Averaging principle for dissipative dynamical systems with ... - Turpion

Sbornik: Mathematics 187:5 635-677

©1996 RAS(DoM) and LMS

Matematicheskii Sbornik 187:5 15-58

UDC 517.9

Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides A. A. Hyin Abstract. We consider two-dimensional Navier-Stokes equations and a damped non-linear hyperbolic equation. We suppose that the right-hand sides of these equations have the form f(u>t), w » 1. We suppose also that / has an average. The main result of the paper is a proof of a global averaging theorem on the convergence of attractors of non-autonomous equations to the attractor of the averaged autonomous equation a s w - K » . Bibliography: 29 titles.

Starting from the fundamental work of Bogolyubov [1] the averaging theory for ordinary differential equations has been developed and generalized in a large number of works (see [2]-[4] and the references therein). Bogolyubov's main theorems have been generalized in [5] to the case of differential equations with bounded operator-valued coefficients. Some problems of averaging of differential equations with unbounded operator-valued coefficients have been considered in [6]-[8] in the framework of abstract parabolic equations. In this paper we shall be dealing with an equation of the form

where / is a given right-hand side and w > 0 is a large dimensionless parameter. We restrict ourselves to a consideration of two characteristic examples: the 2-dimensional Navier-Stokes system (1) from §2 and the damped non-linear hyperbolic equation (1) from § 3; one of these two equations is always meant by the general notation above. We assume that the average exists (in a sense to be specified in each particular case) 1 f* lim t-t-OO t, J/Q f{s)ds = f0. We consider the averaged equation

This work was supported in part by the International Science Foundation (grant no. J6T100). AMS 1991 Mathematics Subject Classification. Primary 34G20, 35Q30, 35L70.

636

A. A. Ilyin

and prove the averaging principle on a finite time interval [0, T], the so-called first Bogolyubov theorem. If the initial data are such that u(0) = u(0), then ||«(t) - ti(t)|| < VT(OJ) -> 0,

u -> oo.

We prove the averaging principle on the entire real axis, the so-called second Bogolyubov theorem: if there exists a stationary solution uo of the averaged equation and this stationary point is hyperbolic, then for u> > u>o in a small neighbourhood of this point there exists a solution u*(t) = u*,,w,At) of the original equation that is bounded on the entire real axis and is conditionally exponentially stable. Moreover, this solution is almost periodic if / is almost periodic. In the autonomous case the semigroups of solution operators St corresponding to the Navier-Stokes system and the hyperbolic equation are typical examples of uniformly compact and asymptotically compact semigroups. A systematic study of their properties as t ->• oo is given in [9]-[14] where it was shown that these semigroups possess a compact attractor A in the appropriate phase space. The theory of attractors for non-autonomous dynamical systems with almost periodic time symbols or, in particular, with almost periodic right-hand sides / ( • ) has recently been developed in [15]-[19] (see also [20] and [21]), where it was shown that the initial non-autonomous equation possesses an attractor A = A(u). We show that the attractors A(u) depend upper semicontinuously on u> as OJ -» oo: dist(.A(u;),.A) := sup inf_||x - y\\ -+ 0,

u -* oo.

x£A(u) y&A

This is a corollary of the first averaging theorem and the Lyapunov stability of the attractor A. One can call this result a global averaging theorem. We observe that this theorem is quite meaningful in the finite-dimensional case as well. The second averaging theorem also has applications to attractors. In the autonomous case all stationary points belong to the attractor A. In the non-autonomous case the corresponding statement is as follows. An attractor of the non-autonomous equation contains the closure of the range of values of the almost periodic solution u*ttu.y

[(J

/ —> oo. The averaging theorem on a finite interval is essential in the proof; moreover, the estimate of proximity of two individual solutions must be uniform with respect to the initial data from bounded sets (absorbing balls of sufficiently large radius) in the corresponding phase space. This can be proved only in a phase space with sufficiently strong norm: Ha(fl), Certain results on the structure of the attractor of the non-autonomous NavierStokes equations in i?a-spaces are contained in Theorem 4.7. A theorem of averaging on the entire axis is used there. Such a theorem for the Navier-Stokes equations was proved by Zhikov [7], [22] giving the proximity of the solutions in the L2-norm. Convergence in the L2-norm is not enough for applications to attractors, therefore we prove the theorem on averaging of the Navier-Stokes equations on the entire axis in a form that is suitable for application to attractors. Our proof is different from that of [7] and is connected rather with the original proof of Bogolyubov [l]-[3]. Furthermore, our approach allows us to prove the theorem on averaging of the damped hyperbolic equation on the entire axis in exactly the same way. § 1. Preliminaries Some results from the theory of semigroups of linear operators [6] and almost periodic functions [7] are collected below. Definition 1. A closed operator A with domain D(A) that is dense in a Banach space X is called a sectorial operator if for some a € E and
v> = {X 6 C, 7T ^ | arg(A - a)\ ^ 0 small enough \\Bx\\ê\\Ax\\+c(e)\\x\\. Then A + B is also a sectorial operator. For a sectorial operator A the analytic semigroup of linear bounded operators in X is defined and denoted by e~At, t ^ 0. We have

638

A. A. Ilyin

Definition 2. Let A be a sectorial operator with Recr(.A) > 0. For a e (0,1) we define fractional powers of A as follows: Aa = {A~a)-1,

where A~a = -j-r

ta-le~Atdt.

f

r(a) Jo

a

The corresponding domains D(A ) are Banach spaces With norm given by IMIa := \\x\\D{A") = \\Aax\\. Theorem 2. The following estimates are valid: \\e-At\\x-+x < Ke'at, \\A°e-M\\x^x < ~e~at,

12 0,

(1)

t > 0.

