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

Global attractors of non-autonomous Navier-Stokes equations 367

and using (11.39) and the Young inequality we obtain

|hB(ω)(ϕ(t, x, ω), ϕ(t, x, ω)), Aϕ(t, x, ω)i| ≤

|ϕ(t, x, ω)|

1/2

kϕ(t, x, ω)k|Aϕ(t, x, ω)|

3/2

≤ (11.42)

|Aϕ(t, x, ω)|

+ c

|ϕ(t, x, ω)|

kϕ(t, x, ω)k

Hence

kϕ(t, x, ω)k

+ |Aϕ(t, x, ω)|

≤ kfk

|Aϕ(t, x, ω)|

|ϕ(t, x, ω)|

kϕ(t, x, ω)k

. (11.43)

From this inequality according to Gronwal lemma we can prove that

|ϕ(t, x, ω)|

D(A)

is uniformly (w.r.t. x and ω) bounded on interval [0, l]. Applying

the uniform Gronwal lemma with g, h, y replaced by

|ϕ(t, x, ω)|

kϕ(t, x, ω)k

, kfk

, kϕ(t, x, ω)k

(11.44)

we obtain that kϕ(t, x, ω)k

is bounded on [l, ∞[ and, consequently, it is bounded

on [0, ∞[ uniformly w.r.t. kxk ≤ r and ω ∈ Ω. The theorem is proved. 

11.2 Attractors of non-autonomous dynamical systems

Lemma 11.3 The non-autonomous Navier-Stokes equation (11.8) generates a

cocycle ϕ ( or more explicit hE, ϕ, (Ω, R, σ)i), where ϕ(t, x, ω) is a unique solution

of equation (11.8) deﬁned on R

with the initial condition ϕ(0, x, ω) = x.

Proof. In fact, according to Theorems 11.2 the mapping ϕ : ×E × Ω →

E ((t, x, ω) → ϕ(t, x, ω)) is continuous and in view of uniqueness of solution

ϕ(t, x, ω) we have the following identity: ϕ(t + τ, x, ω) = ϕ(t, ϕ(τ, x, ω), ωτ) for

all t, τ ∈ R

, x ∈ E and ω ∈ Ω, where ωτ := σ(τ, ω). 

Example 11.2 Let E be a Banach space and C(R × E, E) be a space of all

continuous functions F : R × E → E equipped with the compact-open topology.

Let us consider a parameterized diﬀerential equation

+ Ax = F (σ

ω, x) (ω ∈ Ω)

on a Banach space E with Ω := C(R × E, E), where σ

ω := σ(t, ω) and the linear

operator A is densely deﬁned in E and such that the linear equation

+ Au = 0

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

368 Global Attractors of Non-autonomous Dissipative Dynamical Systems

generates the c

-semigroup of linear bounded operators

−At

: E → E, ϕ(t, x) := e

−At

We will deﬁne σ

: Ω → Ω by σ

ω(·, ·) = ω(t + ·, ·) for each t ∈ R and interpret

ϕ(t, x, ω) as mild solution of the initial value problem

x(t) + Ax = F (σ

ω, x(t)), x(0) = x. (11.45)

Under appropriate assumptions on F : Ω × E → E (or even F : R ×E → E with

ω(t) instead of σ

ω in (11.45)) to ensure forwards existence and uniqueness, then

ϕ is a cocycle on (C(R × E, E), R, σ) with ﬁber E, where (C(R × E, E), R, σ) is a

Bebutov’s dynamical system (see for example

[

]

[

102

]

[

292

]

)

[

300

]

Deﬁnition 11.1 Recall that the cocycle hW, ϕ, (Y, R, σ)i is called compact

(asymptotically compact) if the skew-product dynamical system (X, R

, π) (X =

W × Y, π = (ϕ, σ)) is compact (respectively asymptotic compact).

Let (X, R

, π) be compact dissipative and K be a compact set, which attracts

all compact subsets of X. Suppose

J = Ω(K), (11.46)

where Ω(K) =

t≥0

τ≥t

K. The set J is called the Levinson’s center of the

compact dissipative system (X, R

, π).

Applying general theorems about non-autonomous dissipative systems (see

Chapter 2) to non-autonomous system constructed in the example 11.2, we will

obtain series of facts concerning the equation (11.8). In particular, from Theorems

11.4, 2.19 and 2.20 follows the theorem below.

Theorem 11.6 Let Ω be a compact metric space, (Ω, R, σ) be a dynamical system

on Ω and the conditions (11.18) and (11.19) are fulﬁlled. If the cocycle ϕ generated

by non-autonomous Navier-Stokes equation is asymptotically compact, then for ev-

ery ω ∈ Ω there exists a non-empty compact and connected I

⊂ E such that the

following conditions hold:

(1) the set I = ∪{I

: ω ∈ Ω} is compact and connected in E;

(2) I is connected if Ω is connected;

(3)

lim

t→+∞

sup

ω∈Ω

β(U(t, ω

−t

)M, I) = 0

for any bounded set M ⊂ E, where U (t, ω) = ϕ(t, ·, ω) and β is the semi-

distance of Hausdorﬀ;

(4) U (t, ω)I

= I

ωt

for all t ∈ R

and 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 369

(5) I

consists of those and only those points x ∈ E through which passes the

solution of the equation (11.8) bounded on R.

Theorem 11.7 Under conditions of Theorem 11.6

|ϕ(t, x, ω)| ≤

kfk

for all t ∈ R, ω ∈ Ω and x ∈ I

, where ϕ is a cocycle, generated by non-autonomous

Navier-Stokes equation.

Proof. The set J =

× {ω} : ω ∈ Ω} is a Levinson’s center of dynamical

system (X, R

, π) and according to (11.46) for any point (u

, y

) = z ∈ J there

exists t

→ +∞, u

∈ E and ω

∈ Ω such that the sequence {u

} is bounded,

= lim

n→+∞

ϕ(t

, u

, ω

) and ω

= lim

n→+∞

. From the inequality (11.27) follows

that |u

| ≤

kfk

, i.e. ϕ(t, x, ω) ∈ I

ωt

for all ω ∈ Ω and t ≥ 0, hence |ϕ(t, x, ω)| ≤

kfk

for any t ∈ R, x ∈ I

and ω ∈ Ω. The theorem is proved. 

11.3 Almost periodic and recurrent solutions of non-autonomous

Navier-Stokes equations

Deﬁnition 11.2 An autonomous dynamical system (Ω, T, σ) is said to be pseudo

recurrent if the following conditions are fulﬁlled:

a) Ω is compact;

b) (Ω, T, σ) is transitive, i.e. there exists a point ω

∈ Ω such that Ω =

{ω

t | t ∈ T};

c) every point ω ∈ Ω is stable in the sense of Poisson, i.e.

= {{t

} | ωt

→ ω and |t

| → +∞} 6= ∅.

Lemma 11.4 (

[

105

]

) Let h(X, T

, π), (Ω, T

, σ), hi be a non-autonomous

dynamical system and the following conditions are fulﬁlled:

1) (Ω, T

, σ) is pseudo recurrent;

2) γ ∈ C(Ω, X) is an invariant section of the homomorphism h : X → Ω, i.e.

h(γ(ω)) = ω and γ(σ(t, ω)) = π(t, γ(ω)) for all ω ∈ Ω and t ∈ T

Then the autonomous dynamical system (γ(Ω), T

, π) is pseudo recurrent too.

Deﬁnition 11.3 The solution ϕ(t, x, ω) of non-autonomous Navier-Stokes equa-

tion (11.8) is called recurrent (pseudo recurrent, almost periodic, quasi periodic),

if the point (x, ω) ∈ H ×Ω is a recurrent (pseudo recurrent, almost periodic, quasi

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

370 Global Attractors of Non-autonomous Dissipative Dynamical Systems

periodic) point of skew-product dynamical system (X, R

, π) (X = H × Ω and

π = (ϕ, σ)).

We note (see, for example,

[

238

]

and

[

300

]

[

302

]

) that if ω ∈ Ω is a stationary

(τ-periodic, almost periodic, quasi periodic, recurrent) point of dynamical system

(Ω, R, σ) and h : Ω → X is a homomorphism of dynamical system (Ω, R, σ) onto

(X, R

, π), then the point x = h(ω) is a stationary (τ-periodic, almost periodic,

quasi periodic, recurrent) point of the system (X, R

, π).

Let X := H × Ω and π := (ϕ, σ), then mapping h : Ω → X is a homomorphism

of dynamical system (Ω, R, σ) onto (X, R

, π) if and only if h(ω) = (u(ω), ω) for

all ω ∈ Ω, where u : Ω → H is a continuous mapping with the condition that

u(ωt) = ϕ(t, u(ω), ω) for all ω ∈ Ω and t ∈ R

The following aﬃrmations hold:

Lemma 11.5 Let Ω be a compact metric space, A be a linear operator densely

deﬁned in E such that the equation

+ Ax = 0

generates a c

-semigroup {U(t)}

t≥0

. If the condition (11.18) is fulﬁlled, then

kU(t)k ≤ exp(−αt)

for all t ∈ R

, where U(t) is a Cauchy operator of equation (11.48).

Proof. Let ϕ(t, x) := U(t)x, then according to the inequality 11.18 we have

|ϕ(t, x)|

≤ −2α|ϕ(t, x)|

and, consequently,|ϕ(t, x)| ≤ exp(−αt)|x| for all x ∈ H and t ∈ R

. Thus we have

|U(t)x| ≤ exp(−αt)|x|, therefore kU(t)k ≤ exp(−αt) for all t ∈ R

. 

Lemma 11.6 Suppose that the condition (11.18) is fulﬁlled. Then for every func-

tion f ∈ C(Ω, H) there exists a unique function γ ∈ C(Ω, H) deﬁned by equality

γ(ω) =

−∞

U(−τ )f(ωτ)dτ

such that

γ(ωt) = ϕ(t, γ(ω), ω) (11.48)

for every ω ∈ Ω and t ∈ R

, where ϕ(t, x, ω) is a solution of equation

= Au + f(ωt)

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 371

with the initial condition ϕ(0, x, ω) = x and the following inequality

kγk ≤

kfk

takes place.

Proof. The formulated statement results from Lemma 11.5 and Proposition 7.33

from

[

]

. 

Lemma 11.7 Let Ω be a compact metric space, the cocycle ϕ, generated by the

non-autonomous Navier-Stokes equation (11.8) and α

−2

kfkC

< 1, then the follow-

ing inequality

|ϕ(t, x

, ω) −ϕ(t, x

, ω)| ≤ e

−(α−C

kfk

−x

, x

∈ B[0, r

], r

kfk

, ω ∈ Ω and t ∈ R

)

takes place.

Proof. Let r

kfk

and x

, x

∈ B[0, r

] := {x ∈ E : |x| ≤ r

}. According to

Theorem 11.4 we have |ϕ(t, x

, ω)| ≤ r

for all t ≥ 0, ω ∈ Ω and i = 1, 2. Denote by

ψ(t) := ϕ(t, x

, ω) −ϕ(t, x

, ω), then we obtain

|ψ(t)|

= 2RehAψ(t), ψ(t)i + 2RehB(ωt)(ψ(t), ϕ(t, x

, ω)), ψ(t)i ≤

−2α|ψ(t)|

+ 2C

|ϕ(t, x

, ω)||ψ(t)|

≤

−2α|ψ(t)|

+ 2C

kfk

|ψ(t)|

= −2(α − C

kfk

)|ψ(t)|

and, consequently,

|ψ(t)|

≤ e

−2(α−C

kfk

|ψ(0)|

Thus we have

|ϕ(t, x

, ω) −ϕ(t, x

, ω)| ≤ e

−(α−C

kfk

−x

, x

∈ B[0, r

], ω ∈ Ω and t ∈ R

)

for all x

, x

∈ B[0, r

], ω ∈ Ω and t ∈ R

. The Lemma is proved. 

Theorem 11.8 Let Ω be a compact metric space, ϕ be the cocycle, generated by

the non-autonomous Navier-Stokes equation (11.8) and

kfkC

< 1. Then there

exists a function γ ∈ C(Ω, B[0, r

]) such that:

γ(ωt) = ϕ(t, γ(ω), ω) (11.49)

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

372 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for every ω ∈ Ω and t ∈ R

, where ϕ(t, x, ω) is a solution of equation (11.8)

with the initial condition ϕ(0, x, ω) = x;

kγk ≤

kfk

;

(11.50)

|ϕ(t, x, ω) − γ(ωt)| ≤ e

−(α−C

kfk

|x − γ(ω)| (11.51)

for all x ∈ E, ω ∈ Ω and t ∈ R

, where kγk := sup{|γ(ω)| : ω ∈ Ω}.

Proof. Let Γ := C(Ω, B[0, r

]) (C(Ω, E)) be a space all the continuous functions

f : Ω → B[0, r

] (respectively f : Ω → E ) equipped with the distance

d(f

, f

) = max{|f

(ω) − f

(ω)| : ω ∈ Ω} .

Then (Γ, d) ( respectively (C(Ω, E), d)) is a complete metric space.

Let t ∈ R

. We deﬁne the mapping S

: Γ → C(Ω, E) by the equality

ν)(ω) := U(t, ω

−t

)ν(ω

−t

)

for all ω ∈ Ω, where ω

−t

:= σ(−t, ω) and U (t, ω) := ϕ(t, ·, ω). According to Theorem

11.4 we have S

(Γ) ⊆ Γ for all t ∈ R

. It easy to see that the family of mappings

| t ∈ R

} possesses the following properties:

(1)

= Id

and

(2)

t+τ

= S

for all t, τ ∈ R

Thus {S

| t ∈ R

} forms a commutative semigroup with identity element. Now

we will show that the mapping S

(t > 0) is a contraction. In fact, let ν

, ν

∈ Γ,

then we have

)(ω) − (S

)(ω) = U(t, ω

−t

)ν

(ω

−t

) − U (t, ω

−t

)ν

(ω

−t

). (11.52)

From the lemma 11.7 and the equality (11.52) it follows that

d(S

, S

) ≤ e

−(α−C

kfk

d(ν

, ν

)

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 373

for all t ∈ R

and, consequently, there exists a unique common ﬁxed point γ ∈ Γ,

i.e. S

γ = γ for all t ∈ R

. In particularly

U(t, ω

−t

)γ(ω

−t

) = γ(ω)

for all t ∈ R

and ω ∈ Ω. From this equality follows that

γ(ωt) = U(t, ω)γ(ω) = ϕ(t, γ(ω), ω)

for all t ∈ R

and ω ∈ Ω.

Let x ∈ E, ϕ(t, x, ω) be a unique solution of equation (11.8) with the initial

condition ϕ(0, x, ω) = x and γ ∈ Γ the function with the property (11.49). Denote

by ψ(t) := ϕ(t, x, ω) −γ(ωt), then we have

|ψ(t)|

= 2RehAψ(t), ψ(t)i + 2RehB(ωt)(ψ(t), γ(ωt)), ψ(t)i ≤

−2α|ψ(t)|

+ 2C

|γ(ωt)||ψ(t)|

≤ −2α|ψ(t)|

kfk

|ψ(t)|

= −2(α − C

kfk

)|ψ(t)|

and, consequently,

|ψ(t)|

≤ e

−2(α−C

kfk

|ψ(0)|

Thus we have

|ϕ(t, x, ω) − γ(ωt)| ≤ e

−(α−C

kfk

|x − γ(ω)|

for all x ∈ E, ω ∈ Ω and t ∈ R

. The theorem is proved. 

Corollary 11.1 Under the conditions of Theorem 11.8 there exists a unique func-

tion γ ∈ C(Ω, E) such that

γ(ωt) = ϕ(t, γ(ω), ω) (11.53)

for all t ∈ R

and ω ∈ Ω.

Proof. Let

γ ∈ C(Ω, E) be a function satisfying the equality (11.53) and γ ∈ Γ =

C(Ω, B[0, r

]) the function from Theorem 11.8. Since

γ(ωt) = ϕ(t, γ(ω), ω) for all

t ∈ R

and ω ∈ Ω, then according to the inequality (11.51) we have

γ(ωt) − γ(ωt)| ≤ e

−(α−C

kfk

γ(ω) − γ(ω)| (11.54)

for all t ∈ R

and ω ∈ Ω. In particularly, from (11.54) we obtain

γ(ω) − γ(ω)| ≤ | ≤ e

−(α−C

kfk

γ − γk (11.55)

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

374 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for all t ∈ R

and ω ∈ Ω, where ω

−t

:= σ(−t, ω) and k

γ−γk := max{|γ(ω)−γ(ω)| :

ω ∈ Ω}. Passing to the limit in the inequality (11.55) we obtain

γ(ω) = γ(ω) for all

ω ∈ Ω. 

Corollary 11.2 Under the conditions of Theorem 11.8 the equation (11.8) admits

a compact global attractor {I

: ω ∈ Ω} and I

= {γ(ω)} for all ω ∈ Ω, where

γ ∈ Γ is a function from Theorem 11.8.

Corollary 11.3 Let Ω be a compact minimal set containing only the periodic

(quasi periodic, almost periodic, recurrent, pseudo recurrent) motions, then under

conditions of Theorem 11.8 the non-autonomous Navier-Stokes equation (11.8))

admits a unique periodic (quasi periodic, almost periodic, recurrent, pseudo recur-

rent) solution γ(ωt) and every other solution of this equation is asymptotic peri-

odic (asymptotic quasi periodic, asymptotic almost periodic, asymptotic recurrent,

asymptotic pseudo recurrent).

Proof. Let γ ∈ Γ be a function from Theorem 11.8, then according this theorem

we have ϕ(t, γ(ω), ω) = γ(ωt) for all t ∈ R

and ω ∈ Ω and, consequently, the

solution ϕ(t, γ(ω), ω) is periodic (quasi periodic, almost periodic, recurrent). Let

ϕ(t, x, ω) be a arbitrary solution of equation (11.8), then taking into consideration

the inequality (11.51) we conclude that ϕ(t, x, ω) is asymptotic periodic (asymp-

totic quasi periodic, asymptotic almost periodic, asymptotic recurrent, asymptotic

pseudo recurrent). 

11.4 Uniform averaging for a ﬁnite interval

We shall be dealing with the non-autonomous Navier-Stokes equation

+ εAu + εB(ωt)(u, u) = εf (ωt), (11.56)

where ε ∈ [0, ε

], A is linear and B(ω) is a bilinear operator, f is a forcing term.

Existence of partial averaged. Below we will suppose that B(ω) = B

(ω) +

(ω) (B

, B

∈ C(Ω, L

(E, F )) for all ω ∈ Ω and the average of B

(ω) is equal to

0, that is,

lim

t→∞

(ωτ)dτ = 0 (11.57)

uniformly with respect to ω ∈ Ω.

Remark 11.4 1. The condition (11.57) is fulﬁlled if a dynamical system (Ω, R, σ)

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

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 375

2. According to Lemma 5.1 from

[

104

]

the equality (11.57) takes place if and

only if there exists a positive continuous on R

function k with lim

t→∞

k(t) = 0 such

that

(ωτ)dτk

(E,F )

≤ k(t) (11.58)

for all ω ∈ Ω and t ∈ R

The bilinear operator B

∈ C(Ω, L

(E, F )) satisﬁes the condition

RehB

(ω)(u, v), wi = −RehB

(ω)(u, w), vi (11.59)

for all u, v ∈ E and w ∈ F.

