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

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

Chapter 11

Global attractors of non-autonomous

Navier-Stokes equations

This chapter is devoted to the study of non-autonomous Navier-Stokes equations.

It is proved that such systems admit compact global attractors. This problem is

formulated and solved in the terms of general non-autonomous dynamical systems.

We give conditions of convergence of non-autonomous Navier-Stokes equations . A

test of existence of almost periodic (quasi periodic, recurrent, pseudo recurrent)

solutions of non-autonomous Navier-Stokes equations is given. We prove the global

averaging principle for non-autonomous Navier-Stokes equations.

We consider the two-dimensional Navier-Stokes system

+ q(t)

i=1

∂

u = ν∆u − ∇p + φ(t)

div u = 0, u|

∂D

= 0, (11.1)

where D is an open bounded set with boundary ∂D ∈ C

. This equation can be

written in the following form

+ Au + B(t)(u, u) = f(t) (11.2)

on the corresponding Sobolev’s space E, where −A is a Stokes operator, B(t) is a

bilinear form satisfying the identity

RehB(t)(u, v), wi = −RehB(t)(u, w), vi (11.3)

for all t ∈ R and u, v, w ∈ E, and f is forcing term.

In the work

[

112

]

[

140

]

[

199

]

[

200

]

there is studied a non-stationary equation

(11.2), when f is a function of time t ∈ R. It is shown that the equation with

compact f (in particularly, almost periodic) admits a compact global attractor and

also for small nonlinear (bilinear) term it was proved the existence a unique almost

periodic (quasi periodic, periodic) solution of equation (11.2) if the forcing term f

is almost periodic (quasi periodic, periodic).

The aim of the chapter is to study the equation (11.2) in the case, when the the

bilinear operator B, and the function f are non-stationary. The conditions under

357

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

358 Global Attractors of Non-autonomous Dissipative Dynamical Systems

which a non-stationary equation of type (11.2) admits a compact global attractor

are indicated.

The theorem of ”partial” averaging on ﬁnite interval for ordinary diﬀerential

equations it was proved in the work

[

183

]

. The works

[

140

]

[

199

]

and

[

200

]

are

devoted to generalization of method of averaging for dissipative partial diﬀerential

equations. We prove the theorem of ”partial” averaging for non-autonomous Navier-

Stokes equation (11.2) (i.e. the bilinear form and forcing term are non-stationaries).

This chapter is organized as follows:

In Section 1 we introduce a class of non-autonomous Navier-Stokes equations

and establish its dissipativity (Theorem 11.4).

In Section 2 we prove that non-autonomous Navier-Stokes equations admit a

compact global attractor (Theorem 11.6).

Section 3 is devoted to study of the problem of existence of almost periodic

(quasi periodic, recurrent, pseudo recurrent) solutions of non-autonomous Navier-

Stokes equations (Corollaries 11.3) and we give the conditions of convergence of this

equations (Theorem 11.8).

In Section 4 we prove the uniform averaging principle for the non-autonomous

Navier-Stokes equations on the ﬁnite segment (Theorem 11.9).

Section 5 is devoted to prove the global averaging principle for non-autonomous

Navier-Stokes equations on the semi-axis (Theorems 11.10,11.11 and 11.12).

11.1 Non-autonomous Navier-Stokes equations

Some results from the theory of semigroups of linear operators

[

188

]

and PDEs

[

199

]

[

294

]

[

314

]

are collected below.

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 ∈ R and ϕ ∈ (0,

) the sector

a,ϕ

:= {λ ∈ C, π ≥ |arg(λ − a)| ≥ ϕ} (11.4)

is contained in the resolvent set and for λ ∈ S

a,ϕ

k(λI −A)

−1

X→X

≤

|λ − a| + 1

. (11.5)

For a sectorial operator A the analytic semigroup of linear bounded operators

in X is deﬁned and denoted by e

−At

, t ≥ 0.

Let A be a sectorial operator with Reσ(A) > 0. For α ∈ (0, 1) we deﬁne fractional

powers of A as follows:

:= (A

−α

)

−1

, where A

−α

Γ(α)

∞

α−1

−At

dt.

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 359

The corresponding domains D(A

) are Banach spaces with norm given by

|· |

:= | · |

D(A

)

= |A

·|.

Theorem 11.1 The following estimates are valid:

(1)

−At

X→X

≤ Ce

−at

, t ≥ 0, (11.6)

(2)

−At

X→X

≤ C

−α

−at

, t > 0. (11.7)

Let Ω be a compact metric space, R = (−∞, +∞), (Ω, R, σ) be a dynamical

system on Ω, E be a real or complex Hilbert space, L(E) be the space of all linear

continuous operators on E, L

(E) be the space of all bilinear continuous operators

B : E ×E → F and C(Ω, W ) be a space of all continuous functions f : Ω → W (W

is some metric space), endowed with the topology of uniform convergence. Let us

consider the equation

+ Au + B(ωt)(u, u) = f(ωt), (11.8)

(ω ∈ Ω) where ωt := σ(t, ω), B ∈ C(Ω, L

(E)), f ∈ C(Ω, E) and A is a linear

operator.

Below we will use some notions, denotations and results from

[

200

]

. Let Hilbert

spaces E, F, X satisfy E ⊂ F ; E, F, X ⊂ E, each embedding being dense and

continuous.

Operator A. We further suppose that the linear operator A is densely deﬁned in

E and such that the linear equation

+ Au = 0 (11.9)

generates the c

-semigroup of linear bounded operators

−At

: E → E, ϕ(t, x) := e

−At

which for t > 0 can be extended to the linear bounded operators from F to E

satisfying the following estimates

−At

E→E

≤ Ke

−at

, (11.10)

−At

F →E

≤ Kt

−α

−at

, 0 ≤ α

< 1, (11.11)

kAe

−At

F →E

≤ Kt

−α

−at

, 0 ≤ α

< 2. (11.12)

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

360 Global Attractors of Non-autonomous Dissipative Dynamical Systems

We also suppose that the following condition is satisﬁed

= e

A, (11.13)

in the sense of L(F, E) := {A : F → E |A is linear and bounded } equipped with

the operationel norm.

Bilinear operator B. Denote by L

(E, F ) the space of all bilinear continuous

operators B : E ×E → F with the norm

kBk := sup{|B(u, v)|

: |u|

≤ 1, |v|

≤ 1}.

Let C(Ω, L

(E, F )) be a space of all continuous mappings B : Ω → L

(E, F ) and

:= sup{|B(ω)(u, v)|

: ω ∈ Ω, |u|

≤ 1, |v|

≤ 1},

then the mapping F : Ω × E → F (F (ω, u) := B(ω)(u, u)) satisﬁes the following

inequality

|B(ω)(u

, u

) − B(ω)(u

, u

≤ C

(|u

+ |u

)|u

−u

(11.14)

for all u

, u

∈ E.

From the inequality (11.14) follows that on every ball B[0, R] := {u ∈ E :

|u|

≤ R} we have

|B(ω)(u

, u

) − B(ω)(u

, u

≤ 2C

R|u

−u

(11.15)

for all u

, u

∈ E.

Remark 11.1 The space of all the bilinear continuous operators C(Ω, L

(E, F ))

is a Banach space with the norm kBk := C

Function f. The external force f : Ω → X is continuous, i.e. f ∈ C(Ω, X).

Operators e

−At

. The operators e

−At

(t > 0) can be extended to the linear

bounded operators from X to E satisfying the estimates

−At

X→E

≤ Kt

−β

−at

, 0 ≤ β

< 1, (11.16)

kAe

−At

X→E

≤ Kt

−β

−at

, 0 ≤ β

< 2, (11.17)

and the equation (11.13), this time in the sense of L(X, E) .

We suppose that the following conditions are fulﬁlled:

(1) there exists α > 0 such that

RehAu, ui ≥ α|u|

(11.18)

for all u ∈ E; ;

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 361

(2)

RehB(ω)(u, v), wi = −RehB(ω)(u, w), vi (11.19)

for every u, v, w ∈ E and ω ∈ Ω .

Remark 11.2 a. It follows from (11.19) that

RehB(ω)(u, v), v)i = 0 (11.20)

for every u, v ∈ E and ω ∈ Ω.

|B(ω)(u, v)|

≤ C

|u|

|v|

(11.21)

for all u, v ∈ E and ω ∈ Ω, where C

:= sup{|B(ω)(u, v)|

: ω ∈ Ω, u, v ∈

E, |u|

≤ 1, and |v|

≤ 1}.

The equation (11.8) with conditions (11.18) and (11.19) is called a non-

autonomous Navier-Stokes equation. We will consider the mild solutions of the

equation (11.8), i.e. u ∈ C([0, T ], E) and satisfy the following integral equation

u(t) = e

−At

x +

−A(t−s)

(−B(ωs)(u(s), u(s)) + f(ωs))ds. (11.22)

Theorem 11.2 Let x

∈ E, r > 0 and the conditions (11.10),(11.11) and (11.18)

are fulﬁlled, then there exist positive numbers δ = δ(x

, r) and T = T (x

, r) such

that the equation (11.22) admits a unique solution ϕ(t, x, ω) (x ∈ B[x

, δ] := {x ∈

E | |x − x

| ≤ δ}) deﬁned on the interval [0, T ] with the conditions: ϕ(0, x, ω) = x,

|ϕ(t, x, ω) − x

| ≤ r for all t ∈ [0, T ] and the mapping ϕ : [0, T ] × B[x

, δ] × Ω →

E ((t, x, ω) → ϕ(t, x, ω)) is continuous.

Proof. Let x

∈ E, r > 0, δ > 0 and T > 0. We consider a space C

,r,δ,T

of all

continuous functions ψ : [0, T ] ×B[x

, δ] ×Ω → B[x

, r] equipped with the distance

d(ψ

, ψ

) := sup{|ψ

(t, x, ω) − ψ(t, x, ω)|

: 0 ≤ t ≤ T, x ∈ B[x

, δ], ω ∈ Ω}

is a complete metric space.

We deﬁne the operator Φ acting onto C

,r,δ,T

by the equality

(Φψ)(t, x, ω) = e

−At

x +

−A(t−s)

(−B(ωs)(ψ(s, x, ω), ψ(s, x, ω)) + f(ωs))ds.

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

362 Global Attractors of Non-autonomous Dissipative Dynamical Systems

There exist δ

= δ

, r) > 0 and T

= T

, r) > 0 such that ΦC

,r,δ,T

⊆

,r,δ,T

for all δ ∈ (0, δ

] and T ∈ (0, T

]. In fact,

|(Φψ)(t, x, ω) −x

≤ |e

−At

x − x

−A(t−s)

B(ωs)(ψ(s, x, ω), ψ(s, x, ω)))ds|

+ |

−A(t−s)

f(ωs)ds|

≤

m(δ, T ) +

−a(t−s)

(t − s)

−α

|ψ(s, x, ω)|

ds +

−a(t−s)

(t − s)

−β

kfkds ≤ m(δ, T ) + K(|x

+ r)

1−α

1 − α

+Kkf k

1−β

1 − β

:= d

, r, δ, T ) → 0

as δ + T → 0, where m(δ, T ) := sup{|e

−tA

x − x

: t ∈ [0, T ], x ∈ B[x

, r]}

and kf k := sup{|f(ω)|

: ω ∈ Ω}. Thus there exist δ

= δ

, r) > 0 and

= T

, r) > 0 such that d

, r, δ, T ) ≤ r for all δ ∈ (0, δ

] and T ∈ (0, T

Let now ψ

, ψ

∈ C

,r,δ,T

, then

|(Φψ

)(t, x, ω)) − (Φψ

)(t, x, ω))|

[B(ωs)(ψ

(s, x, ω), ψ

(s, x, ω)) − B(ωs)(ψ

(s, x, ω), ψ

(s, x, ω))]|

≤

(|x

+ r)T d(ψ

, ψ

)

and, consequently, d(Φψ

, Φψ

) ≤ L(x

, r, T )d(ψ

, ψ

), where L(x

, r, T ) :=

(|x

| + r)T → 0 as T → 0. Thus there exists T

= T

, r) > 0 such

that L(x

, r, T ) < 1 for all T ∈ (0, T

]. Denote by δ(x

, r) := δ

, r) and

T (x

, r) := min(T

, r), T

, r)), then the mapping Φ : C

,r,δ,T

→ C

,r,δ,T

is a contraction and, consequently, there exists a unique function ϕ ∈ C

,r,δ,T

satisfying the equation (11.22) on the interval [0, T ]. The theorem is proved. 

Remark 11.3 The theorem 11.2 is true and for the equation

+ Au = F(ωt, u)

if the continuous function F : Ω × E → F satisﬁes the following conditions:

(1)

sup{|F(ω, 0)|

: ω ∈ Ω} < ∞

(Ω ,generally speaking, is not compact);

(2) F is locally Lipschitz, i.e. for every r > 0 there exists L(r) > 0 such that

|F(ω, u

) − F(ω, u

≤ L(r)|u

−u

for all u

, u

∈ E with condition: |u

≤ r (i = 1, 2).

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 363

Theorem 11.3 Let K be a family of solutions of equation (11.22) satisfying the

following condition: there exists a positive constant M such that |x(t)|

D(A)

≤ M

for all t ∈ R

(|x|

D(A)

:= |Ax|

). If there exists

> 0 such that

|B(ω)(u, v)|

≤

|u|

D(A)

|v|

D(A)

for all u, v ∈ D(A) , then this family of functions is uniform equicontinuous on

, i.e. for every ε > 0 there exists δ(ε) > 0 such that |t

− t

| < δ implies

|x(t

) − x(t

)| < ε for all t

, t

∈ R

and x ∈ K.

Proof. Let ψ ∈ K and x := ψ(0), then ψ(t) = ϕ(t, x, ω) for all t ∈ R

and we have

|ϕ(t, x, ω) − x|

≤ |e

−At

x − x|

+ |

−A(t−τ)

(−B(ωs)(ϕ(s, x, ω),

ϕ(s, x, ω)) + f(ωs))ds|

≤

−as

−α

|x|

D(A)

ds +

−as

(t − s)

−α

|ϕ(s, x, ω)|

D(A)

ds +

−a(t−s)

(t − s)

−β

kfkds ≤

1−α

1 − α

M + C

1−α

1 − α

+ kfk

1−β

1 − β

From the last inequality we obtain

sup{|ϕ(t, x, ω) − x|

| |x|

D(A)

≤ M, ω ∈ Ω} → 0

as t → 0 and, consequently,

|ϕ(t

, x, ω) − ϕ(t

, x, ω)|

= |ϕ(t

−t

, ϕ(t

, x, ω), ωt

) − ϕ(t

, x, ω)|

≤

sup{|ϕ(t

−t

, x, ω) − x|

: |x|

D(A)

≤ M, ω ∈ Ω} → 0

as t

−t

→ 0. The theorem is proved. 

Example 11.1 We consider the two-dimensional Navier-Stokes system

+ q(t)

i=1

∂

u = ν∆u − ∇p + φ(t) (11.23)

div u = 0, u|

∂D

= 0,

where D is an open bounded set with smooth boundary ∂D ∈ C

The functional setting of the problem is well known

[

228

]

[

311

]

. We denote

by H and V the closure of the linear space {u ∈ C

∞

(D)

, div u = 0} in L

(D)

and H

(D)

, respectively. Denote by P the corresponding orthogonal projection

P : L

(D)

→ H. We further set

A := −νP ∆, B(t)(u, v) := q(t)P (

i=1

∂

v).

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

364 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The Stokes operator A is self-adjoint positive with domain D(A) dense in H.

The inverse operator is compact. We deﬁne the Hilbert spaces D(A

), α ∈ (0, 1] as

the domains of the powers of A in the standard way. Furthermore, V := D(A

1/2

and |u|

D(A

1/2

)

= |∇u|.

Applying P we write (11.23) as the evolution equation of the following form

+ Au + B(t)(u, u) = F(t), F(t) := P φ(t). (11.24)

Let F ∈C(R,H) (X := H) and B ∈C(R,L

(H,D(A

−δ

)) (F := D(A

−δ

)). De-

note by Y := C(R, H)×C(R,L

(H,D(A

−δ

))) and (Y, R, σ) a dynamical system of

translations (Bebutov’s dynamical system, see for example,

[

300

]

[

302

]

and

[

292

]

Let Ω := H(B, F) =

{(B

, F

)| τ ∈ R}, where B

(t) := B(t + τ) ( respectively

(t) := F(t + τ)) for all t ∈ R, by bar we denote a closure in the compact-open

topology and (Ω, R, σ) be a dynamical system of translations on Ω.

Along with the equation (11.24) we consider its H-class

+ Au +

B(t)(u, u) =

F(t), (11.25)

where (

F) ∈ H(B, F). Let B : Ω → L

(H, D(A

−δ

)) (respectively f : Ω → H) be

a mapping deﬁned by equality

B(ω) = B(

F) :=

B(0) (f(ω) = f(

F) :=

F(0)),

where ω = (

F) ∈ Ω, then the equation (11.24) and its H-class can be written in

the form (11.21).

We now set in the notation above E := D(A

1/2

), X := H, F := D(A

−δ

) and see

that (11.10)-(11.12), (11.18) and (11.16)-(11.17) are valid with α

= 1/2 + δ, β

1/2, β

= 3/2.

According to Theorem 11.2 through every point x ∈ H passes a unique solution

ϕ(t, x, ω) of equation (11.8) at the initial moment t = 0. And this solution is deﬁned

on some interval [0, t

(x,ω)

). Let us note, that

(t) = 2Rehϕ

(t, x, ω), ϕ(t, x, ω)i = 2RehA(ωt)ϕ(t, x, ω), ϕ(t, x, ω)i

+2RehB(ωt)(ϕ(t, x, ω), ϕ(t, x, ω)), ϕ(t, x, y)i + 2Rehf(ωt), ϕ(t, x, ω)i

= 2RehA(ωt)ϕ(t, x, ω), ϕ(t, x, ω)i + 2Rehf(ωt), ϕ(t, x, ω)i

≤ −2α|ϕ(t, x, ω)|

+ 2kfk|ϕ(t, x, ω)|

, (11.26)

where kfk := max{|f(ω)|

: ω ∈ Ω} and w(t) = |ϕ(t, x, ω)|

. Then

≤ −2αw + 2kf kw

(11.27)

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 365

and consequently

w(t) ≤ v(t) (11.28)

for all t ∈ [0, t

(x,ω)

), where v(t) is an upper solution of equation

= −2αv + 2||f||v

, (11.29)

satisfying condition v(0) = w(0) = |x|

. Thus

v(t) = [(|x|

−

kfk

−αt

kfk

]

(11.30)

and consequently

|ϕ(t, x, ω)|

≤ (|x|

−

kfk

−αt

kfk

(11.31)

for all t ∈ [0, t

(x,ω)

). It follows from the inequality (11.27) that solution ϕ(t, x, ω) is

bounded and therefore it may be prolonged on R

= [0, +∞).

Thus we have proved the following theorem.

Theorem 11.4 Let the conditions (11.18) and (11.19) are fulﬁlled. Then the

following statements hold:

(i) Every solution ϕ(t, x, ω) of non-autonomous Navier-Stokes equation (11.8) is

bounded and therefore it may be prolonged on R

(ii)

|ϕ(t, x, ω)|

≤ C(|x|

), (11.32)

for all t ≥ 0, ω ∈ Ω and x ∈ E, where C(r) = r if r ≥ r

kfk

and

C(r) = r

if r ≤ r

;

(iii)

lim sup

t→+∞

sup{|ϕ(t, x, ω)|

: |x|

≤ r, ω ∈ Ω} ≤

||f||

(11.33)

for every r > 0.

Lemma 11.1 Under the conditions of Theorem 11.4 we have

t+l

|ϕ(τ, x, ω)|

dτ ≤

2α

lkf k := M (r) (11.34)

for all t ≥ 0 and r ≥ r

Proof. From the equality (11.27) after integration in t between t and t+l we obtain

2α

t+l

|ϕ(τ, x, ω)|

dτ ≤ |ϕ(t, x, ω)|

+ 2rlkfk (11.35)

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

366 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and,consequently,

t+l

|ϕ(τ, x, ω)|

dτ ≤

2α

lkf k := M(r). (11.36)



Lemma 11.2 (

[

314, Ch.3

]

) (The Uniform Gronwall Lemma). Let g, h, y, be

three positive locally integrable functions on ]t

, ∞[ such that y

is locally integrable

on ]t

, ∞[, and which satisfy

≤ gy + h for t ≥ t

t+l

g(s)ds ≤ a

t+l

h(s)ds ≤ a

t+l

y(s)ds ≤ a

for t ≥ t

where l, a

, a

, are positive constants. Then

y(t + l) ≤ (

+ a

∀ t ≥ t

Theorem 11.5 Under the conditions of Theorem 11.4 if

|hB(ω)(u, v), wi| ≤ C|u|

1/2

|Au|

1/2

|v|

1/2

|w| (11.37)

∀ u ∈ D(A), v ∈ V, w ∈ H,

then

|ϕ(t, x, ω)|

D(A)

≤ K(r) ∀|x| ≤ r (r ≥ r

) (11.38)

for all t ≥ 0 and ω ∈ Ω, where K(r) is some positive constant depending only on r.

Proof. Since

hAϕ(t, x, ω), ϕ(t, x, ω)i =

|ϕ(t, x, ω)|

(11.39)

by taking the scalar product of (11.8) with Au we ﬁnd

|ϕ(t, x, ω)|

+ |Aϕ(t, x, ω)|

+ (11.40)

hB(ϕ(t, x, ω), ϕ(t, x, ω)), Aϕ(t, x, ω)i = hf(ωt), Aϕ(t, x, ω)i.

Taking into account the inequality

|hf(ωt), Aϕ(t, x, ω)i| ≤ |f (ωt)|

|Aϕ(t, x, ω)|

≤

|Aϕ(t, x, ω)|

+ kfk

(11.41)