(2)

Theorem 3. Given two sectorial operators A and B in X, let D(A) = D(B), Rea(A) > 0, Re 0, and for some a € [0,1) let the operator (A - B)A~a be bounded in X. Then for every /3 £ [0,1) the two norms being equivalent. We now turn to the almost periodic functions. Definition 3. A continuous function / : K -* X is called almost periodic (a.p.), if for every e > 0 there exists a number I = l(e) > 0 such that each interval (T, T + l) contains a point r = TE (called an almost period) satisfying the inequality

If / depends on other arguments, then the above inequality holds uniformly with respect to them. It follows from the theory of a.p. functions that there exists a countable set of numbers Xa for which

These numbers {Aa} are called the Fourier exponents of / . Definition 4. A countable set of numbers {uja} is called the frequency basis of an a.p. function / and is denoted by M/ if every Xa can be uniquely represented by a linear combination of the numbers u)a with integer coefficients. Definition 5. For a given a.p. function / the sequence {tm} is called f-recurrent if p||/(

m

) / ( ) | |

m->oo.

t€R

Theorem 4. Given two almost periodic functions f and g, suppose that every f -recurrent sequence is also a g-recurrent sequence. Then the frequency basis of g is contained in that of f : Mg C Mf.

Averaging principle for dissipative dynamical systems

639

§ 2. Navier-Stokes equations 1. Averaging on a finite time interval. We consider the two-dimensional Navier-Stokes system 2

dtu + ^2 UidiU = uAu - Vp + f(ut),

(1)

i=l

divu = 0,

w| 9 n =0,

where ft 0. This amounts to

(B( u,v),h) =b(u,v,h) = / UidiVjhj However, Hs C L2+fi, /x > 0, and this is enough to apply Holder's inequality to the integral with u G Lp, p -* 00, dv € L2, h 6 L2+Ii. Therefore B(u,v) C #_,$ C

D{A-S'2), 6 > 0. The estimate (8') follows from the embeddings £>(A1/2+ Tg is valid. Then

6. Let £o be so small that

if r < T$. Let (3 > — | . Since r^ does not depend on e, we let 0. We obtain lT [Tfat)- 0 when e ->• 0. (15) Jo 1/2 The second term on the right in (14) is less than ^3/2- / 3 e 1 / 2 + / 3 f m i n ( M / 3 , ^ ( u ) ) ^ - 1 / 2 d« Jo ^ K3/2_0M0^2+0 r^ Jo -l

=: G02(T,e),

(16)


643

where for any n > 0 we have chosen rM so large that O0(T) < /x when T > T^. Letting /i -> 0 and then e->0we obtain )Ô,

e-K).

Thus, by (12)-(16) we obtain the following inequality: \\Z(T)\\I/2

< K{6)e1'2-5 f(T-s)-^2-s\\z(s)\\1/2ds

+ G0(T,6),

(17)

where K{8) = 2RcsK1/2+s and G73 = G0i + G02 -> 0, e ->• 0. Lemma 2 (see [6]). Let 7 E (0,1] and /or t 6 [0, T] u{t) â + b

Jo

(t-s)'y-1u(s)ds.

Then where the function Ey(z) is monotone increasing and E~,{z) ~ 7~ 1 e z as z —> CXD.

Applying this lemma to the inequality (17) on r e [0, T/e] we obtain

Since S is at our disposal, we finally obtain ll*(r)||i/2 < G0(T,e)

min El/2_S{T(T{\

- S)2RcsK1/2+sf(1-2S)).

(18)

0 0 and u(0) = u(0) 6 BD{A)(Ro), \\U(T) - U(T)\\D(A)

e -K).

then < i#(e) -> 0,

£ -> 0.

2. Averaging on the entire axis. Consider equation (9) once again: dTu + eB(u, u) - -evAu + e(p(r).

(20)

All the hypotheses concerning the data of the problem are the same as those in subsection 1; in particular, the average exists in the sense of (10). It is known that equation (11) has at least one stationary solution u0 6 D{A1/2): B(uo,uo) + uAuo = 0. / / a - 7 < 1, tfien tfte equation (30) /10s 0 unique solution h bounded in D{Aa) : (

( ) ) -

(31)

Proof. We set

G(t-s)f(s)ds,

(32)

J—oo

where

Using (29) we see that

J—oo OO

c2a Jt

However, on H- the inequality Re«r/f_(L_) > a > 0 holds and L\0H- C Hence, ReaH-(L\0) > a + Ao > 0. It is also clear that D(L_) = I Therefore by Theorem 3 from § 1, for ft € JT_ and 7 > 0


649

Using this together with (2) from § 1, (29) and the boundedness of the projection P_ in D(Ay), we estimate the first term:

The operator L+ is finite-dimensional, hence, the second term is bounded by #2(c*,7)ll/llz,~,(R;D(^)), w h i c h P r o v e s ( 31 )One can easily verify that ft is a solution of (30), that is, a continuous solution of the integral equation corresponding to (30). We claim that this solution is unique. This amounts to proving that the homogeneous equation dth + Lh = 0 has only the trivial solution. Let ft0 = ft(0) ^ 0. Then h{t) = /»_(') + h+(t) = e-L-%- + e L As t -> oo the first term is bounded, and since h(t) is bounded, it follows that the second term is also bounded. As t -> -oo the second term is bounded, hence, the first one is also bounded, that is, both h+(t) and ft_(£) are bounded uniformly in t € R. However, Rea(L+) < -a < 0 and Rea(L_) > a > 0. Therefore

HM0)|| < tfe-°'IIM-*)ll. o o , ||M0)|K Ke-at\\h+(t)l

*>0.

Letting t —> oo, we see that /i+(0) = h-(0) = 0, which proves the lemma. R.everting in (23) to the original time variable t = er, we obtain dth + Lh = Q(h,e,t), where Q(h,e,t) - F(h,e,t/e),

(33)

and by (27),

IIQ(fti) - Q{h2)h < *ay(e,p)\\hi ~ h2\\a, \\Q(0,e,t/e)\\7 < Na7(e), where ||/ii,2|U ^ P,an^ AQ7(e,/£»), Nai(e)

{3

'

-> 0, as e,p ->• 0.

Lemma 6. Let Q satisfy (34) and let a — 7 < 1, 7 ^ 0. Then for e < £0 equation (33) has a unique bounded solution h* with the following properties:

1. Wh'Www-))

^ 0.

(35)

2. There exists an initial manifold M_ in D(Aa), codimM_ = iV, such that if ft0 G M_ and \\ho\\a ^ p for p small enough, then the solution h(t) of equation (33) with h(0) = h0 satisfies the estimate ||ft(t)-ft*(*)IU^Jft:e-al 0 ,

(36)

as t -¥ +00. In particular, if N = 0, then the solution h* is asymptotically stable. There exists an initial manifold M+ in D(Aa), dimM+ = N, such that (36) holds as t ->• - 0 0 . 3. / / the function Q{h,e, •): E -¥ D(A'r) is almost periodic, then the solution h* is almost periodic with frequency basis contained in that of Q.

650

A. A. Ilyin

Proof. As in Lemma 5 we reduce equation (33) to the equivalent integral equation h{i) = f

G(t- s)Q(h(s),e,s)ds =: 7h(t).

(37)

J—oo

By (31) and (34) )

< sup /

G(t - s) (Q(0,e, s) + {Q(h{s),e, s) - Q(0,e, s)j) ds\

teR||J-oo

< K(a,j){Naê)

||a

+ X^(e,p)\\h\\Cb{R.MAa))).

(38)

Similarly,

)

(39)

for e, p small enough. Thus we can choose p and e so small that 3" is a contraction map taking a ball in Cft(R; D(Aa)) of radius p into itself. By the Banach contraction principle, 3" has a unique fixed point 1h* = h\ in other words, (37) holds and /i*is a D(.Aa)-bounded solution of the equation (37). It also follows from (38) that (35) holds, which proves the first statement of the theorem. Let us prove statement 3. Let {tm} be a Q-recurrent sequence (see Definition 5 from §1). Then sup \\h*(t + tm) - h*(t)\\a teR

= sup|/

G(t - s)[Q{h*(s + tm),e,s + tm) - Q(h*(s + tm),e,s) t

i*(s + tm),e,s)-Q(h*(s),e,s)]ds

(a,y)sup \\Q(h*(s + tm),e,s + tm)- Q(h*{s + tm),e,s)\U seR

+ K(a, j)\ay(e,p)

sup \\h*(t + tm) - h*(t)\\a, R

whence, taking into account (39), we obtain t€R

< 2AT(a,7)sup \\Q(h*{t + tm),e,t + tm) - Q{h"{t + tm),e,t)\\y ^ 2K(a, 7)e m -+ 0,

m -* oo,


651

that is, the sequence {tm} is h*-recurrent. Hence by Theorem 4 from § 1, statement 3 follows. Let us prove statement 2. We set h(t) = h*(t) + y(t). Then y satisfies the equation ,t), (40) where

#(y,e,0 = Q(h* +y,e,t) - Q(h\e,t). The function $ has the following properties for ||yi,2||a ^ P#(0,6,0 = 0,

||#(l/i)-*(» 2 )|| 7 0,

(42)

where the operator-valued function G(t) is defined in (32) and satisfies the estimate f-ocp—a't

for some a' > 0. For technical reasons we rewrite this estimate in the form ||A a G(0l|ij-nr < Ka\

M

'

'

(43)

for some a, b > 0, a + b < a'. Suppose that for e < EQ, p < po the Lipschitz constant A a7 (e, p) is so small that

( ~ 1 r ( 7 - a + 1) + J L ) ^ I , and let We solve (42) using the method of successive approximations:

yo{t,a) = 0 , yp(t,a)=G{t)a+

f°° / G{t -

(44)

652

A. A. Ilyin

Then

\\yi\\a4Ko\\a\\ae-at, hence, for p = 1

Using (43), (41) and (44), we find that

\\yP+i(t,a)-yp(t,a)\\a AaG(t - s) [*(«, yp(s, a)) - *(«, j / p _ ! («, a))] ds

< AQ7irQ_7 (6-1-^(7 - a + 1) + (a + 6)"1) J^-||o|U e - ot

Therefore yp(t,a) -> y(t,a) and \\y(t,a)\\Q
oo. We put t = 0 in (42) and obtain y(0 ,a)

= a-+

r°° G(-s)$(y($,a),s)ds € 5_, Jo

where S- is some manifold, namely, the graph of a function in H = H+ + Hover H-:

G(-a)*(y(8,a),a)d8, o_|, Thus, for j/o 6 SL the solution j/(f) of equation (40) with j/(0) = j/o tends to 0 as t ->• oo, that is, (36) holds with Af_ = 5_ + A*(0). In the case when t -»• — oo we consider the following integral equation with a = a+ and £ < 0:

y(t, a) = G(t)a+ + /

G(t - s)$(y(s, a), s) ds.


653

Using a similar argument we find that the solution y(t) of equation (40) with 2/(0) ES+CH++H-, where

S+ = |-a+, J

G(-s)*(y(s,a),s)ds\,

tends exponentially to 0 as t -* - c o . The manifolds 5+ and 5_ (in fact hypersurfaces over H- and H+) contain the origin and intersect H+ and i/_ at a small angle tending to 0 as e -» 0. The proof is complete. Theorem 2 (Averaging principle on the entire axis). Suppose that in the NavierStokes system (9) written in the standard form, the function
0,

e ->0.

2. In the ball Ba(p) there exists a stable initial manifold M-, codimM_ = N, such that if u(0) G M- and T -¥ oo, then \\u(r) - u*(T)\\a < ce-«^\\u(0) - «*(0)|| a ;

(45)

and there exists an unstable manifold M+, dimM + = N, such that if u(0) € M+, then the inequality (45) holds as T -* -oo. In particular, if N = 0, then the solution u* is asymptotically stable. 3. If the right-hand side - \ , then the theorem is valid for every a € (|,/3 + 1); if 0 > 0, then the theorem is valid for a — 1. Proof. All the assertions of the theorem follow from the representation

u(r,e) = uo + h(r,e) - ev(r,e) and Lemmas 3, 6. Remark 1. In terms of equation (4) we merely have to put r = u>t in all the statements of Theorem 2. All the statements remain valid for u*(uit). For example, if V?(T) is T-periodic, then the solution u*(u}t) of equation (4) is T/w-periodic. To conclude this section we present a result which is a global analogue of Theorem 2 when the averaged right-hand side is sufficiently small.

654

A. A. Ilyin

Theorem 3. Suppose that the average (10) exists and let fl ^ 0. Suppose further that the averaged right-hand side is such that ~ « 2-4> \

(46) K

Co

'

where Co is the constant from the estimate (2) and Ai is the first eigenvalue of the Stokes operator A. Then the stationary solution uo is unique, Rea(L) > c > 0 ana", in addition to the assertions of Theorem 2, for every R > 0 and VQ, \\vo\\a ^ R, the following inequality is valid:

||tt(r)-u*(T)|U^C(fl)c- OT ||u(0)-ti*(0)|| a ,

T->OO,

(47)

where U(T) = u(r,e) is the solution of equation (9) with u(0) = vo and e < eo(R)If/3 = 0, then the theorem is valid for every a e (|, 1); if/3 > 0, then the theorem is valid for every a € (§, 1]. Proof. We claim that for the averaged equation (11') \\u(t)-uo\\^ce-kt\\u(0)-uo\\,

k>0,

t>0,

(48)

where uo is the stationary solution of equation (11). In fact, v(t) = u(t) — UQ satisfies the equation

dtv + vAv + B(v, v) + B(v, u0) + B(uo,v) = 0. Taking the scalar product with v we obtain dt\\vf

+ 2v\\Vvf

= 2b(v,v,u0) < 2cb||v|| ||V«|| ||Vtio||

It is well known that the stationary solution uo satisfies the estimate

where the Poincare inequality was used: \ 2 (0- Q )|| /||2

0. In fact, we already know that the spectrum is discrete. Let vAh + B(h, u0) + B(uQ, h) = Xh,

h = hi + ih2.

Taking the scalar product with h* = hi — ih2 and separating out the real part, we find: = H|V/i||2 - b(hlthuuo)

-

b(h2,h2,u0)

This shows that if (46) holds, then for 6 £ (0, u) small enough the expression in brackets is positive; hence, ReA^-^->0.

(49)

ZAl

In view of (48), the point UQ is the (H, H)-attractor of the semigroup generated by the system (11')- (Definitions of attractors and their basic properties are given in §4.) It is shown in [9] that for (yla))-attractor. In fact, there exists a local solution St: D{Aa) -> D(Aa), t < t0. This follows from (6') and the contraction mapping principle by analogy with [6], Theorem 3.3.3. However, the D(Aa)-noTm of the solution Stu0 is bounded for t > 0, and therefore the solution exists for all t, since St maps sets that are bounded in H D D(Aa) into sets that are compact in D(A) C D{Aa) [9]. This also provides the existence of a compact absorbing set in D(Aa). Hence, for every ball Ba(R) \\StBa(R) - uo\\a < l*(t)-* 0,

t-K».

Let for t > To, fJ,(t) < p/2, where p is the same as in Theorem 2. We consider trajectories of the systems (10) and (11) starting from a point Vo £ Ba(R). By Theorem 1 on the interval T € [0, T0/e] the inequality ||u(r) - w(r)|| a ^ p/2 holds if e < e0 is small enough. Therefore, for r 0 = T0/e the inequality ||u(r0) -u o || Q < p is valid. Taking into account the fact that (49) implies that the ball Ba(uo, p) belongs to the stable manifold M_, we see by Theorem 2 that from this moment onwards \\U*{T -

r0) -

U(T

- ro)\\a ^ ce-a^-^\\u*(T0)

- u(ro)\\a.

On the finite interval [0, To/e] we majorize (47) by some large constant C{R) and obtain (47) for all r > 0. The proof is complete.

656

A. A. Ilyin

Remark 2. In the course of the proof of Theorem 3 we have shown that the semigroup St generated by the autonomous equation is uniformly bounded in D(Aa) and has an absorbing set there. If /? ^ 0, then the estimates (13)-(16) are valid with respect to the D(Aa)-noTm as well. Therefore the theorem on averaging on a finite interval (Theorem 1) is valid in the space D(Aa) for every a G [|, 1). Remark 3. The first eigenvalue satisfies Ai ^ 27r/|fi| where |fi| is the area of Q [27]. Therefore (46) holds for (3 = 0 if

V*

Co

§ 3. Non-linear dissipative hyperbolic equation 1. Averaging on a finite interval. In this section we shall be dealing with the non-autonomous damped second-order hyperbolic equation with rapidly oscillating right-hand side d?u 4- 7&u = Au — f(u) + (p(u)t), , , (1) u\9n=0, fi 0 . We consider the case of a three-dimensional domain Q with dCl 6 C 2 . Suppose that the non-linear interaction term / satisfies the formulae / G C 2 (E), /(0) = 0, and that the following inequalities are valid [9]: F(u) ^ -(Ai - e)u2 - C,

e > 0,

(2X)

F(u) < Cf(u)u + Ci + i(Ax - £)u2,

e > 0,

(2 2 ) (23) (24)

/? < oo, where F(u) = /

f(v)dv,

(25)

C,C\ > 0 and Ai > 0 is the first eigenvalue of the

Dirichlet problem for the operator - A in fl. All the conditions above are satisfied for a third order polynomial with a positive leading coefficient and for f(u) = sin(u) as well. It is convenient to rewrite equation (1) as the system (3)

a p = - 7 P + Au-/(u) +¥ >M) or dty - -Ay-g(y)

+ il)(iJt),

(4)

where V = (u,p) , A

0i|

= ~\A

-7/ '

_/..\

/r>

il

\\T

9(y) = (o,f(u)) = (0Mt))T


657

The superscript T will be omitted in the sequel. Changing the variable r = ut, LJ~X = e, we obtain (5)

We define the following function spaces E0 = {y = («,p); u e H^(n), p €

\\y\\2Eo = liv«|| 2 + ||p|| 2 , EI = {y = («,p); u € # 2 (fi) n Hl((l), p G HQ' Hyll^ = ||An||2 + ||Vp||2, E2 = {y = (u,p); u G F 3 n HQ1 n {Au 6 Fp1}, p € H2 n Suppose that the average exists in the following sense: y(r), ¥>o G H2(SI)C\HQ (ft) and i

J

rt rt+T /

||

(l+T^r 1 )- 1 / 2 ,

Jt

(6)

where M > 0, CT(T) -»• 0, T -»• oo. Since Aip — (—V'l

(1 + T'Af

+T it follows from (6) that t+T

(7)

where t/>0 = (0,Vo)Along with (5) we consider the averaged system dTy = e(-Ay-g(y)+ip0),

(8)

»(0) = 2/o or

(8')

tf(0) = Ifti. The evolution operator of the linear homogeneous equation dty + Ay = 0, V(O) = 0

(9)

satisfies the following estimates [9], [13], [14]: i = 0,l,2,

a > 0.

(10)

658

A. A. Ilyin

Lemma 1. // conditions (2) hold, then the function g(y) is a bounded Lipschitz map from Ei into E\: \\g{yi) - 0 - *I>(T - t))dtds

Jo

Ei

: e2K f e~£ass min(M, a(s)) ds Jo . e2KMs0(S)2 5K_ a Letting S -» 0 and then e -> 0 we obtain (25). Finally, let {r m } be a V'-recurrent sequence. Then /•oo Tm)

V{T)

= / Jo

e ~ e A s (ip(T - s ) -

)) ds.

Hence, Sup \\v(T + Tm) - V{T)\\EX r€R

^ SUp \\ip(r + Tm)

-

,K{ea)-\

r€R

which proves the lemma. Thus, we have obtained the following equation for h:

dTh- -eLh + eF(h, e, r), F(h, e, T) = -g(y0 + h-ev)+ g(y0) + Dg(yo)h.

(27)

662

A. A. Ilyin

Lemma 4. the function F is a Lipschitz map from E\ into E\. Ifhi,h2 then \\F(hue,r)

- F(h2,e,T)\\El

G BEl (p),

< A(£,p)||/n - h2\\El,

(28)

\\F(0, e,r)\\El^N(e),

(29)

where A(e, p), N(e) -¥ 0 as e, p -*• 0.

Proof. We denote by III the projection III (u,p) = u and we set Ilit; = w, IIi/ii,2 = Ti,2- Then \\F(hi) -

F(h2)\\El

= || V ( / ( « 0 -SW + 7Ti) - f(u0 -£W + 7T2) - /'(u O )(7Tl - 7T2)) ||

^ ||(/'(uo - en) + TTI) - /'(«o -ew + 7r2))V(uo - ew) - / " K ) ( T T I - 7r2)Vu0|| + | | / ' ( " 0 ~ew + 7Ti)V7Ti - f'(u0 -SW + 7r2)V7r2 - /'(U O )V(7T 1 - 7T2)||

=:A + B. Next, for some function 9 = 9{x) € [0,1], A = || (f"(u0 - ew + m + 0(ir2 - vn)) - /"(««,))Vt*)fa - TT2) + f"(uQ - ew + 7Ti + flfa - 7Ti))£:Vio(7ri - 7r2)|| ^ ||/"(«o -ew + 7T! +0(7T2 -TTi)) - / " («o) ^ || V«o|| |kl - ^Hoo + ||/"(uo - ew + TTj + ^(7r2 - iri))||J|eVu;|| hi - n2\U Using the continuity of / " , the embedding theorems and (25), we obtain A^Xl(e,p)\\h1-h2\\B1,

Ai(e,p)->0,

e.p-^0.

The term B satisfies a similar inequality, which gives (28). Inequality (29) is obtained in an analogous way. The proof is complete. We now turn to the analysis of the linear equation 8ty = -Ly,

j/(0) = y0 € £ i

(30)

or

dfu + jdtu = Auu(0)=u0,

dtu(0)=po,

f'{uo)u, (u o ,po)e£i.

This equation has a unique solution y € C([0,T];Ei). We denote by St the corresponding semigroup (group) of continuous linear operators. Definition 1 (see [9]). A semigroup of continuous linear operators St'. X ~> X acting in a Banach space X is called almost stable if (1) \\St\\x-^x^C(T),te[0,T}; (2) the function Stv is weakly continuous in t; (3) there exists a number ro, 0 < ro < 1, such that outside the circle |£| ^ ro there are only finitely many points of the spectrum of the operator Si; (4) the invariant subspace corresponding to this part of the spectrum is finitedimensional. dimensional


663

Lemma 5. The semigroup St generated by (30) is almost stable in E\. Proof. Using the main idea from [9]—[11] we decompose St as

St = Sf + Du where 5° is the linear semigroup generated by the linear homogeneous equation d\v + jdtv = (A - /'(uo) - M)v = -Aov,

v{0)=u0,

dtv(Q)=po,

M>0,

and the linear operators Dt are defined as follows: Dt(uo,Po) = where w(t) is the solution of the equation

[

'

(w(t),dtw(t)),

dfw + -ydtw = -Aow + Mu(t),

w(o) = o, dtw(o) = o,

(33)

with u(t), of course, being the solution of equation (31). Let the constant M, which is at our disposal, satisfy M > max xS n |/'( u o)|- Then (Aov,v) > a||Vw||2, a > 0, and for such an M the following estimate is valid [9], [13], [14]: bt IISJIIE,-*,- *S Kje- , b > 0, j = 0,1,2. (34) We represent the solution of (31) by the Duhamel integral (u(t),dtu{t)) = f S°_T(0,Mu(T))dT + S?(uo,po). Jo

(35)

This implies that the solution (u(t), dtu(t)) € C([0,T];Ei) and the first two conditions of Definition 1 are satisfied. Analogously, the solution of (33) is given by (W(t),dtw(t))

= //

S?_r(0,Afu(T))dT.

J

Since (0,Mu) € C([0,T];E2), we see that (w,dtw) € C([0,T];E2) as well. The embedding E2 C E\ is compact, hence, the operator Dt: E\ -> E\ is compact. Using (34) and the spectral radius theorem we see that the spectrum of 5° is contained inside the disc |(| ^ e~bt < 1. The operator Dt is compact; hence, the essential spectrum cre(St) of the operator St = 5° + Dt is contained inside the same disc \(\ ^ e~bt. The operator St is bounded; hence the connected component of the set C \ ae(St) containing infinity contains the exterior of our disc, that is, the set |C| > e~bt [28]. Therefore each point in the exterior of this disc either belongs to the resolvent set of the operator St or is an isolated eigenvalue of it with finite multiplicity. Thus, outside this circle there are only finitely many eigenvalues counting multiplicities. The proof is complete.

A. A. Ilyin

664

We denote by (J41/2))) the ball BD(A){R2) is uniformly absorbing for the family of processes Ug(t, r) acting in DiA1'2). Next, the set Ug(t,T)BD{A)(R2) for T + 6 0 is bounded, say, in D(A3/2~'r), 7 > 0 and compact in D(A) uniformly in g G "K(f). This is proved in the same way as in the previous example taking into account that now u{s) e Loo(0,1;D(A)) and therefore F(s) £ L o o ^ . l ; ! ? ^ 1 / 2 ) ) . The continuity of this family from D(A) x "K{f) to D(A) is proved as in the previous example and we see that in D(A) there exists a uniform attractor ./ lying in BD(A)(R2) for all u>. We observe that the results of Example 2, 3 above are valid for the Navier-Stokes system on a compact 2-dimensional manifold [29]. Example 4. Consider equation (1) from §3 with n = dimfi = 1 (see Remark 2 from §3). Let ip be a.p. in L2. Then in the space EQ — H° x L2 there exists a uniform attractor of the family of processes generated by equation (1) from § 3. This was proved in [17], where the attractor A^^w.)) is (uniformly in w > 0) contained in the ball BE(RQ) with RQ = •Ro(||y|U00(R;L2))T h e o r e m 2 (Global averaging theorem). We consider the equation

dtu = N(u) + f(cjt),

(2)

where f is a.p. in X, and let /o € X be the average of f. Suppose that this equation generates a family of processes in E Ug(t,T)uT = u(t),

g = g(u•)

Suppose further that the averaged equation dtu = N(u) + /o

(3)

generates in E a semigroup StUo = u(t) having an attractor A in E. Suppose, in addition, that the semigroup St is uniformly bounded for t ^ 0 and has an absorbing set in E. Finally, suppose that the theorem on averaging on a finite time interval is valid. In other words, if U(T) = U(T) € BE(Ro), then for te[r,T + T], \\Ug(t,T)u(T) - Stu(T)\\E ^ *?T,fl0M -> 0,

W ->• 0 0 ,

uniformly with respect to g € We now consider two cases. 1. Suppose that the family of processes Ug(t,r) possesses a uniform attractor A. Then the attractor Ay^f^.)) depends upper semicontinuously on w : distE(AK(f(u.)),A)

-+ 0,

w -^ 00.


671

2a. / / it is not known a priori that the attractor ^.j£(/(w •)) *s bounded in E uniformly in u (or it does not exist at all), then for every sufficiently large R > 0 and every e > 0 there exists U>Q — u>o(R, £) such that for LJ > LJO the e-neighbourhood of the attractor A uniformly absorbs bounded sets from BE(R), in other words, this neighbourhood is an R-locally absorbing set. 2b. If the family of processes has an R-local attractor A^,,,-.,, then it depends upper semicontinuously on u : distj J (./l& (/(u ,. )) ,;A) - • 0,

u ->• oo.

Proof. Let us prove the first part of the theorem. In accordance with part (3) of Definition 2, it is sufficient to prove that every neighbourhood of the attractor A in E is uniformly absorbing for the family of processes Ug(t, r ) , g £ Ji(f(u> •)) for u> large enough. Let n > 0 be arbitrary small. We denote by 0 M (A) an arbitrary /x-neighbourhood of A: Qp(A) = {u£

E; distE{u,A)

< /x}.

Let 6, 0 < S < n/2 be so small that St0s(A) C 0 M/2 (A),

t > 0.

(4)

Such a S exists, since the attractor A is a stable set in the sense of Lyapunov [9], Proposition II.1.3. We consider the absorbing ball BE(RQ) for the semigroup St. Taking if necessary a larger Ro, we may assume that for every u > 0 attractors of the non-autonomous equation lie inside this ball Aj^f^-)) C BE{RQ), UI > Q. It is also clear that A C BE(RQ). Let T = T(Ro,6) be so large that the absorbing ball BE(RO) satisfies the inclusions: StBE(Ro) C 0s/2(A), StBE(Ro) C BE{R)

t > T, V i > 0 , R^Ro,

(5) (6)

where we have used the attraction property and the uniform boundedness of Style now fix this T > 0. We consider a point u0 E BE(RQ). By the averaging theorem there exists an LJ0 = LJO(RO,T,S) such that for u > w0 the inequality rjTtRo(oj) < 8/2 is valid. Let two trajectories of systems (2) and (3) start at u0: u(t) = Ug(t,0)uo, u(t) = Stu0. These trajectories will diverge on the interval t € [0, T] by a distance less than 5/2, and by (5) the end-point u(T) £ 0s/2(A); hence, u(T) 6 OS(A). We repeat this once more, that is, we take ui = u(T) g O$(A) as the initial point, consider the trajectory Stu\, t € [0,T], starting from it, and continue the trajectory u(t) to the interval [T,2T]: u(t + T) = Ug{t + T,0)u0. By (5) and (4) Srui £ OS/2(A), and on the whole interval t £ [0,T] Stui € 0^/2{A). Again, using the theorem on averaging we see that \\Ug(t+T, 0)uo—StUi \\E ^ S/2, t e [0,T] and therefore the end-point u2 = Ug(2T,0)u0 € 0s(A), while on the interval t £ [0,T] Ug(T + t,0)uo £

672

A. A. Ilyin

We can repeat this procedure infinitely many times. However, neither T nor u>o depends on the choice of the particular initial point t»o G BB(RO)- It is also clear that the above construction is applicable to the trajectory Ug(t + T,T)UQ, t > 0, r e R as well. Finally, all our estimates are uniform in g € Jt(f). Thus, we have proved that for u > u0 and t - r > T* E [0, T] Va(t,T)B cOv(A) for any g G "K(f) and any bounded set B C BE(RQ). The proof of the first part of the theorem is complete. The proof of the second part is completely analogous. Some applications of the first part of Theorem 2 are given below. Theorem 3. Suppose that in the Navier-Stokes system (1) from § 2 the right-hand side f is a.p. in H. Then for every a G [^, 1) )),.A) -> 0,

u ->• oo.

Theorem 4. We consider the Navier-Stokes system (1) from § 2 with spaceperiodic boundary condition (or on a closed 2-dimensional manifold). Suppose that f is a.p. in D(Al>2). Then d i s t D ( j 4 ) ( A w ( / ( t a ) . ) ) , .A) -» 0,

w-4oo.

Theorem 5. We consider the hyperbolic equation (1) from §3 with n = 1. Let the right-hand side ip - (0, tp) be a.p. in Ex = H2 D H£ x H&. Then in Eo = H% x L2 we have distBo {AK(V(U.)), ^l) ^ 0, w-Mx). Theorems 3, 4, 5 immediately follow from Theorem 2 and Examples 2-4 examined above (in Theorem 3 we also use Remark 2 from § 2). We now consider an example of application of the second part of Theorem 2. It is concerned with the hyperbolic equation (1) from § 3 in a 3-dimensional domain. We have been unable to prove that the family of processes generated by this equation has an attractor that is bounded in E\ uniformly in u. Nevertheless, the following i?-local result is valid. Theorem 6. Consider the damped hyperbolic equation (4) from § 3 in the phase space Ei. Let ip = (0, ip) be a.p. in Ei. Then for every sufficiently large R in the ball BEX (R) there exists an R-local attractor Ajt(\i>(w •)) and dist^ (A%Mu).)), .A) -> 0,

w->oo.

Proof. As was already stated, the autonomous equation has an attractor A in Ei [12], [9] (this also follows from the proof given below). Let R be so large that A C BEX (R)- Then all the hypotheses of Theorem 2, part 2a are satisfied and,


673

hence, for u>o = u>0(R,e) the ^neighbourhood of A uniformly absorbs the ball BEl(R): U Ux(t,r)B C Qe(A) for t - T > T = T(oj,e,R) and VB C BEl{R). It remains to prove that for w > w0 the family of processes Ux(t,r), X 6 !K(V>(w-))> n a s a n -R-local attractor. Let us verify conditions (1) and (2) of Theorem 1. To verify condition (1), that is, to prove the existence of a uniformly ii-locally attracting set compact in Ei, we proceed in the following way. Let \ € JK(V0- Any solution y(t) of (4) from §3: dty = -Ay - g{y) + x(ut),

y(r) = yT £ BEl (R)

can be written in the form y(t) = Ux(t,T)y(T) =

U1(t,T)y(T)+y(t,T,X)+wy(t),

where z(t) = C/i(i, r)y(r) is the solution of the linear homogeneous equation dtz = -Az,

Z(T) = 2/(T),

which by (10) from §3 satisfies the estimate

Further, y(t, r, x) is the solution of the linear inhomogeneous equation

and, as in [17], we can show that for t ^ r, r € M, and for all \ € Ji(ip) the solution y{t,TJ X) € Di, where Di is a compact set in E\. Finally, w(t) = wy(t) is the solution of the equation dtw = -Aw-

g(y(t)),

W(T)=0.

We now see that H?/^)!!^ < Ri = Ri(R) for * ^ r and y(r) G BEl(R). This follows from the facts that the semigroup St of the autonomous system is uniformly bounded (with respect to t ^ 0) in E\ and the ball BEX{R) is absorbing for 5 t , and also from Theorem 1 from § 3. Therefore the following estimate is valid for 9 = (0,f)

\\g(v(t))\\Ba < W)

for

tzr,

x

By the Duhamel principle and (10) from § 3 we see that < K2R2a-1 =: R3. \E2

674

A. A. Ilyin

Thus, for r G R, t ^ r and x £ ^(VO the inclusion w(t) G BE2(R3) is valid. The embedding Ei -4 Ei is compact; hence all the solutions Ux{t,r)y(j) for y(r) G BE1(R) are uniformly attracted (with respect to x € !H(t/>)) to the set [Z?i -I- BE2(R3)]E1, which is compact in E\. Finally, the continuity condition from Theorem 1 is verified in the same way as in Example 2, using the analogue of (12) from §3. The proof is complete. We have considered applications to attractors of the theorem on averaging over a finite time interval. Theorems on averaging over the entire axis also have applications to attractors. In the following theorem, equation (2) and the spaces E and X have the same meaning as in Examples 2-5. Theorem 7. Let the right-hand side f of equation (2) be a.p. in X and let f satisfy all the additional conditions guaranteeing the existence of the attractor ~4? •) = fh, fh(t) = f(ut + h). Since the bounded solution u* is unique in a small neighbourhood of a fixed stationary point u0, it follows that V/i G R the solution u*.h, which by Theorem 2 from §2 and Theorem 2 from §3 corresponds to the right-hand side fh, may only be (and in fact is) the translation of the solution u*j\ u*fh = T{h)u). Therefore u*ho(0) = T{ho)u*f(O) = u*f(h0) - v0. Obviously, g = fh° G Ji(f) and hence, v0 6 A^f)On the other hand, if v0 = limôo u*f(hk), hk G R, then again by the uniqueness of the solution u* we see that v0 = limfc_+oou*hfc(0), where fhk = T(hk/u)f. By Bochner's criterion, some subsequence fhk -> uj in Cb(R;E) if fh Ug(0) in E and consequently Vo = u*(0), g G !H(/). The proof is complete.


675

Corollary 1. / / the right-hand side rj) of the hyperbolic equation (4), (2!)-(2s) from §3 in a ^-dimensional domain is a.p. in E2, then for every sufficiently large R for u) > LJ(R) the inclusion

t€R

is valid. In some cases we have equality in (7). Corollary 2. / / the right-hand side f in the Navier-Stokes system (1) from § 2 is a.p. in D(A13), 0^0 and the averaged function f0 satisfies (46) , then for the attractor Ax(f) considered as a subset of D(Aa), a € (5,1), the following equality holds:

Proof. For each a € [|>1) we have ||^l>c(/(w))l|a ^ Ra uniformly in u> (see Example 2). Wefixa € (|)1)- Let w > uo{Ra), where cjo(Ra) (more precisely eo(Ra)) was defined in Theorem 3 from § 2. We claim that in the large ball Ba(Ra) there exists only one solution bounded on the entire axis, namely, the solution u j constructed in Theorem 2 from §2. In fact, any other solution bounded in D(Aa) on the entire axis would, in view of (47) from §2, necessarily coincide with u j , since they are both bounded as t —¥ —00. Hence, the kernel K consists of a single element K — u*j. Let some vo € Ba(Ra) belong to the attractor vo G-AM(/)- Then by Theorem 1 v0 = ubg(0), where ubg(t) is a bounded complete trajectory corresponding to some g 6 3i(f). But such a trajectory can only be u*(t). Therefore, v0 = u*(0). Then g = lim/t-xx, fhk and, as in the previous theorem, we see that u*,hk —> u* in Cb(R;D(Aa)). Hence, u*f(hk) = u*fhk{0) ->• u*(0) = v0. The proof is complete. Remark 1. The following estimate of the Hausdorff dimension of the attractor of equation (11) from §2 is known [14]: 11 r 11

(9) where c(Q) is a dimensionless constant depending on the shape of the domain fi. This estimate was made more precise in [27]:

di.i