The forcing term f (ω) = f

(ω) + f

(ω) for all ω ∈ Ω (f

, f

∈ C(Ω, X)) and

the function f

has the average which is equal to 0, i.e. there exists a positive

continuous on R

function k

with lim

t→∞

(t) = 0 such that

(ωτ)dτ|

≤ k

(t) (11.60)

for all ω ∈ Ω and t ∈ R

Along with equation (11.56) we consider also the partial averaged equation

+ εAx + εB

(ωt)(u, u) = εf

(ωt). (11.61)

If we introduce the ”slow time” τ := εt (ε > 0), then the equations (11.56) and

(11.61) can be written in the following way

+ Au + B(ω

)(u, u) = f(ω

) (11.62)

and

¯u

+ A¯u + B

(ω

)(¯u, ¯u) = f

(ω

). (11.63)

We will consider the mild solutions u(t) and ¯u(t) of the equations (11.62) and

(11.63), i.e. u, ¯u ∈ C([0, T ], E) and satisfy the following integral equations

u(τ) = e

−Aτ

x +

−A(τ−s)

(−B(ω

)(u(s), u(s)) + f(ω

))ds, (11.64)

and

¯u(τ) = e

−Aτ

x +

−A(τ−s)

(−B

(ω

)(¯u(s), ¯u(s)) + f

(ω

))ds. (11.65)

Denote by ϕ(τ, x, ω, ε) ( ¯ϕ(τ, x, ω, ε)) a unique solution of equation (11.64) (re-

spectively (11.65)). According to Theorem 11.4 the cocycle ϕ(·, ·, ·, ε) ( ¯ϕ(·, ·, ·, ε)),

generated by equation (11.64) (respectively (11.65)), has an absorbing ball

B[0, R

] (B[0,

]) in E, where R

kfk

(

). This means that for

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

376 Global Attractors of Non-autonomous Dissipative Dynamical Systems

every ball B[0, R] (respectively B[0,

R]) there exists a positive number L = L(R)

(respectively

L =

R)) such that

U(t, ω, ε)B[0, R] ⊆ B[0, R

] (11.66)

(

U(t, ω, ε)B[0,

R] ⊆ B[0,

]) (11.67)

for all t ≥ L (t ≥

L), ε ∈ [0, ε

] and ω ∈ Ω, where U(t, ω, ε) := ϕ(t, ·, ω, ε)

U(t, ω, ε) := ¯ϕ(t, ·, ω, ε).

According to Theorem 11.4 the cocycle ϕ(·, ·, ·, ε) ( ¯ϕ(·, ·, ·, ε)) is uniformly

bounded for t ≥ 0, that is, for every ball B[0, R

] (B[0,

]) there exists a ball

B[0, R

] (B[0,

]) such that

U(t, ω, ε)B[0, R

] ⊆ B[0, R

] (11.68)

(

U(t, ω, ε)B[0,

] ⊆ B[0,

]) (11.69)

for all t ≥ 0, ε ∈ [0, ε

] and ω ∈ Ω.

Let C(R×E, E) be the space of all continuous functions f : R×E → E equipped

with compact open topology and let F ⊆ C(R ×E, E).

Theorem 11.9 Let L > 0 be arbitrary but ﬁxed. If ϕ(0, x, ω, ε) = ¯ϕ(0, x, ω, ε) =

x ∈ B[0,

], that is, the initial points coincide and belong to the absorbing ball of

equation (11.63) and the condition (11.32) is fulﬁlled, then the following relation

takes place

sup{|ϕ(t, x, ω, ε) − ¯ϕ(t, x, ω, ε)|

: 0 ≤ t ≤ L, |x|

≤

, ω ∈ Ω} → 0 (11.70)

as ε → 0.

Proof. The proof below goes along the same lines as the proofs of the correspond-

ing results from

[

140

]

[

199

]

and

[

200

]

. We set v(t) := ϕ(t, x, ω, ε) − ¯ϕ(t, x, ω, ε).

Subtracting the equation (11.64) from the equation (11.65), we obtain

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

(ωs)ds (11.71)