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

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Global attractors of non-autonomous Navier-Stokes equations 377

According to Theorem 11.4 |ϕ(t, x, ω, ε)|, |¯ϕ(t, x, ω, ε)| ≤ r

for all t ≥ 0, where

:= max{

kfk

}. In view of (11.71) v(t) satisﬁes the inequality

|v(t)|

≤ |

(t−s)A

(B(ω

)(v(s), ϕ(s, x, ω, ε)) +

B(ω

)( ¯ϕ(s, x, ω, ε), v(s)))ds|

+ |

(t−s)A

(ω

)( ¯ϕ(s, x, ω, ε),

¯ϕ(s, x, ω, ε))ds|

+ |

(t−s)A

(ω

)ds|

, (11.72)

for all t ∈ [0, L]. By (11.11) and (11.12) we see that the ﬁrst term on the right-hand

side of (11.72) is less than

−a(t−s)

(t − s)

|v(s)|

ds . (11.73)

We now show that the sum of the second and third terms in (11.72) tends to

0 as ε → 0 uniformly w.r.t. t ∈ [0, L], |x| ≤ R

and ω ∈ Ω. Before estimating the

second term, we must integrate it by parts. We have

−(t−s)A

(ω

)( ¯ϕ(s, x, ω), ¯ϕ(s, x, ω)ds =

−tA

(ω

)( ¯ϕ(s, x, ω), ¯ϕ(s, x, ω)ds + (11.74)

−(t−s)A

(ω

)( ¯ϕ(τ, x, ω), ¯ϕ(τ, x, ω)dτds.

Hence

−(t−s)A

(ω

)( ¯ϕ(s, x, ω), ¯ϕ(s, x, ω)ds|

≤

−tA

(ω

)( ¯ϕ(s, x, ω), ¯ϕ(s, x, ω)ds|

+ (11.75)

−(t−s)A

(ω

)( ¯ϕ(τ, x, ω), ¯ϕ(τ, x, ω)dτds ≤

−aL

(ε) +

1−α

1 − α

(ε) := c(ε),

where

(ε) := sup{|

(ω

)( ¯ϕ(s, x, ω), ¯ϕ(s, x, ω)ds|

ω ∈ Ω, |x|

≤ R

, 0 ≤ t ≤ L}.

(11.76)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

378 Global Attractors of Non-autonomous Dissipative Dynamical Systems

According to Lemma 6.13 k

(ε) → 0 as ε → 0, because according to Theorems

11.5 and 11.3 the family of functions K := {¯ϕ(·, x, ω) : |x|

≤ R

, ω ∈ Ω} is

equicontinuous on R

We begin with the third term. Integrating by part in s and taking into account

the inequalities (11.11)-(11.12), (11.16)-(11.17) and (11.59) we ﬁnd

(t−s)A

(ω

)ds|

= |e

(ω

)ds +

(t−s)A

(ω

)dτds|

≤ ke

Aτ

X→E

(ω

)ds|

kAe

A(t−s)

X→E

(ω

)ds|

≤ Kt

1−β

−at

(

) +

K(t −s)

1−β

−a(t−s)

(

t − s

)ds. (11.77)

Let α ∈ [0, 1), ν ∈ (0, 1) and β ∈ [0, 2). Since

(

) ≤ sup

0≤t≤ε

(

) + sup

≤t≤L

(t) ≤

αν

(0) + L

(ε

ν−1

) → 0 as ε → 0

and

1−β

(

)ds =

1−β

(

)ds +

1−β

(

)ds (11.78)

≤ k

(0)

ν(2−β)

2 − β

+ k(ε

ν−1

)

2−β

−ε

ν(2−β)

)

2 − β

≤ k

(0)

ν(2−β)

2 − β

+ k(ε

ν−1

)

2−β

−ε

ν(2−β)

)

2 − β

as ε → 0

uniformly w.r.t. t ∈ [0, L] and ω ∈ Ω, then

sup

0≤t≤L

(t−s)A

(ω

)ds|

→ 0 as ε → 0. (11.79)

Thus,

|v(t)|

≤ C(ε) + D

(t − s)

−α

|v(s)|

ds (11.80)

for all t ∈ [0, L], where C(ε) := Ke

−aL

1−β

+ L

1−β

(

) + c(ε) → 0 as ε → 0

and D := 2R

We now use the known inequality

[

188, Ch.7

]

. If

u(t) ≤ a + b

(t − s)

β−1

u(s)ds, 0 < β ≤ 1, (11.81)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

Global attractors of non-autonomous Navier-Stokes equations 379

then

u(t) ≤ aG

([bΓ(β)]

1/β

t), (11.82)

where G

(x) is a monotone function, while Γ(β) is a gamma function.

