text: 14 A. KOSTIANKO, E. TITI AND S. ZELIK → Passing now to the limit t t in the (cid:12)right-hand side of (3.27), we see that, for almost all ∈ 1 2 (cid:12) t [0, T ], 2 d (cid:107) (cid:107) ≤ v(t) 2 2(∂ u(t ), v(t )) dt H t=t2 t 2 2 (recall that →the existence of the derivativ(cid:12)e has already been proved). Analogously, passing to (cid:12) the limit t t in the left-hand side of inequality (3.27), we get 2 1 d (cid:107) (cid:107) ≥ v(t) 2 2(∂ u(t ), v(t )) dt H t=t1 t 1 1 ∈ for almost all t [0, T ]. The last two inequalities imply (3.24) and ﬁnish the proof of the 1 (cid:3) lemma. Now, it is not diﬃcult to ﬁnish the proof of the theorem. Indeed, the already proved lemma justiﬁes the derivation of the key inequality (3.23) for the case where the solution u(t) is smooth (which will be the case if we start from smooth initial data). The validity of the desired estimate ∈ (3.14) for any u(0) H follows then by the usual approximation arguments. Thus, Theorem 0 (cid:3) 3.6 is proved. Corollary 3.9. Let the assumptions of Theorem (3.6) hold. Then, the Kuramoto-Sivashinsky A equation (1.3) possesses a global attractor (L), in the phase space H . Moreover, this global KS 0 attractor is smooth and (cid:107)A (cid:107) ≤ (3.28) (L) C (L), KS Hs s for some positive constant C dependent on s, a and L. In particular, C (L) = CL with some 0 C independent of L, but dependent on the parameter a. Proof. Indeed, analogously to (3.2), we have the smoothing property on a ﬁnite interval for the parabolic equation (1.3) (see, e.g., [26]): (cid:107) (cid:107) ≤ (cid:107) (cid:107) (3.29) u(t + 1) C(1 + u(t) Ns) Hs H for some positive C and N . Moreover, the analogue of the Lipschitz estimate (3.3) also holds s for this equation (and can be proved analogously to the proof given in Theorem 3.1). Thus, the proof of the corollary repeats word by word the proof of Corollary 3.4 and, for this reason, is (cid:3) omitted. A Remark 3.10. As before, one can show that the global attractor is analytic in the sense KS of Gevrey class. However, in contrast to the cases of Ginzburg-Landau equations, its Gevrey → ∞ norm is not uniformly bounded as L . As we will see in the next section, the norms of the A → ∞ global attractors (L) indeed grow, as L . KS → ∞ 4. The limit L The main aim of this section is to study the dependence of the above constructed global → ∞ attractors on L, as L . We start with the case of complex Ginzburg-Landau equations (cid:40) (1.1) and (1.2). To this end, we change the dependent variable u as follows H u(t) = (t)w(t) for the case of equation (1.2), L (4.1) F u(t) = (t)w(t) for the case of equation (1.1), L ##LTLine## ##LTLine##

text: A.

text: KOSTIANKO,

text: E.

text: TITI

text: AND

text: S.

text: ZELIK

text: Passing

text: now

text: to

text: the

text: limit

text: t

text: t

text: in

text: the

text: side

text: of

text: we

text: see

text: that,

text: for

text: almost

text: all

text: t

text: T

text: d

text: v(t)

text: u(t

text: v(t

text: dt

text: H

text: t

text: (recall

text: that

text: →the

text: existence

text: of

text: the

text: has

text: already

text: been

text: proved).

text: Analogously,

text: passing

text: to

text: the

text: limit

text: t

text: t

text: in

text: the

text: left-hand

text: side

text: of

text: inequality

text: we

text: get

text: d

text: v(t)

text: u(t

text: v(t

text: dt

text: H

text: t

text: for

text: almost

text: all

text: t

text: T

text: The

text: last

text: two

text: inequalities

text: imply

text: and

text: ﬁnish

text: the

text: proof

text: of

text: the

text: lemma.

text: Now,

text: it

text: is

text: not

text: diﬃcult

text: to

text: ﬁnish

text: the

text: proof

text: of

text: the

text: theorem.

text: Indeed,

text: the

text: already

text: proved

text: lemma

text: justiﬁes

text: the

text: derivation

text: of

text: the

text: key

text: inequality

text: for

text: the

text: case

text: where

text: the

text: solution

text: u(t)

text: is

text: smooth

text: (which

text: will

text: be

text: the

text: case

text: if

text: we

text: start

text: from

text: smooth

text: initial

text: data).

text: The

text: validity

text: of

text: the

text: desired

text: estimate

text: for

text: any

text: H

text: follows

text: then

text: by

text: the

text: usual

text: approximation

text: arguments.

text: Thus,

text: Theorem

text: is

text: proved.

text: Corollary

text: Let

text: the

text: assumptions

text: of

text: Theorem

text: hold.

text: Then,

text: the

text: Kuramoto-Sivashinsky

text: A

text: equation

text: possesses

text: a

text: global

text: attractor

text: (L),

text: in

text: the

text: phase

text: space

text: H

text: Moreover,

text: this

text: global

text: KS

text: attractor

text: is

text: smooth

text: and

text: (L)

text: C

text: (L),

text: KS

text: Hs

text: s

text: for

text: some

text: positive

text: constant

text: C

text: dependent

text: on

text: s,

text: a

text: and

text: L.

text: In

text: particular,

text: C

text: (L)

text: CL

text: with

text: some

text: C

text: independent

text: of

text: L,

text: but

text: dependent

text: on

text: the

text: parameter

text: a.

text: Proof.

text: Indeed,

text: analogously

text: to

text: we

text: have

text: the

text: smoothing

text: property

text: on

text: a

text: ﬁnite

text: interval

text: for

text: the

text: parabolic

text: equation

text: (see,

text: e.g.,

text: u(t

text: u(t)

text: Ns)

text: Hs

text: H

text: for

text: some

text: positive

text: C

text: and

text: N

text: Moreover,

text: the

text: analogue

text: of

text: the

text: Lipschitz

text: estimate

text: also

text: holds

text: s

text: for

text: this

text: equation

text: (and

text: can

text: be

text: proved

text: analogously

text: to

text: the

text: proof

text: given

text: in

text: Theorem

text: Thus,

text: the

text: proof

text: of

text: the

text: corollary

text: repeats

text: word

text: by

text: word

text: the

text: proof

text: of

text: Corollary

text: and,

text: for

text: this

text: reason,

text: is

text: omitted.

text: A

text: Remark

text: As

text: before,

text: one

text: can

text: show

text: that

text: the

text: global

text: attractor

text: is

text: analytic

text: in

text: the

text: sense

text: KS

text: of

text: Gevrey

text: class.

text: However,

text: in

text: contrast

text: to

text: the

text: cases

text: of

text: Ginzburg-Landau

text: equations,

text: its

text: Gevrey

text: norm

text: is

text: not

text: uniformly

text: bounded

text: as

text: L

text: As

text: we

text: will

text: see

text: in

text: the

text: next

text: section,

text: the

text: norms

text: of

text: the

text: A

text: global

text: attractors

text: (L)

text: indeed

text: grow,

text: as

text: L

text: KS

text: The

text: limit

text: L

text: The

text: main

text: aim

text: of

text: this

text: section

text: is

text: to

text: study

text: the

text: dependence

text: of

text: the

text: above

text: constructed

text: global

text: attractors

text: on

text: L,

text: as

text: L

text: We

text: start

text: with

text: the

text: case

text: of

text: complex

text: Ginzburg-Landau

text: equations

text: and

text: To

text: this

text: end,

text: we

text: change

text: the

text: dependent

text: variable

text: u

text: as

text: follows

text: H

text: u(t)

text: (t)w(t)

text: for

text: the

text: case

text: of

text: equation

text: L

text: F

text: u(t)

text: (t)w(t)

text: for

text: the

text: case

text: of

text: equation

text: L

text: ##LTLine##

text: ##LTLine##


text: 18 A. KOSTIANKO, E. TITI AND S. ZELIK Moreover, by the deﬁnition of the operator N, F = 0 (cid:90) 0 (cid:90) (cid:88) (cid:88) (cid:101) (cid:101) and, by the Parseval identity | | π | | | | π | | F (t, x) 2 = 2π H(τ, w (t, x)) 2 dτ, ∂ F (t, x) 2 = 2π ∂ H(τ, w (t, x)) 2 dτ. n − ε x n − x ε ∈Z π ∈Z π n n (cid:90) Integrating these equalitie(cid:88)s in x, we get (cid:101) (cid:107) (cid:107) π (cid:107) (cid:107) (4.16) F (t) 2 = 2π H(τ, w (t)) 2 dτ. n H1 − ε H1 ∈Z π n Since w are uniformly bounded (cid:88)in H1, we have ε (cid:107) (cid:107) ≤ ∞ (4.17) F (t) 2 C < , n H1 ∈Z n where the constant C is independent of ε and t (here we have implicitly used (2.17)). Fur- thermore, using the fact that ∂ w (t) are uniformly bounded in L2([T, T + 1], H), and w (t) are t ε ε uniformly bounded in L2([T, T + 1], H2), with respect to ε, and the explicit formula (2.7) for (cid:88) the nonlinearity F , arguing analogously, we can prove that (cid:107) · (cid:107) ≤ (4.18) ∂ F ( ) 2 C, t n L2([T,T+1],H) ∈Z n (cid:98) where the constant C is independent of ε and T . (cid:88) We ﬁx approximate solution θ of (4.14) in the form (cid:98) − − (4.19) θ(t) := eint/ε( (1 + iγ)∂2 + 1 + in/ε) 1F (t) x n (cid:54) n=0 (we solve equation (4.14) with respect to the “fast” variable considering the “slow” variable t − as a parameter). Then, since the operator (1 + iγ)∂2 generates an analytic semigroup in Hs, x we have (cid:107) − − (cid:107) ≤ ε (4.20) ( (1 + iγ)∂2 + 1 + in/ε) 1 → C , x Hs Hs s n see, e.g., [26]. Estimate (4.20) together with (4.17) gives   (cid:88) (cid:88) (cid:88) (cid:98)     (cid:107) (cid:107) 1/2 1/2 (cid:107) (cid:107) ≤ F (t) ≤ 1 (cid:107) (cid:107) ≤ (cid:48) (4.21) θ(t) Cε n H1 Cε F (t) 2 C ε. H1 (cid:98) (cid:54) n (cid:54) n2 (cid:54) n H1 n=0 n=0 n=0 − We now set θ˜ := θ θ. Then, this function solves (cid:98) − − (4.22) ∂ θ˜ (1 + iγ)∂2θ˜+ θ˜ = h(t), θ˜(0) = θ(0), t x ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: A.

text: KOSTIANKO,

text: E.

text: TITI

text: AND

text: S.

text: ZELIK

text: Moreover,

text: by

text: the

text: deﬁnition

text: of

text: the

text: operator

text: N,

text: F

text: and,

text: by

text: the

text: Parseval

text: identity

text: π

text: π

text: F

text: (t,

text: x)

text: 2π

text: H(τ,

text: w

text: (t,

text: x))

text: dτ,

text: F

text: (t,

text: x)

text: 2π

text: H(τ,

text: w

text: (t,

text: x))

text: dτ.

text: n

text: ε

text: x

text: n

text: x

text: ε

text: ∈Z

text: π

text: ∈Z

text: π

text: n

text: n

label: equation

text: Integrating

text: these

text: in

text: x,

text: we

text: get

text: π

text: F

text: (t)

text: 2π

text: H(τ,

text: w

text: (t))

text: dτ.

text: n

text: ε

text: ∈Z

text: π

text: n

text: Since

text: w

text: are

text: uniformly

text: bounded

text: we

text: have

text: ε

text: F

text: (t)

text: C

text: n

text: ∈Z

text: n

text: where

text: the

text: constant

text: C

text: is

text: independent

text: of

text: ε

text: and

text: t

text: (here

text: we

text: have

text: implicitly

text: used

text: Fur-

text: thermore,

text: using

text: the

text: fact

text: that

text: w

text: (t)

text: are

text: uniformly

text: bounded

text: in

text: T

text: H),

text: and

text: w

text: (t)

text: are

text: t

text: ε

text: ε

text: uniformly

text: bounded

text: in

text: T

text: with

text: respect

text: to

text: ε,

text: and

text: the

text: explicit

text: formula

text: for

text: the

text: nonlinearity

text: F

text: arguing

text: analogously,

text: we

text: can

text: prove

text: that

text: F

text: C,

text: t

text: n

text: ∈Z

text: n

text: where

text: the

text: constant

text: C

text: is

text: independent

text: of

text: ε

text: and

text: T

text: We

text: ﬁx

text: approximate

text: solution

text: θ

text: of

text: in

text: the

text: form

text: θ(t)

text: eint/ε(

text: in/ε)

text: (t)

text: x

text: n

text: (we

text: solve

text: equation

text: with

text: respect

text: to

text: the

text: “fast”

text: variable

text: considering

text: the

text: “slow”

text: variable

text: t

text: as

text: a

text: parameter).

text: Then,

text: since

text: the

text: operator

text: generates

text: an

text: analytic

text: semigroup

text: in

text: Hs,

text: x

text: we

text: have

text: ε

text: in/ε)

text: C

text: x

text: Hs

text: Hs

text: s

text: n

text: see,

text: e.g.,

text: Estimate

text: together

text: with

text: gives

text: F

text: (t)

text: θ(t)

text: Cε

text: n

text: Cε

text: F

text: (t)

text: C

text: ε.

text: n

text: n

text: We

text: now

text: set

text: θ˜

text: θ

text: θ.

text: Then,

text: this

text: function

text: solves

text: θ˜

text: iγ)∂2θ˜+

text: θ˜

text: h(t),

text: t

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: LARGE DISPERSION 3 → ∞ grows proportionally to L, as L . Moreover, we provide here rigorous justiﬁcation to the numerical investigation, reported in [22]. The paper is organized as follows. The resonances and the corresponding inﬁnite dispersion limit equations are presented in section for all three cases of equations (1.1), (1.2) and (1.3). The existence of the corresponding global attractors for these limit equations, as well as the upper bounds for the diameter of these global attractors, are veriﬁed in section . The singular → ∞ limit, L , is studied in section . In particular, the convergence of the global attractors of the Ginzburg-Landau equations to the corresponding global attractors of the inﬁnite dispersion limit equations, as well as the growing lower bounds for the diameter of the global attractor of the Kuramoto-Sivashinsky equation, are established in section . Finally, the gradient structure of the inﬁnite dispersion limit equations, corresponding to the Ginzburg-Landau equations (1.1) and its consequences, are presented in section . 2. Preliminaries In this section, we introduce the groups of solution operators associated with the auxiliary linear dispersion equations: (2.1) ∂ v = L∂3v, t x (2.2) ∂ v = iL∂2v t x R − on , subject to periodic boundary conditions with fundamental periodic domain ( π, π), and formulate their simple properties. Moreover, we compute here some averages of the non- linearities with respect to the rapid time oscillations for large values of L, generated by these groups, which are crucial for what follows. − We denote by H = L2 ( π, π), the space of complex-valued 2π-periodic square integrable per ∈ R functions, and introduce the family Hs, s , of Sobolev spaces of periodic functions with − (cid:88) ∈ Z periodic fundamental domain ( π, π). Let e := einx, n , be the standard orthogonal basis n in H and let v = v einx. n ∈Z n (cid:88) Neglecting the scalar factor 2π, we deﬁne the Hs-norm of v as follows: (cid:107) (cid:107) | | | | (2.3) v 2 := ( n 2 + 1)s v 2. Hs n ∈Z n H F Lemma 2.1. The groups of solution operators (t) and (t), associated with equations (2.1) L L ∈ R and (2.2) respectively, are isometries on the Sobolev spaces Hs, for any s . Moreover, e = einx are eigenfunctions satisfying n (2.4) H (t)e = e−iLn3te , F (t)e = e−iLn2te , n ∈ Z. L n n L n n Finally, these groups of solution operators are 2π/L-periodic with respect to time. Indeed, formulas (2.4) follow immediately from equations (2.1) and (2.2) and all other asser- tions of the lemma are immediate corollaries of these explicit expressions. We are going to change the variable u in equations (1.2) and (1.3) using the transformation H (2.5) u(t) = (t)w(t) L

text: LARGE

text: DISPERSION

text: grows

text: proportionally

text: to

text: L,

text: as

text: L

text: Moreover,

text: we

text: provide

text: here

text: rigorous

text: justiﬁcation

text: to

text: the

text: numerical

text: investigation,

text: reported

text: in

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: The

text: resonances

text: and

text: the

text: corresponding

text: inﬁnite

text: dispersion

text: limit

text: equations

text: are

text: presented

text: in

text: section

text: for

text: all

text: three

text: cases

text: of

text: equations

text: and

text: The

text: existence

text: of

text: the

text: corresponding

text: global

text: attractors

text: for

text: these

text: limit

text: equations,

text: as

text: well

text: as

text: the

text: upper

text: bounds

text: for

text: the

text: diameter

text: of

text: these

text: global

text: attractors,

text: are

text: veriﬁed

text: in

text: section

text: The

text: singular

text: limit,

text: L

text: is

text: studied

text: in

text: section

text: In

text: particular,

text: the

text: convergence

text: of

text: the

text: global

text: attractors

text: of

text: the

text: Ginzburg-Landau

text: equations

text: to

text: the

text: corresponding

text: global

text: attractors

text: of

text: the

text: inﬁnite

text: dispersion

text: limit

text: equations,

text: as

text: well

text: as

text: the

text: growing

text: lower

text: bounds

text: for

text: the

text: diameter

text: of

text: the

text: global

text: attractor

text: of

text: the

text: Kuramoto-Sivashinsky

text: equation,

text: are

text: established

text: in

text: section

text: Finally,

text: the

text: gradient

text: structure

text: of

text: the

text: inﬁnite

text: dispersion

text: limit

text: equations,

text: corresponding

text: to

text: the

text: Ginzburg-Landau

text: equations

text: and

text: its

text: consequences,

text: are

text: presented

text: in

text: section

text: Preliminaries

text: In

text: this

text: section,

text: we

text: introduce

text: the

text: groups

text: of

text: solution

text: operators

text: associated

text: with

text: the

text: auxiliary

text: linear

text: dispersion

text: equations:

text: v

text: t

text: x

text: v

text: t

text: x

text: R

text: on

text: subject

text: to

text: periodic

text: boundary

text: conditions

text: with

text: fundamental

text: periodic

text: domain

text: π,

text: π),

text: and

text: formulate

text: their

text: simple

text: properties.

text: Moreover,

text: we

text: compute

text: here

text: some

text: averages

text: of

text: the

text: non-

text: linearities

text: with

text: respect

text: to

text: the

text: rapid

text: time

text: oscillations

text: for

text: large

text: values

text: of

text: L,

text: generated

text: by

text: these

text: groups,

text: which

text: are

text: crucial

text: for

text: what

text: follows.

text: We

text: denote

text: by

text: H

text: π,

text: π),

text: the

text: space

text: of

text: complex-valued

text: square

text: integrable

text: per

text: R

text: functions,

text: and

text: introduce

text: the

text: family

text: Hs,

text: s

text: of

text: Sobolev

text: spaces

text: of

text: periodic

text: functions

text: with

text: Z

text: periodic

text: fundamental

text: domain

text: π,

text: π).

text: Let

text: e

text: einx,

text: n

text: be

text: the

text: standard

text: orthogonal

text: basis

text: n

text: in

text: H

text: and

text: let

text: v

text: v

text: einx.

text: n

text: ∈Z

text: n

text: Neglecting

text: the

text: scalar

text: factor

text: we

text: deﬁne

text: the

text: Hs-norm

text: of

text: v

text: as

text: follows:

text: v

text: n

text: v

text: Hs

text: n

text: ∈Z

text: n

text: H

text: F

text: Lemma

text: The

text: groups

text: of

text: solution

text: operators

text: (t)

text: and

text: (t),

text: associated

text: with

text: equations

text: L

text: L

text: R

text: and

text: respectively,

text: are

text: isometries

text: on

text: the

text: Sobolev

text: spaces

text: Hs,

text: for

text: any

text: s

text: Moreover,

text: e

text: einx

text: are

text: eigenfunctions

text: satisfying

text: n

text: H

text: (t)e

text: F

text: (t)e

text: n

text: Z.

text: L

text: n

text: n

text: L

text: n

text: n

text: Finally,

text: these

text: groups

text: of

text: solution

text: operators

text: are

text: with

text: respect

text: to

text: time.

text: Indeed,

text: formulas

text: follow

text: immediately

text: from

text: equations

text: and

text: and

text: all

text: other

text: asser-

text: tions

text: of

text: the

text: lemma

text: are

text: immediate

text: corollaries

text: of

text: these

text: explicit

text: expressions.

text: We

text: are

text: going

text: to

text: change

text: the

text: variable

text: u

text: in

text: equations

text: and

text: using

text: the

text: transformation

text: H

text: u(t)

text: (t)w(t)

text: L


text: 22 A. KOSTIANKO, E. TITI AND S. ZELIK (cid:107) (cid:107) ≤ ∗ (4.36) sup P w(τ) C ε, ∈R N H τ ∗ where C is independent of ε. Let us consider now the Q -component of w. Then, the analogous arguments lead to the N inequality (cid:107) (cid:107) ≤ (cid:107) (cid:107) − Q w(t + τ) C Q w(τ) e αt + C εeKt. N H 1 N H 2 ∈ R Taking again the supremum over τ , we arrive at (cid:107) (cid:107) ≤ eKt ln(C + 1) sup Q w(τ) C ε − − , for all t > 1 . ∈R N H 2 1 C e αt α τ 1 ∞ Minimizing the right-hand side with respect to t over [ln(C + 1)/α, ) we obtain 1 (cid:107) (cid:107) ≤ ∗ (4.37) sup Q w(τ) C ε. ∈R N H τ Thus, estimate (4.33) is veriﬁed. We see that any trajectory w(t) of equation (4.30) which is contained in the ball of radius R ≡ in H should belong to the Cε-neighbourhood of zero equilibrium w 0 (where the constant C depends on R, but is independent of ε = 1/L). It remains to note that, since the equilibrium ≡ | w 0 is hyperbolic and H˜ (τ, 0) = 0, D H˜ (τ, w) = 0, the usual implicit function theorem w w=0 shows that the only solution w(t) of (4.30) which belongs to some small r-ball of H for any t (r ≡ ≡ (cid:3) is independent of ε!) is w 0. Thus, u 0 if ε is small enough and the theorem is proved. The proof of Theorem 4.6 indicates particularly that the diameter of the global attractor A → ∞ (L) (say, in the H-norm) indeed expands as L : KS (cid:107)A (cid:107) ∞ (4.38) lim (L) = , →∞ KS H L but gives no information about the rate of expansion. The next proposition removes this drawback and shows that the upper bound given by estimate (3.14) is optimal. The proof is based on the well-known fact on the existence of rotating waves for the perturbed KdV equation with periodic boundary conditions, see [34] (see also [22] for the numerical study of the stability of these waves as well as the related attractors). A Proposition 4.7. Let a > 1 and L be large enough. Then, the global attractor (L) satisﬁes KS the following estimate: ≤ (cid:107)A (cid:107) ≤ (4.39) C L (L) C L, 1 KS H 2 where the positive constants C , C are independent of L. 1 2 Proof. Indeed, the upper bound is an immediate corollary of estimate (3.14), so we only need to establish the lower one. To this end, we do change of variables u(t) = Lv(t) where v is a new dependent variable. Then, equation (1.3) reads − − ε∂ v = ε( ∂4v a∂2v) + v∂ v + ∂3v, t x x x x and, ﬁnally, introducing the fast variable τ = Lt, we end up with − − (4.40) ∂ v = ε( ∂4v a∂2v) + v∂ v + ∂3v, τ x x x x ##LTLine## ##LTLine##

text: A.

text: KOSTIANKO,

text: E.

text: TITI

text: AND

text: S.

text: ZELIK

text: sup

text: P

text: w(τ)

text: C

text: ε,

text: ∈R

text: N

text: H

text: τ

text: where

text: C

text: is

text: independent

text: of

text: ε.

text: Let

text: us

text: consider

text: now

text: the

text: Q

text: -component

text: of

text: w.

text: Then,

text: the

text: analogous

text: arguments

text: lead

text: to

text: the

text: N

text: inequality

text: Q

text: w(t

text: τ)

text: C

text: Q

text: w(τ)

text: e

text: αt

text: C

text: εeKt.

text: N

text: H

text: N

text: H

text: R

text: Taking

text: again

text: the

text: supremum

text: over

text: τ

text: we

text: arrive

text: at

text: eKt

text: ln(C

text: sup

text: Q

text: w(τ)

text: C

text: ε

text: for

text: all

text: t

text: ∈R

text: N

text: H

text: C

text: e

text: αt

text: α

text: τ

text: Minimizing

text: the

text: right-hand

text: side

text: with

text: respect

text: to

text: t

text: over

text: [ln(C

text: we

text: obtain

text: sup

text: Q

text: w(τ)

text: C

text: ε.

text: ∈R

text: N

text: H

text: τ

text: Thus,

text: estimate

text: is

text: veriﬁed.

text: We

text: see

text: that

text: any

text: trajectory

text: w(t)

text: of

text: equation

text: which

text: is

text: contained

text: in

text: the

text: ball

text: of

text: radius

text: R

text: in

text: H

text: should

text: belong

text: to

text: the

text: Cε-neighbourhood

text: of

text: zero

text: equilibrium

text: w

text: (where

text: the

text: constant

text: C

text: depends

text: on

text: R,

text: but

text: is

text: independent

text: of

text: ε

text: It

text: remains

text: to

text: note

text: that,

text: since

text: the

text: equilibrium

text: w

text: is

text: hyperbolic

text: and

text: H˜

text: (τ,

text: D

text: H˜

text: (τ,

text: w)

text: the

text: usual

text: implicit

text: function

text: theorem

text: w

text: shows

text: that

text: the

text: only

text: solution

text: w(t)

text: of

text: which

text: belongs

text: to

text: some

text: small

text: r-ball

text: of

text: H

text: for

text: any

text: t

text: (r

text: is

text: independent

text: of

text: ε!)

text: is

text: w

text: Thus,

text: u

text: if

text: ε

text: is

text: small

text: enough

text: and

text: the

text: theorem

text: is

text: proved.

text: The

text: proof

text: of

text: Theorem

text: indicates

text: particularly

text: that

text: the

text: diameter

text: of

text: the

text: global

text: attractor

text: A

text: (L)

text: (say,

text: in

text: the

text: H-norm)

text: indeed

text: expands

text: as

text: L

text: KS

text: lim

text: (L)

text: KS

text: H

text: L

text: but

text: gives

text: no

text: information

text: about

text: the

text: rate

text: of

text: expansion.

text: The

text: next

text: proposition

text: removes

text: this

text: drawback

text: and

text: shows

text: that

text: the

text: upper

text: bound

text: given

text: by

text: estimate

text: is

text: optimal.

text: The

text: proof

text: is

text: based

text: on

text: the

text: well-known

text: fact

text: on

text: the

text: existence

text: of

text: rotating

text: waves

text: for

text: the

text: perturbed

text: KdV

text: equation

text: with

text: periodic

text: boundary

text: conditions,

text: see

text: (see

text: also

text: for

text: the

text: numerical

text: study

text: of

text: the

text: stability

text: of

text: these

text: waves

text: as

text: well

text: as

text: the

text: related

text: attractors).

text: A

text: Proposition

text: Let

text: a

text: and

text: L

text: be

text: large

text: enough.

text: Then,

text: the

text: global

text: attractor

text: (L)

text: satisﬁes

text: KS

text: the

text: following

text: estimate:

text: C

text: L

text: (L)

text: C

text: L,

text: KS

text: H

text: where

text: the

text: positive

text: constants

text: C

text: C

text: are

text: independent

text: of

text: L.

text: Proof.

text: Indeed,

text: the

text: upper

text: bound

text: is

text: an

text: immediate

text: corollary

text: of

text: estimate

text: so

text: we

text: only

text: need

text: to

text: establish

text: the

text: lower

text: one.

text: To

text: this

text: end,

text: we

text: do

text: change

text: of

text: variables

text: u(t)

text: Lv(t)

text: where

text: v

text: is

text: a

text: new

text: dependent

text: variable.

text: Then,

text: equation

text: reads

text: ε∂

text: v

text: ε(

text: v∂

text: v

text: t

text: x

text: x

text: x

text: x

text: and,

text: ﬁnally,

text: introducing

text: the

text: fast

text: variable

text: τ

text: Lt,

text: we

text: end

text: up

text: with

text: v

text: ε(

text: v∂

text: v

text: τ

text: x

text: x

text: x

text: x

text: ##LTLine##

text: ##LTLine##


text: LARGE DISPERSION 29 (cid:40) and the dynamics on this manifold is given by the equations − | | | | y˙ = βy (1 + iω)[y( y 2 + 4 v 2) + 2y¯v2], (5.28) − − − | | | | v˙ = (β 1 iγ)v (1 + iω)[v(2 y 2 + 3 v 2) + v¯y2]. √ √ − Substituting y = reiϕ, v = ρeiψ, η = 2(ψ ϕ) and scaling time by the factor of 2, we  arrive at the following 3-dimensional system:   − − − −  r˙ = r[β r 4ρ 2ρ(cos η ω sin η)], − − − − (5.29) ρ˙ = ρ[β 1 2r 3ρ r(cos η + ω sin η)], − − − η˙ = γ + ω(ρ r) + r(sin η ω cos η) + 2ρ(sin η + ω cos η)], see [36] for the details. As shown there, there are values of parameters (γ , β , ω ) for which 0 0 0 equations (5.29) possess an equilibrium (r , ρ , η ) with 3 zero eigenvalues. Then, the standard 0 0 0 bifurcation analysis (also performed in [36]) shows that this system possesses the Shilnikov saddle-focus homoclinic loop at certain values of the parameters and, as a result there is an open region in the space of parameters (β, γ, ω) for which the corresponding dynamics generated by (5.29) is chaotic. Thus, in contrast to (4.5), the averaged equations (4.4) cannot be transformed to a gradient system. Moreover, the standard perturbation arguments show that the chaotic dynamics of the limit averaged equations (4.4) persists at the initial equations (1.2) if L is large enough. The work of S.Z. was partially supported by the Russian Foundation of Basic Researches (projects 14-01-00346 and 15-01-03587) and by the grant 14-41-00044 of RSF. The work of E.S.T. was supported in part by the ONR grant N00014-15-1-2333 and the NSF grants DMS- 1109640 and DMS-1109645. References [1] M. Al-Jaboori and D. Wirosoetisno, Navier-Stokes equations on the β-plane, Discrete Contin. Dyn. Syst. Ser. B 16 (2011), no. 3, 687––701. [2] V. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics, Vol. 60), New York, NY: Springer, 1997. [3] Z. Artstein, I. Kevrikidis, M. Slemrod, and E.S. Titi, Slow observables of singularly perturbed diﬀerential equations, Nonlinearity 20 (2007), 2463––2481. [4] Z. Artstein, J. Linshiz, and E.S. