Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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Dissipativity of some classes of equations 257

If we introduce the step functions ˜x

(τ) := x(τ

) (τ

k−1

≤ τ ≤ τ

; τ

−τ

k−1

; k = 1, 2, ..., n and x ∈ K), then from the equality (6.80), we have the following

relation

lim

ε→0

, ˜x

(s))ds =

(˜x

(s))ds. (6.81)

Under our assumption the family of functions K is equicontinuous on [0, T

] and,

consequently,

sup

x∈K

sup

0≤τ≤T

k˜x

(τ) − x(τ )k → 0 (6.82)

as n → +∞. Using the condition of Lipschitz (6.74) for the family of functions F

we obtain the estimate

, x(s))ds −

(x(s))ds| ≤

|f(

, x(s)) −f(

, ˜x

(s))|ds+ (6.83)

[f(

, ˜x

(s)) − f

( ˜x

(s))]ds| +

(x(s)) −f

( ˜x

(s))|ds ≤

2LT

sup

x∈K

sup

0≤τ≤T

|˜x

(τ) − x(τ)| + |

[f(

, ˜x

(s)) − f

( ˜x

(s))]ds|.

From (6.80) - (6.83) immediately we obtain the results in the lemma. 

Proof. Now we will prove Theorem 6.20. Denote by ψ(τ, x, f, ε) (respectively

ψ(τ, x)) a unique solution of equation

= f(

, x)

(respectively (6.77)) passing through the point x ∈ B[0, r] at the moment τ = 0

and deﬁned on [0,

]. The functions ψ(τ, x, f, ε) and

ψ(τ, x) satisfy the integral

equations

ψ(τ, x, f, ε) = x +

, ψ(s, x, f, ε))ds

and

ψ(τ, x) = x +

(

ψ(s, x))ds,

respectively. Using the condition of Lipschitz (6.74), we obtain the estimate

|ψ(τ, x, f, ε) −

ψ(τ, x)| = |

[f(

, ψ(s, x, f, ε)) − f

(

ψ(s, x))]ds| ≤

|f(

, ψ(s, x, f, ε)) − f(

ψ(s, x))|ds + |

[f(

ψ(s, x)) − f

(

ψ(s, x))]ds| ≤

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258 Global Attractors of Non-autonomous Dissipative Dynamical Systems

|ψ(s, x, f, ε) −

ψ(s, x)|ds + c(ε),

where

c(ε) := sup

0≤τ≤T

,x∈K

, x(s)) −f

(x(s))]ds|.

According to the Gronwall-Bellman inequality (see, for example,

[

137

]

[

186

]

), we

can now conclude that

|ψ(τ, x, f, ε) −

ψ(τ, x)| ≤ exp(2Lτ)c(ε)

and it remains only to note that by virtue of Lemma 6.13, c(ε) → 0 as ε → 0 and

|x(t, ε) − y(εt)| = |ψ(τ, x, f, ε) −

ψ(τ, x)| ≤ exp(2Lτ)c(ε) = exp(2Lεt))c(ε)

for all t ∈ [0,

]. The theorem is thus proved. 

In the next subsection, we will also need the following lemma.

Lemma 6.14 Let F be a transitive subset of C(R × H, H), i.e. there exists a

function g ∈ F such that F = H(g), where H(f) if the hull of g. Then the following

two assertions are equivalent:

1. There exists f

∈ C(H, H) such that

lim

T →+∞

f(t, x)dt = f

(x)

uniformly w.r.t. f ∈ F and x ∈ B[0, r];

2. There exists f

∈ C(H, H) such that

lim

T →+∞

t+T

g(τ, x)dτ = f

(x)

uniformly w.r.t. t ∈ R and x ∈ B[0, r].

Proof. It is evident that 1. implies 2. because g

∈ F for all t ∈ R and, consequently,

t+T

g(τ, x)dτ =

g(t + τ, x)dτ → f

(x)

as T → +∞ uniformly w.r.t. t ∈ R and x ∈ B[0, r].

Let now ε > 0 and f ∈ F = H(g), then there exists a sequence {t

} ⊂ R and

L(ε) > 0 such that g

→ f and

g(τ + t

, x)dτ − f

(x)| < ε (6.84)

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Dissipativity of some classes of equations 259

for all T > L(ε). Passing to limit as n → +∞ in the inequality (6.84) we obtain

f(τ, x)dτ − f

(x)| ≤ ε

for all T > L(ε). From the latter inequality, the required statement immediately

follows. This proves the lemma. 

Remark 6.6 All the results of this subsection are true for arbitrary Banach space

too, not only for Hilbert space.

6.6.5 Global averaging principle for the non-autonomous Lorenz

systems

Now we consider a global averaging principle for the non-autonomous Lorenz

systems. Let Ω be a compact metric space and (Ω, R, σ) be a dynamical system on

Ω. We consider the ”perturbed” non-autonomous Lorenz equation

= εA(ωt)x + εB(ωt)(x, x) + εf(ωt), (6.85)

where ε ∈ [0, ε

] (ε

> 0) is a small parameter. Suppose that the conditions (6.59)–

(6.62) are fulﬁlled and the following averaging values exist uniformly w.r.t. ω ∈ Ω:

A = lim

T →+∞

−T

A(ωt)dt, (6.86)

B = lim

T →+∞

−T

B(ωt)dt, (6.87)

and

f = lim

T →+∞

−T

f(ωt)dt. (6.88)

Remark 6.7 The conditions (6.86) − (6.88) are fulﬁlled if a dynamical system

(Ω, R, σ) is strictly ergodic, i.e. there exists on Ω a unique invariant w.r.t. (Ω, R, σ)

measure µ.

Along with equation (6.85), we will also consider the averaged equation

= ε

Ax + εB(x, x) + εf. (6.89)

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260 Global Attractors of Non-autonomous Dissipative Dynamical Systems

If we introduce the ”slow time” τ := εt (ε > 0), then the equations (6.85) and (6.89)

can be written as

dτ

= A(ω

)x + B(ω

)(x, x) + f(ω

) (6.90)

and

dτ

Ax + B(x, x) + f. (6.91)

Remark 6.8 a. From the conditions (6.61) and (6.87) it follows that

Reh

B(u, v), vi = 0 (6.92)

for all u, v ∈ H;

b. From the inequality (6.59) it follows that

Reh

Ax, x)i ≤ −α|x|

(6.93)

for all x ∈ H.

Theorem 6.21 Assume the conditions enumerated above are all satisﬁed. Then

for all T > 0 and ρ ≥ r

kfk

> 0, the solution for the non-autonomous Lorenz

equation (6.85) approaches the solution of the averaged Lorenz equation (6.89) in

the following sense:

max{|ϕ(t, x, ω, ε) −

ϕ(t, x, ε)| : 0 ≤ t ≤ T /ε, |x| ≤ ρ, and ω ∈ Ω} → 0 (6.94)

as ε → 0, where ϕ(t, x, ω, ε) ( respectively

ϕ(t, x, ε)) is a solution of equation (6.85)

(respectively (6.89)), passing through the point x at the initial moment t = 0.

Proof. According to Theorem 6.14, we have |ϕ(t, x, ω, ε)| ≤ ρ and |¯ϕ(t, x, ε)| ≤ ρ

for all t ≥ 0, |x| ≤ ρ, ω ∈ Ω and ε ∈ (0, ε

]. If we take F := {f

| ω ∈ Ω} ⊂

C(R × H, H), where f

(t, x) := A(ωt)x + B(ωt)(x, x) + f(ωt) for all t ∈ R and

x ∈ H, then the relation (6.94) follows from Theorem 6.20. This completes the

proof. 

Theorem 6.22 (Global averaging principle for non-autonomous Lorenz systems)

Let ϕ

be a cocycle generated by the equation (6.85). Assume the conditions enumer-

ated above are all satisﬁed. If the cocycle ϕ

( ε ∈ [0, ε

]) is asymptotically compact,

then the following assertions hold:

(1) The averaged equation (6.91) admits a compact global attractor

I ⊂ H;

(2) For every ε ∈ (0, ε

] the equation (6.85) has a compact global attractor {I

| ω ∈

Ω};

(3) The set I := ∪{I

| ε ∈ [0, ε

]} is bounded, where I

I and I

:= ∪{I

| ω ∈

Ω};

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Dissipativity of some classes of equations 261

(4)

lim

ε→0

sup

ω∈Ω

β(I

I) = 0

and, in particular,

lim

ε→0

β(I

I) = 0.

Proof. The ﬁrst three statements of the theorem follow from Theorems 6.14, 6.15

and Remark 6.5. Now we will prove the fourth statement of the theorem. To this

end, we will use the same arguments as in

[

70, 199

]

. Let λ > 0 and B(

I, λ) = {x ∈

H | ρ(x,

I) < λ}. According to orbital stability of the set I (see, for example,

[

179,

Ch.I

]

or Theorem 1.6), for given λ there exists δ = δ(λ) > 0 (we may consider

δ(λ) < λ/2) such that

ϕ(t, B(I, δ)) ⊂ B(I, λ/2)

for all t ≥ 0. By virtue of boundedness of the set I = ∪{I

| 0 ≤ ε ≤ ε

} we may

choose ρ ≤ r

such that I ⊂ B(0, ρ) = {x ∈ H | |x| < ρ}. Since

I is a compact global

attractor of the system (6.91), then for the closed ball B[0, ρ] := {x ∈ H||x| ≤ ρ}

and the number δ > 0 there exists T = T (ρ, δ) > 0 such that

ϕ(t, B[0, ρ]) ⊂ B(I, δ/2), t ≥ T. (6.95)

Let x ∈ B[0, ρ]. Then by virtue of Theorem 6.21 for the numbers ρ ≥ r

and

T (ρ, δ) > 0 there exists µ = µ(ρ, δ) > 0 such that 0 < ε ≤ µ, m(ε) < λ/2 (see

(6.94)), i.e.

|ϕ(t, x, ω, ε) −

ϕ(t, x)| < δ/2 (6.96)

for all x ∈ B[0, ρ], ω ∈ Ω, t ∈ [0, T/ε] and 0 < ε ≤ µ. According to (6.95)

we have

ϕ(T/ε, x, ω, ε) ∈ B(I, δ/2). Thus, taking into account (6.96), we obtain

ϕ(T/ε, x, ω, ε) ∈ B(

I, δ). Let us take the initial point x

:= ϕ(T/ε, x, ω, ε) and we

will repeat for this point the same reasoning as above. Taking into consideration

the equality ϕ(t, x

, σ(T/ε, ω), ε) = ϕ(t + T/ε, x, ω, ε), we will have

|ϕ(t + T/ε, x, ω, ε) −

ϕ(t, x

)| < δ/2 (6.97)

for all t ∈ [0, T/ε], x ∈ B[0, ρ] and ω ∈ Ω, where x

= ϕ(T/ε, x, ω, ε).

By the inequality (6.97) we obtain again x

:= ϕ(2T/ε, x, ω, ε) ∈ B(

I, δ) and,

consequently,

ϕ(t + T/ε, x, ω, ε) ∈ B(

I, λ/2 + δ/2) ⊂ B(I, λ).

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262 Global Attractors of Non-autonomous Dissipative Dynamical Systems

If we continue this process and later (by virtue of uniformity w.r.t. |x| ≤ ρ and

ω ∈ Ω of the estimation (6.96) it is possible), we will obtain

ϕ(t, x, ω, ε) ∈ B(

I, λ) (6.98)

for all t ≥ T/ε, x ∈ B[0, ρ], ω ∈ Ω and o ≤ ε ≤ µ and, consequently,

ϕ(t, x, σ(−t, ω), ε) ∈ B(

I, λ)

for all t ≥ T/ε and |x| ≤ ρ. Since I = ∪{I

| 0 ≤ ε ≤ ε

} ⊆ B(0, ρ), then according

to Theorem 2.24

t≥0

[

τ≥t

ϕ(τ, B[0, ρ], σ(−τ, ω), ε).

Therefore, from (6.98) we have I

⊂ B(

I, λ) for all ω ∈ Ω and 0 < ε < µ. Note

that λ is arbitrarily chosen. Hence from the last inclusion we obtain the equality

(6.95). The theorem is proved. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

Chapter 7

Upper semi-continuity of attractors

7.1 Introduction

The problem of upper semi-continuity of global attractors for small perturbations

is well studied (see, for example,

[

179

]

and references therein) for autonomous and

periodical dynamical systems. In the works

[

]

and

[

]

this problem was studied

for non-autonomous and random dynamical systems.

This chapter is devoted to a systematic study of the problem of upper semi-

continuity of compact global attractors and compact pullback attractors of abstract

non-autonomous dynamical systems for small perturbations. Several applications

of our results are given for diﬀerent classes of evolutionary equations.

The chapter is organized as follows. In section 2 we study some general prop-

erties of maximal compact invariant sets of dynamical systems. In particular, we

prove that the compact global attractor and pullback attractor are maximal com-

pact invariant sets (Theorem 7.2).

Section 3 contains the main results about upper semi-continuity of compact

global attractors of abstract non-autonomous dynamical systems for small pertur-

bations (Lemmas 7.3, 7.4 and Theorems 7.3, 7.4, 7.5 and 7.6).

In section 4 we give conditions for connectedness and component connectedness

of global and pullback attractors (Theorem 7.7).

Section 5 is devoted to an application of our general results obtained in sections

2-4, to the study of diﬀerent classes of non-autonomous diﬀerential equations (quasi-

homogeneous systems, monotone systems, non-autonomously perturbed systems,

non-autonomous 2D Navier-Stokes equations and quasi-linear functional-diﬀerential

equations).

7.2 Maximal compact invariant sets

Let W be a complete metric space, T = R or Z, Ω be a compact metric space,

(Ω, T, σ) be a group dynamical system on Ω and hW, ϕ, (Ω, T, σ)i be a cocycle with

263

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264 Global Attractors of Non-autonomous Dissipative Dynamical Systems

ﬁber W , i.e. the mapping ϕ : T

× W × Ω → W is continuous and possesses the

following properties: ϕ(0, x, ω) = x and ϕ(t + τ, x, ω) = ϕ(t, ϕ(τ, x, ω), ωt), where

ωt = σ(t, ω).

We denote by X = W ×Ω, g = pr

: X 7→ W, (X, T

, π) a semi-group dynamical

system on X deﬁned by the equality π = (ϕ, σ), i.e. π

x = (ϕ(t, u, ω), σ(t, ω)) for

every t ∈ T

and x = (u, ω) ∈ X = W × Ω. Let h(X, T

, π), (Ω, T, σ), hi be a

non-autonomous dynamical system, where h = pr

: X 7→ Ω.

Deﬁnition 7.1 A family {I

| ω ∈ Ω}(I

⊂ W ) of nonempty compact subsets of

W is called a maximal compact invariant set of cocycle ϕ, if the following conditions

are fulﬁlled:

(1) {I

| ω ∈ Ω} is invariant, i.e. ϕ(t, I

, ω) = I

ωt

for every ω ∈ Ω and t ∈ T

;

(2) I =

| ω ∈ Ω} is relatively compact;

(3) {I

| ω ∈ Ω} is maximal, i.e. if the family {I

| ω ∈ Ω} is relatively compact

and invariant, then I

⊆ I

for every ω ∈ Ω.

Lemma 7.1 The family {I

| ω ∈ Ω} is invariant w.r.t. cocycle ϕ if and only

if the set J =

| ω ∈ Ω}(J

:= I

× {ω}) is invariant with respect to the

skew-product dynamical system (X, T

, π).

Proof. Let the family {I

| ω ∈ Ω} be invariant, J =

|ω ∈ Ω} and J

×{ω}. Then

J =

[

{π

| ω ∈ Ω} =

[

{(ϕ(t, I

, ω), ωt) | ω ∈ Ω} (7.1)

[

ωt

×{ωt} | ω ∈ Ω} =

[

ωt

| ω ∈ Ω} = J

for all t ∈ T

. From the equality (7.1) follows that the family {I

| ω ∈ Ω} is

invariant w.r.t. cocycle ϕ if and only if a set J is invariant w.r.t. dynamical system

(X, T

, π). 

Theorem 7.1 Let the family of sets {I

| ω ∈ Ω} be maximal, compact and

invariant. Then it is closed.

Proof. We note that the set J =

| ω ∈ Ω}(J

:= I

× {ω}) is relatively

compact and according to Lemma 7.1 it is invariant. Let K =

J, then K is compact.

We will show that K is invariant. If x ∈ K, then there exists {x

} ⊂ J such that

x = lim

n→+∞

. Thus x

∈ J = π

J for all t ∈ T

, then for t ∈ T

there exists

∈ J such that x

= π

. Since J is relatively compact, then we can suppose that

the sequence {

} is convergent. We denote by x = lim

n→+∞

,then x ∈ J, x = π

and, consequently, x ∈ π

J for all t ∈ T

, i.e. J = π

J. Let I

= pr

K, then we

have I

| ω ∈ Ω}, where I

= {u ∈ W | (u, ω) ∈ K} and K

:= I

× {ω}.

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Upper semi-continuity of attractors 265

Since the set K is invariant, then according to Lemma 7.1 the set I

is also invariant

w.r.t. cocycle ϕ. The set I

is compact, because K is compact and pr

: X 7→ W

is continuous. According to the maximality of the family {I

| ω ∈ Ω} we have

⊆ I

for every ω ∈ Ω and, consequently, I

⊆ I. On the other hand I = pr

J = I

and, consequently, I

= I. Thus the set I is compact. The theorem is proved. 

Denote by C(W ) the family of all compact subsets of W .

Deﬁnition 7.2 Recall that a family {I

| ω ∈ Ω} (I

⊂ W ) of nonempty compact

subsets of W is called

− a compact pullback attractor of the cocycle ϕ, if the following conditions are

fulﬁlled:

a. I =

| ω ∈ Ω} is relatively compact ;

b. I is invariant w.r.t. cocycle ϕ, i.e. ϕ(t, I

, ω) = I

σ(t,ω)

for all t ∈ T

and

ω ∈ Ω;

c. for every ω ∈ Ω and K ∈ C(W )

lim

t→+∞

β(ϕ(t, K, ω

−t

), I

) = 0, (7.2)

where β(A, B) = sup{ρ(a, B) : a ∈ A} is a semi-distance of Hausdorﬀ and

−t

:= σ(−t, ω).

− a compact global attractor, if the following conditions are fulﬁlled:

d. a family {I

| ω ∈ Ω} is compact and invariant;

f. for every K ∈ C(W )

lim

t→+∞

sup

ω∈Ω

β(ϕ(t, K, ω), I) = 0, (7.3)

where I =

| ω ∈ Ω}.

Theorem 7.2 A family {I

| ω ∈ Ω} of nonempty compact subsets of W will

be maximal compact invariant set w.r.t. cocycle ϕ, if one of the following two

conditions is fulﬁlled :

a. {I

| ω ∈ Ω} is a compact pullback attractor w.r.t. cocycle ϕ;

b. {I

| ω ∈ Ω} is a compact global attractor w.r.t. cocycle ϕ.

Proof. a. Let the family {I

| ω ∈ Ω} be a compact pullback attractor. If the

family {I

| ω ∈ Ω} is a compact and invariant set of cocycle ϕ, then we have

β(I

, I

) = β(ϕ(t, I

−t

, ω

−t

), I

) ≤ β(ϕ(t, K, ω

−t

), I

) → 0

as t → +∞, where K =

| ω ∈ Ω}, and, consequently, I

⊆ I

for every ω ∈ Ω,

i.e. {I

| ω ∈ Ω} is maximal.

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266 Global Attractors of Non-autonomous Dissipative Dynamical Systems

b. Let the family {I

| ω ∈ Ω} be a compact global attractor w.r.t. cocycle

ϕ, then according to Theorem 2.24 it is a uniform compact pullback attractor and,

consequently, the family {I

| ω ∈ Ω} is maximal compact invariant set of the

cocycle ϕ. 

Remark 7.1 The family {I

| ω ∈ Ω} (I

⊂ W ) is a maximal compact invariant

w.r.t. cocycle ϕ if and only if the set J =

| ω ∈ Ω}, where J

= I

×{ω}, is

a maximal compact invariant in the dynamical system (X, T, π).

7.3 Upper semi-continuity

Lemma 7.2 Let {I

| ω ∈ Ω} be a maximal compact invariant set of cocycle

ϕ, then the function F : Ω 7→ C(W ), deﬁned by equality F (ω) := I

is upper

semi-continuous, i.e. for all ω

∈ Ω

β(F (ω

), F (ω

)) → 0,

if ρ(ω

, ω

) → 0.

Proof. Let ω

∈ Ω, ω

→ ω

and suppose there exists ε

> 0 such that

β(F (ω

), F (ω

)) ≥ ε

Then there exists x

∈ I

such that

ρ(x

, I

) ≥ ε

. (7.4)

As the set I is compact, without loss of generality we can suppose that the sequence

} is convergent. Denote by x = lim

k→+∞

, then by virtue of Theorem 7.1 the set

I =

| ω ∈ Ω} is compact and hence there exists ω

∈ Ω such that x ∈ I

⊂ I.

On the other hand, according to the inequality (7.4) x /∈ I

. This contradiction

shows that the function F is upper semi-continuous. 

Lemma 7.3 Let Λ be a compact metric space and ϕ : T

× W × Λ × Ω 7→ W

verify the following conditions:

1. ϕ is continuous;

2. for every λ ∈ Λ the function ϕ

:= ϕ(·, ·, λ, ·) : T

× W × Ω 7→ W is a cocycle

on Ω with the ﬁber W ;

3. the cocycle ϕ

admits a pullback attractor {I

| ω ∈ Ω} for every λ ∈ Λ;

4. the set

| λ ∈ Λ} is relatively compact, where I

| ω ∈ Ω},

then the equality

lim

λ→λ

,ω→ω

β(I

, I

) = 0 (7.5)