In our case we have

|v(t)|

≤ C(ε)G

([bΓ(β)]

1/β

t) ≤ (11.83)

C(ε)G

([bΓ(β)]

1/β

L) := d(ε) → 0 (β := 1 −α

∈ (0, 1])

as ε → 0 uniformly w.r.t. ω ∈ Ω, x ∈ B[0, R

] and t ∈ [0, L] for every L > 0. The

theorem is proved. 

11.5 The global averaging principle for Navier-Stokes equations

Let Ω be a compact metric space, (Ω, R, σ) be a dynamical system on Ω, E be a

Banach space and hE, ϕ, (Ω, R, σ)i be a cocycle on (Ω, R, σ) with ﬁber E.

Deﬁnition 11.4 A family of nonempty compact sets { I

| ω ∈ Ω} (I

⊂ E)

is called a local attractor (local forward attractor) if the followings conditions are

fulﬁlled:

(1)

I =

[

: ω ∈ Ω}

is compact;

(2)

(t, I

, ω) = I

σ(t,ω)

for all t ∈ R

and ω ∈ Ω;

(3) there exists R

> 0 such that I ⊂ B(0, R

) := {x ∈ E| |x| < R

} and

lim

t→∞

sup

ω∈Ω

β(ϕ(t, B[0, R

], ω), I) = 0

(respectively lim

t→∞

sup

ω∈Ω

β(ϕ(t, B[0, R

], ω), I

ωt

) = 0)

Theorem 11.10 Let Λ be a compact metric space, E be a banach space and

(λ ∈ Λ) be a cocycle on (Ω, R, σ) with ﬁber E. Suppose that the following

conditions are fulﬁlled:

(1) the cocycle φ

admits a local forward attractor,

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

380 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) the following relation takes place

(λ) := sup{|ϕ

(t, x, ω) − ϕ

(t, x, ω)| :

0 ≤ t ≤ L, ω ∈ Ω, |x| ≤ R

} → 0 (11.84)

as λ → λ

for every positive number L;

(3) every cocycle ϕ

is asymptotically compact.

Then the next statements are valid:

a. there exists a positive number µ such that for all λ ∈ B[λ

, µ] := {λ ∈ Λ :

ρ(λ, λ

) ≤ µ} the cocycle ϕ

admits in B[0, R

] a forward attractor {I

: ω ∈

Ω};

sup

ω∈Ω

β(I

, I

) → 0

as λ → λ

Proof. Let ρ > 0 be an arbitrary small number such that B[I

, ρ] ⊂ B[0, R

]. We

choose L = L(

) according to the condition

(t, B[0, R

], ω) ⊂ B[I

]

for all ω ∈ Ω and t ≥ L(

). Now we choose ε

= ε

(L) so that m(λ) <

for all

λ ∈ B[λ

, ε

Let t

:= L, then we have ϕ

, x, ω) ∈ B[I

ωt

] and ϕ

, x, ω) ∈

B[0, R

]. We take the point x

:= ϕ

, x, ω) as the initial point and we con-

sider ϕ

(t, x

, ωt

) on the segment [0, L],

(t, x

, ωt

); ϕ

(t, x

, ωt

) = ϕ

(t, ϕ

, x, ω), ωt

) = ϕ

(t + t

, x, ω).

On this segment ϕ

(t, x

, ωt

) and ϕ

(t, x

, ωt

) will diverge by the value less than

. Since ϕ

(t, x

, ωt

) ∈ B[I

ωt

], we get ϕ

(2t

, x, ω) ∈ B[I

ω2t

2ρ

If we take the point x

:= ϕ

(2t

, x, ω) as the initial one, then we see that the

situation is similar to that occurred at the previous step.

Repeating this process, we arrive at a conclusion that ϕ

(t, x, ω) ∈ B[I

ωt

, ρ] ⊂

B(0, R

) for all t ≥ L(

) and ω ∈ Ω. Since the cocycle ϕ

is asymptotic compact

then it admits a forward attractor {I

: ω ∈ Ω} such that I

: ω ∈

Ω} ⊆ B[I

, ρ] and, consequently, β(I

, I

) → 0 as λ → λ

Below we proved the inclusion ϕ

(t, B[0, R

], ω) ⊆ B[I

ωt

] for all t ≥ L and

ω ∈ Ω and, consequently, we obtain

(t, B[0, R

], ω

) ⊆ B[I

] (11.85)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

Global attractors of non-autonomous Navier-Stokes equations 381

for all t ≥ L and ω ∈ Ω. Taking onto consideration that

t≥0

[

τ≥t

(τ, B[0, R

], σ(−τ, ω)) (11.86)

from (11.85) and (11.86) it follows that I

⊆ B[I

, ρ] for all ω ∈ Ω and λ ∈ B[λ

, ε

]

and, consequently, sup

ω∈Ω

β(I

, I

) → 0 as λ → λ

. The theorem is proved.



Remark 11.5 1. The second condition of Theorem 11.10 is fulﬁlled, for example,

if the space E is ﬁnite-dimensional and the mapping ϕ : R

× E × Ω × Λ → E,

deﬁned by the equality ϕ(t, x, ω, λ) := ϕ

(t, x, ω), is continuous.

In fact, if we suppose that it is not true, then there exist L

> 0, λ

→ λ

, x

∈

B[0, R

], t

∈ [0, L

] and ω

∈ Ω such that

|ϕ

, x

, ω

) − ϕ

, x

, ω

)| ≥ ε

> 0. (11.87)

Since the sets B[0, R

], Ω and [0, L

] are compacts, we can suppose that the se-

quences {x

}, {t

}and {ω

}are convergent. Denote by t

:= lim

k→∞

, x

:= lim

k→∞

and ω

:= lim

k→∞

. Passing to limit in the equality (11.87) and taking into account

the continuity of the mapping ϕ we obtain 0 ≥ ε

. The obtained contradiction prove

our statement.

2. Under the conditions of Theorem 11.10 if we suppose that the cocycle ϕ

admits a compact global forward attractor {I

: ω ∈ Ω}, i.e.

lim

t→∞

sup

ω∈Ω

β(ϕ

(t, B[0, R], ω), I

ωt

) = 0

for every R > 0, then should be naturally to hope that for the λ suﬃciently close to

the cocycle ϕ

also will admits a compact global forward attractor {I

: ω ∈ Ω}

in the small neighborhood of I

. Unfortunately, generally speaking, this assertion

is not true.

In fact, let ϕ

be a cocycle (dynamical system) generated by the equation x

−x and ϕ

be a cocycle generated by the equation x

= −x + λx

(λ > 0). It is

clear that the cocycle ϕ

(ϕ

) admits a compact global attractor I

= {0} (I

[−λ

−1/2

, λ

−1/2

]). In the small neighborhood of the attractor I

= {0} the cocycle

( for small λ) admits a local (but not global) attractor I

= {0}.

Theorem 11.11 Let Λ be a compact metric space, (Ω, R, σ) be a dynamical system

on the compact metric space Ω, E be a Banach space and ϕ

(λ ∈ Λ) be a cocycle

on (Ω, R, σ) with ﬁber E. Suppose that the following conditions are fulﬁlled:

(1) the cocycle ϕ

admits a compact global forward attractor;

(2) the following relation takes place

(λ) := sup{|ϕ

(t, x, ω) −ϕ

(t, x, ω)| :

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

382 Global Attractors of Non-autonomous Dissipative Dynamical Systems

0 ≤ t ≤ L, ω ∈ Ω, |x| ≤ R

} → 0

as λ → λ

for every positive number L;

(3) every cocycle ϕ

admits a compact global attractor {I

: ω ∈ Ω};

(4) the set I :=

: λ ∈ Λ} is bounded in E.

Then the following equality

lim

λ→λ

sup

ω∈Ω

β(I

, I

) = 0

is fulﬁlled and, in particularly,

lim

λ→λ

β(I

, I

) = 0.

Proof. Suppose that the conditions of the theorem are fulﬁlled. According to the

condition 4. there exists a positive number R

such that I ⊂ B(0, R

). Reasoning

as in Theorem 11.10 for all ρ > 0 we will ﬁnd a L = L(

) > 0 and δ

= δ

(ρ) > 0

such that

(t, I

, ω) ⊆ B[I

ωt

, ρ]

for all t ≥ L and ω ∈ Ω and, consequently,

= ϕ

(t, I

σ(−t,ω)

, σ(−t, ω)) ⊆ B[I

, ρ]

for all ω ∈ Ω and ρ(λ, λ

) < δ

. The theorem is proved. 

Theorem 11.12 Let ε ∈ (0, ε

), Ω be compact and connected and ϕ

( ¯ϕ

) be a

cocycle generated by the equation (11.56) (respectively by the equation (11.61)).

Suppose that the following conditions are fulﬁlled:

(1) B(ω) := B

(ω) + B

(ω) (ω ∈ Ω), B

, B

∈ C(Ω, L

(E, F ));

(2) the bilinear forms B and B

satisfy the condition (11.19);

(3) the average of B

(ω) is equal to 0, i.e. lim

t→∞

(ωs)ds = 0 uniformly w.r.t.

ω ∈ Ω;

(4) the bilinear form B

satisﬁes the condition (11.32);

(5) the cocycles ϕ

and ¯ϕ

(ε ∈ (0, ε

])) are asymptotically compact .

Then the following statements are true:

a. for every ε ∈ (0, ε

]) and ω ∈ Ω the set I

:= {x ∈ E : the solution ϕ

(t, x, ω)

of equation (11.57) is deﬁned and bounded on R} (respectively

:= {x ∈

E | the solution ¯ϕ

(t, x, ω) of equation (11.61) is deﬁned and bounded on R})

is nonempty, compact and connected;

b. the cocycle ϕ

( ¯ϕ

) admits a compact global attractor {I

: ω ∈ Ω} (respectively

{

: ω ∈ Ω});

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

Global attractors of non-autonomous Navier-Stokes equations 383

c. the set I

(respectively

) is compact and connected;

d. the set I :=

: ε ∈ [0, ε

]} (respectively

I :=

{

: ε ∈ [0, ε

]},

where

{

: ω ∈ Ω}) is compact, where I

: ω ∈ Ω},

{

: ω ∈ Ω} and {

: ω ∈ Ω} is a compact global attractor of

equation (11.61), when ε = 1;

e. lim

ε→0

β(I

) = 0, where β is a semi-distance of Hausdorﬀ;

f. If a dynamical system (Ω, R, σ) is periodic, i.e. there exists ω

∈ Ω such that

τ = ω

and Ω = {ω

t : t ∈ [0, τ )}, then

lim

ε→0

sup

ω∈Ω

{β(I

, I

)} = 0.

Proof. Let ε ∈ (0, ε

), Ω be compact and connected and ϕ

( ¯ϕ

) be a cocycle

generated by the equation (11.56) (respectively by the equation (11.61)), then we

have

(t, x, ω) = ϕ(εt, x, ω, ε) (11.88)

(respectively ¯ϕ

(t, x, ω) = ¯ϕ(εt, x, ω, ε)

for all t ∈ R

, x ∈ E and ω ∈ Ω, where ϕ(·, ·, ·, ε) (respectively ¯ϕ(·, ·, ·, ε)) is a

cocycle generated by the equation (11.62) (respectively (11.63)). From the equality

(11.88) it follows that {I

: ω ∈ Ω} (respectively {

: ω ∈ Ω},) is a compact

global attractor of the equation (11.62) (respectively (11.63)). Now to ﬁnish the

proof of theorem it is suﬃcient to apply Theorems 11.9, 11.11, 7.7 and Lemmas 7.3,

7.4. The theorem is proved. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 11

384 Global Attractors of Non-autonomous Dissipative Dynamical Systems

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

Chapter 12

Global attractors of V -monotone

dynamical systems

Diﬀerential equations with monotone right hand side make one of the most studied

classes of nonlinear equations (see, for example,

[

]

[

185

]

[

240

]

[

316

]

[

333

]

and

the literature quoted there).

The problem of existence of almost periodic solutions of monotone nonlinear

almost periodic equation has been studied by many authors (see

[

113

]

[

114

]

[

184

]

[

189

]

[

238

]

[

316

]

[

333

]

and others).

The purpose of this chapter is to study the global attractors of general V - mono-

tone non-autonomous dynamical systems and their applications to diﬀerent classes

of diﬀerential equations (ODEs, ODEs with impulse, some classes of evolutionary

partial diﬀerential equations).

For autonomous equations an analogous problem was studied before (see, for

example,

[

]

[

]

[

314

]

), but for non-autonomous dynamical systems this problem

is considered for the ﬁrst time in our book.

12.1 Global attractors of V -monotone NDS

Lemma 12.1 Let (X, S

, π) be a compactly k-dissipative dynamical system with

the Levinson’s center J. Then the dynamical system (X × X, S

, π × π) (where

π×π(t, (x

, x

)) := (π(t, x

), π(t, x

)) for all t ∈ S and x

, x

∈ X) is also compactly

k-dissipative and J ×J is its Levinson’s center.

Proof. Let (X, S

, π) be a compactly k-dissipative dynamical system. Then there

exists a nonempty compact K ⊆ X such that

lim

t→+∞

β(π

M, K) = 0 (12.1)

for every M ∈ C(X). Let K

:= K × K and M

∈ C(X × X). Then from the

equality (12.1) follows that

lim

t→+∞

β((π × π)

, K

) = 0

385

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

386 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and, consequently, (X ×X, S

, π × π) is compactly k-dissipative and the set J

Ω(K

) is its Levinson’s center. We note that Ω(K × K) ⊆ Ω(K) × Ω(K) and,

consequently J

⊆ J ×J, because J = Ω(K). On the other hand, the set J ×J is a

compact invariant set for the dynamical system (X ×X, S

, π×π) and, consequently,

J × J ⊆ J

because J

is a maximal compact invariant set in (X × X, S

, π × π).

The Lemma is proved. 

Let Ω be a compact topological space , (E, h, Ω) be locally trivial Banach strat-

iﬁcation

[

]

and | · | be a norm on (E, h, Ω) co-ordinate with the metric ρ on E

(that is ρ(x

, x

) = |x

−x

| for any x

, x

∈ X such that h(x

) = h(x

)).

Recall that a non-autonomous dynamical system h(X,T

,π), (Ω,S,σ),hi is said

to be uniformly stable in positive direction on compacts of X

[

]

, if for arbitrary

ε > 0 and K ⊆ X there is δ = δ(ε, K) > 0 such that the inequality ρ(x

, x

) <

δ (h(x

) = h(x

)) implies that ρ(π

, π

) < ε for t ∈ T

Denote by X

×X = {(x

, x

) ∈ X × X | h(x

) = h(x

) }.

Deﬁnition 12.1 The non-autonomous dynamical system h(X, S

, π), (Ω, S, σ),

hi is called (see

[

113

]

[

114

]

and

[

333

]

[

238

]

) V -monotone, if there exists a function

V : X

×X → R

with the following properties:

a. V is continuous.

b. V is positively deﬁned, i.e. V (x

, x

) = 0 if and only if x

= x

c. V (x

t, x

t) ≤ V (x

, x

) for all (x

, x

) ∈ X

×X and t ∈ S

Theorem 12.1 Let h(X, S

, π), (Ω, S, σ), hi be V -monotone and compactly dissi-

pative, then it is uniformly stable in positive direction on compacts from X.

Proof. Let h(X, S

, π), (Ω, S, σ), hi be a V -monotone non-autonomous dynamical

system and let it be not uniformly stable in positive direction on compacts from X.

Then there are ε

> 0, a sequence {t

} ⊆ S

→ +∞ as n → +∞), a sequence

→ 0 (δ

> 0), a compact K

⊆ X and sequences {x

} ⊆ K

(i = 1, 2) such that

ρ(x

, x

) < δ

and ρ(x

, x

) ≥ ε

(12.2)

for all n ∈ N. Since the dynamical system h(X, S

, π), (Ω, S, σ), hi is com-

pactly dissipative, then we may suppose without loss of generality that the se-

quences {x

} (i = 1, 2) and {x

} (i = 1, 2) are convergent. We denote by

= lim

n→+∞

(i = 1, 2) and ¯x

= lim

n→+∞

(i = 1, 2). According to the in-

equality (12.2) we obtain x

= x

and ¯x

6= ¯x

. On the other hand, in view of V -

monotonicity of h(X, S

, π), (Ω, S, σ), hi, we have

V (x

, x

) ≤ V (x

, x

) (12.3)