary this equation can be supplemented by homogeneous
Dirichlet boundary conditions for both c and v s Dc , namely,
c x , y , t s 0, v x , y , t s 0 on › D , 3Ž . Ž . Ž .
together with an appropriate initial condition,
v x , y , 0 s v x , y on D. 4Ž . Ž . Ž .0
ŽThe global well-posedness i.e., existence and uniqueness of smooth
. Ž . Ž .solutions of the dissipative model 2 ] 4 can be obtained similarly as in,
w xfor example, 1, 11, 19; see also 2 . Steady wind forcing has been used in
w x w xnumerical simulations 5 and Duan 10 has shown the existence of time
periodic quasi-geostrophic response of time periodic wind forcing by
means of a Leray]Schauder topological degree argument and Browder's
Ž .principle. In this paper it is assumed that the wind forcing f x, y, t is
w xtemporally almost periodic and a concept of pullback attraction 7, 15 is
used to establish the existence of a unique temporally almost periodic
Ž . Ž .solution of 2 and 3 under appropriate constraints on the model parame-
ters. The main result is
THEOREM 1. Assume that
< <r pn 1 D
q ) b q 1 ,ž /< <2 D 2 p
DUAN AND KLOEDEN76
Ž .and that the wind forcing f x, y, t is temporally almost periodic with its
2Ž .L D -norm bounded uniformly in time t g R by
3r2< <p r r pn 1 D
f ?, ? , t F q y b q 1 .Ž . ( ž /< < < <D 2 D 2 p
Ž . Ž .Then the dissipati¤e quasi-geostrophic model 2 and 3 has a unique
temporally almost periodic solution that exists for all time t g R.
The necessary terminology is presented as required in the text and proof
that follow. Some mathematical preliminaries are stated below, while
dissipativity and strong contraction properties of QG flow are established
in Section 2. Background material on pullback attraction for nonau-
tonomous systems is presented in Section 3 and that for almost periodicity
in Section 4, where it is applied to the QG model under consideration to
complete the proof of Theorem 1.
2 2Ž . k kŽ .Standard abbreviations L s L D , H s H D , k s 1, 2, . . . , are used0 0
w x ² :for the common Sobolev spaces in fluid mechanics 18 , with ? , ? and
5 5 2? denoting the usual scalar product and norm, respectively, in L . We
Ž w x.need the following properties and estimates see also 11 of the Jacobian
operator J: H 1 = H 1 “ L1:0 0
J f , g h dx dy s y J f , h g dx dy , 5Ž . Ž . Ž .H H
D D
J f , g g dx dy s 0, 6Ž . Ž .H
D
5 5 5 5J f , g dx dy F =f =g , 7Ž . Ž .H
D
for all f , g, h g H 1, and0
< <2 D
5 5 5 5 5 5J D f , g Dh dx dy F D f D g Dh , 8Ž . Ž .(H
pD
2 w xfor all f , g, h g H , as well as the Poincare inequality 13 :Â0
< < < <D D2 2 225 5 < < 5 5g s g x , y dx dy F =g dx dy s =g , 9Ž . Ž .H H
p pD D
ALMOST PERIODIC QUASI-GEOSTROPHIC MOTION 77
1 w xfor g g H , and Young 's inequality 13 :0
e 1
2 2AB F A q B , 10Ž .
2 2e
where A, B are nonnegative real numbers and e ) 0.
2. DYNAMICS OF DISSIPATIVE QG FLOW
Ž . Ž .We first show that the equation 2 with boundary conditions 3 is a
Žw x.dissipative system in the sense 14, 18; see also 10 that all solutions
Ž . 2v x, y, t approach a bounded set in the phase space L as time goes to
infinity provided that the L2 norm of the forcing term is uniformly
bounded in time and that the system parameters satisfy the inequality of
Theorem 1. Then we show that the system is strongly contracting under
the restriction on the magnitude of the L2 norm of the forcing term
assumed in Theorem 1.
2.1. Dissipati¤e property
2 2 Ž .Define the solution operator S : L “ L by S v [ v t for t G t ,t, t t, t 0 00 0
Ž . 2 2where v t is the solution of the QG equation in L starting at v g L at0
Ž . Ž .time t . Since the dissipative QG model 2 and 3 are strictly parabolic,0
the solution operators S exist and are compact for all t ) t ; see, fort, t 00w x kexample, 18 . In fact, the S are compact in H for all k G 0 and so, int, t 00
particular, S B is a compact subset of L2 for each t ) t and everyt, t 00
closed and bounded subset B of L2.
Ž .Multiplying 2 by v and integrating over D, we obtain
1 d 2 2 25 5 5 5 5 5v s yn =v y r v q f x , y , t v dx dyŽ .H2 dt D
y J c , v v dx dy y b c v dx dy. 11Ž . Ž .H H x
D D
Ž . Ž .Now H J c , v v dx dy s 0 by 5 and from the Young and PoincareÂD
inequalities we have
1
2 2b c v dx dy F b c dx dy q v dx dyH H Hx xž /2D D D
< <1 D
2 2F b v dx dy q v dx dy ,H Hž /2 p D D
DUAN AND KLOEDEN78
that is
< <1 D 25 5b c v dx dy F b q 1 v , 12Ž .H x ž /2 pD
and by the Poincare inequality again we also haveÂ
pn2 25 5 5 5yn =v F y v . 13Ž .
< <D
Ž .Now assume that the square-integral of the wind forcing f x, y, t with
Ž .respect to x, y g D is uniformly bounded in time, i.e.,
f ?, ? , t F M 14Ž . Ž .
Žfor some positive constant M. This is a mild assumption because a
w x .temporally almost periodic function is bounded in time, see 3 and later .
Then
1 r
2 2f x , y , t v dx dy F f x , y , t dx dy q v dx dyŽ . Ž .H H H2 r 2D D D
M 2 r 25 5F q v .
2 r 2
Ž . Ž . Ž .Putting 12 ] 15 into 11 we obtain
1 d M 22 25 5 5 5v q a v F , 15Ž .
2 dt 2 r
where
< <r pn 1 D
a s q y b q 1 . 16Ž .ž /< <2 D 2 p
Then a ) 0 if we assume that
< <r pn 1 D
q ) b q 1 , 17Ž .ž /< <2 D 2 p
which is in fact the first constraint of Theorem 1. Thus, by the Gronwall
inequality, we have
M 22 2 y2 a t y2 a t5 5 5 5v F v e q 1 y e . 18Ž . Ž .0 2 ra
ALMOST PERIODIC QUASI-GEOSTROPHIC MOTION 79
Hence all solutions v enter the closed and bounded set
M
5 5B s v : v F½ 5'2 ra
in finite time and stay there. The set B is thus an absorbing set of the
system and is positively invariant in the sense that S B ; B for allt, t0
t G t and t g R.0 0
For later purposes note that the solution operator S satisfies S s idt, t t , t0 0 0
 4and S (S s S for any t F t F t , that is S : t G t , t g R ist , t t , t t , t 0 1 2 t, t 0 02 1 1 0 2 0 0
a nonautonomous process or cocycle mapping. In addition, it follows from
Ž .existence and uniqueness theory that t, t , v “ S v is continuous.0 0 t, t 00
Hence, in particular, when the forcing f is independent of time there
exists a global autonomous attractor defined by
A s S B,F0 t , 0
tG0
which is a nonempty compact subset of L2, and is invariant under the
 4autonomous semigroup S : t G 0 in the sense that S A s A for allt, 0 t, 0 0 0
t G 0.
2.2. Strong contraction property
Now consider two trajectories v Ž i. corresponding to initial values v Ž i. g0
B, i s 1 and 2. Note that these trajectories remain inside B. Their
difference dv s v Ž1. y v Ž2. satisfies the equation
dv q J c Ž1. , v Ž1. y J c Ž2. , v Ž2. q b dc s n Dd v y r dv .Ž . Ž .t x
Similarly to the proof above it can be shown from this equation that
1 d 2 2 25 5 5 5 5 5dv q d J dv dx dy q b dc dv dx dy s yn =v y r dv ,H H x2 dt D D
19Ž .
where
d J c , v [ J c Ž1. , v Ž1. y J c Ž2. , v Ž2. .Ž . Ž . Ž .
DUAN AND KLOEDEN80
Ž . Ž .Now from the properties 5 ] 8 of the Jacobian J we have
Ž1. Ž1. Ž2. Ž2. Ž1. Ž2.d J dv dx dy s J c , v y J c , v v y v dx dyŽ .Ž . Ž .Ž .H H
D D
Ž1. Ž1. Ž2. Ž2. Ž2. Ž1.s J c , v v dx dy q J c , v v dx dyŽ . Ž .H H
D D
Ž1. Ž1. Ž2. Ž2. Ž1. Ž2.s J c , v v dx dy y J c , v v dx dyŽ . Ž .H H
D D
Ž1. Ž2. Ž1. Ž1. Ž2.s J c y c , v v y v dx dyŽ .Ž .H
D
Ž1.s J dc , v dv dx dyŽ .H
D
Ž1.s J v , dc dv dx dyŽ .H
D
Ž1.s J Dc , dc Dd c dx dyŽ .H
D
< <2 D
Ž1. 5 5 5 5F Dc Dd c Dd c(
p
< <2 D 2Ž1. 5 5s v dv ,(
p
Ž . Ž1.where in the last two steps, we have used 8 with f s c , g s h s dc ,
and the fact Dd c s dD c s dv. Using this and noting that v Ž1. is in the
Ž1. '5 5positively invariant absorbing set B so v F Mr 2 ra , we have
< <2 D 2Ž1. 5 5d J dv dx dy F v dv(H
pD
< <2 D M 25 5F dv( 'p 2 ra
< <D 25 5s M dv . 20Ž .(
p ra
ALMOST PERIODIC QUASI-GEOSTROPHIC MOTION 81
Ž . Ž . Ž .Then from Eq. 19 , using 20 and 12 , we obtain
1 d 2 2 25 5 5 5 5 5dv s yn =dv y r dv y d J dv dx dyH2 dt D
y b dc dv dx dyH x
D
2 25 5 5 5F yn =dv y r dv q d J dv dx dyH
D
q b dc dv dx dyH x
D
< <pn D2 2 25 5 5 5 5 5F y dv y r dv q M dv(< <D p ra
< <1 D 25 5q b q 1 dvž /2 p
5 5 2F yg dv ,
where
< < < <pn 1 D D
g [ r q y b q 1 y M .(ž /< <D 2 p p ra
< <'Note that g ) a y D rp ra M. Thus, g ) 0 if we assume that
p r
3r2f ?, ? , t F M - a , 21Ž . Ž .( < <D
Ž . Ž .for all t g R. Here a is defined in 16 , so 21 holds because of the
assumption on the L2 norm of f in Theorem 1. This gives
2 y2g t5 5dv t F v e “ 0 as t “ ‘,Ž . 0
for solutions starting within the positively invariant absorbing set B. This
is the desired strong contractive condition. This means there is a unique
U Ž .solution v t in B to which all other solutions converge. This solution
U Ž .v t can be determined by the pullback convergence to be discussed in
the following two sections.
DUAN AND KLOEDEN82
3. NONAUTONOMOUS DYNAMICAL SYSTEMS
In order to show existence of temporally almost periodic solutions, we
need some results from the theory of nonautonomous dynamical systems.
Consider first an autonomous dynamical system on a metric space P
 4described by a group u s u of mappings of P into itself.t t g R
Let X be a complete metric space and consider a continuous mapping
F : Rq= P = X “ X
satisfying the properties
F 0, p , ? s id , F t q t , p , x s F t , u p , F t , p , xŽ . Ž . Ž .Ž .X t
for all t, t g Rq, p g P, and x g X. The mapping F is called a cocycle on
X with respect to u on P.
The appropriate concept of an attractor for a nonautonomous cocycle
system is the pullback attractor. In contrast to autonomous attractors it
consists of a family subsets of the original state space X that are indexed
by the cocycle parameter set.
Ã  4DEFINITION 1. A family A s A of nonempty compact sets of Xp pg P
is called a pullback attractor of the cocycle F on X with respect to u on Pt
if it is F-invariant, i.e.,
F t , p , A s A p for all t g Rq, p g P ,Ž .p u t
and pullback attracting, i.e.,
lim HU F t , u p , D , A s 0 for all D g K X , p g P ,Ž . Ž .Ž .X yt p
t“‘
Ž .where K X is the space of all nonempty compact subsets of the metric
Ž .space X, d .X
Here HU is the Hausdorff semi-metric between nonempty compactX
subsets of X, i.e.,
HU A , B [ max dist a, BŽ . Ž .X ag A
s max min d A , bŽ .ag A bg B X
Ž .for A, B g K X .
The following theorem combines several known results. See Crauel and
w x w x w x w xFlandoli 9 , Flandoli and Schmalfuû 12 , and Cheban 6 as well as 7, 15
for this and various related proofs.
ALMOST PERIODIC QUASI-GEOSTROPHIC MOTION 83
THEOREM 2. Let F be a continuous cocycle on a metric space X with
respect to a group u of continuous mappings on a metric space P. In addition,
suppose that there is a nonempty compact subset B of X and that for e¤ery
Ž . Ž . qD g K X there exists a T D g R , which is independent of p g P, such
that
F t , p , D ; B for all t ) T D . 22Ž . Ž . Ž .
Ã  4Then there exists a unique pullback attractor A s A of the cocycle Fp pg P
on X, where
A s F t , u p , B . 23Ž . Ž .F Dp yt
q t)ttgR
qtgR
Moreo¤er, the mapping p ‹ A is upper semicontinuous.p
w xMoreover, in 7 it is shown that the pullback attractor consists of a
single trajectory when the cocycle dynamics are in fact strongly contract-
ing.
THEOREM 3. Suppose that the cocycle F in Theorem 2 is strongly
contracting inside the absorbing set B. Then the pullback attractor consists of
 U Ž .4 U Ž .singleton ¤alued sets; i.e., A s a p , and the mapping p ‹ a p isp
continuous.
4. ALMOST PERIODICITY
Ž .A function w : R “ X, where X, d is a metric space, is called almostX
w xperiodic 3 if for every « ) 0 there exists a relatively dense subset M of R«
such that
d w t q t , w t - « ,Ž . Ž .Ž .X
for all t g R and t g M . A subset M : R is called relati¤ely dense in R if«
there exists a positive number l g R such that for every a g R the interval
w x w xa, a q l l R of length l contains an element of M, i.e., M l a, a q l
/ B for every a g R.
The QG solution operators S form a cocycle mapping on X s L2t, t0
with parameter set P s R, where p s t , the initial time, and u t s t q t,0 t 0 0
the left shift by time t. However, the space P s R is not compact here.
Though more complicated, it is more useful to consider P to be the
 4closure of the subset u f , t g R , i.e., the hull of f , in the metric spacet
2 Ž 2Ž .. 2Ž . 2Ž .L R, L D of locally L R -functions f : R “ L D with the metricloc
‘
N 2yNd f , g [ 2 min 1, f t y g t dt ,Ž . Ž . Ž .Ý H(P ½ 5yNNs1
DUAN AND KLOEDEN84
Ž . Ž .with u defined to be the left shift operator, i.e., u f ? [ f ?q t . By at t
w xclassical result 3, 17 , a function f in the above metric space is almost
periodic if and only if the hull of f is compact and minimal. Here minimal
means nonempty, closed, and invariant with respect to the autonomous
dynamical system generated by the shift operators u such that with not
proper subset has these properties. The cocycle mapping is defined to be
Ž .the QG solution v t starting at v at time t s 0 for a given forcing0 0
mapping f g P, i.e.,
F t , f , v [ S f v ,Ž .0 t , 0 0
where we have included a superscript f on S to denote the dependence on
Ž .the forcing term f. This dependence is in fact continuous . The cocycle
property here follows from the fact that S f v s Su t0 f v for all t G t ,t t 0 tyt , 0 0 00 0
t g R, v g L2, and f g P.0 0
THEOREM 4. Let the assumptions of Theorem 1 hold. Then the dissipati¤e
Ž . Ž . UQG model 2 ] 3 has a unique almost periodic solution v defined by
vU t [ aU u f , t g R,Ž . Ž .t
 U Ž .4where a p is the singleton ¤alued pullback attractor-trajectory of the
Ž . Ž . 2 Ž 2Ž ..cocycle F t, f , v on L D , P is the hull in the metric space L R, L D0 2 loc
of the almost periodic forcing term f and the u are the left shift operators on P.t
This is proven as follows. By Theorems 2 and 3 the pullback attractor
 U Ž .4exists, consists of singleton valued components a p and the mapping
U Ž . U Ž .p ‹ a p is continuous on P. In fact, the mapping p ‹ a p is uni-
2 Ž 2Ž ..formly continuous on P because P is a compact subset of L R, L Dloc
due to the assumed almost periodicity. That is, for every « ) 0 there exists
Ž . 5 U Ž . U Ž .5 Ž .a d « ) 0 such that a p y a q - « whenever d p, q - d . NowP
Ž .let the point p s f , the given temporal forcing function be almost
Ž .periodic and for d s d « ) 0 denote by M the relatively dense subset ofd
Ž .R such that d u p, u p - d for all t g M and t g R. From this andP tqt t d
the uniform continuity we have
U Ua u p y a u p - « ,Ž . Ž .tqt t
U UŽ . Ž .for all t g R and t g M . Hence t ‹ v t [ a u p is almost peri-d Ž« . t
odic. It is unique as the single-trajectory pullback attractor is the only
trajectory that exists and is bounded for the entire time line.
This completes the proof of the main result, Theorem 1.
REFERENCES
1. V. Barcilon, P. Constantin, and E. S. Titi, Existence of solutions to the Stommel]Charney
Ž .model of the gulf stream, SIAM J. Math. Anal. 19 1988 , 1355]1364.
ALMOST PERIODIC QUASI-GEOSTROPHIC MOTION 85
2. A. F. Bennett and P. E. Kloeden, The dissipative quasigeostrophic equations, Mathe-
Ž .matika 28 1981 , 265]285.
3. A. Besicovitch, ``Almost Periodic Functions,'' Dover, New York, 1954.
4. J. Brannan, J. Duan, and T. Wanner, Dissipative quasigeostrophic dynamics under
Ž .random forcing, J. Math. Anal. Appl. 228 1998 , 221]233.
5. P. Cessi and G. R. Ierley, Symmetry-breaking multiple equilibria in quasigeostrophic,
Ž .wind-driven flows, J. Phys. Oceanography 25 1995 , 1196]1205.
6. D. N. Cheban, Global attractors of infinite-dimensional nonautonomous dynamical
Ž .systems, Iz¤ . Akad Nauk RM. Mat. 25, No. 3 1997 , 42]57.
7. D. N. Cheban, P. E. Kloeden, and B. Schmalfuû, Pullback attractors in dissipative
nonautonomous differential equations under discretization, to appear.
8. H. Crauel, A. Debussche, and F. Flandoli, Random attractors, J. Dyn. Differential
Ž .Equations 9 1997 , 307]341.
9. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory
Ž .Related Fields 100 1994 , 1095]113.
10. J. Duan, Time-periodic quasigeostrophic motion under dissipation and forcing, Appl.
Ž .Math. Comput., 102 1999 , 121]127.
11. V. P. Dymnikov and A. N. Filatov, ``Mathematics of Climate Modeling,'' Birkhauser,
Boston, Cambridge, MA, 1997.
12. F. Flandoli and B. Schmalfuû, Weak solutions and attractors of the 3D Navier Stokes
Ž .equation with nonregular force, Stochastics Stochastic Rep. 59 1996 , 21]45.
13. D. Gilbarg and N. S. Trudinger, ``Elliptic Partial Differential Equations of Second
Order,'' 2nd ed., Springer-Verlag, BerlinrNew York, 1983.
14. J. K. Hale, ``Asymptotic Behavior of Dissipative Systems,'' Am. Math. Soc., Providence,
RI, 1988.
15. P. E. Kloeden and B. Schmalfuû, Nonautonomous systems, cocycle attractors and variable
Ž .time-step discretization, Numer. Algorithms 14 1997 , 141]152.
16. J. Pedlosky, ``Geophysical Fluid Dynamics,'' 2nd ed., Springer-Verlag, BerlinrNew York,
1987.
17. G. R. Sell, ``Lectures on Topological Dynamics and Differential Equations,'' Van Nos-
trand-Reinbold, London, 1971.
18. R. Temam, ``Infinite-Dimensional Dynamical Systems in Mechanics and Physics,''
Springer-Verlag, New York, 1988.
19. J. Wu, ``Inviscid Limits and Regularity Estimates for the Solutions of the 2-D Dissipative
Quasigeostrophic Equations,'' IMA Preprint Series 1441, Inst. of Math. Appl., Southend-
on-Sea, 1996.


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Dissipative Quasigeostrophic Motion under Temporally Almost Periodic Forcing

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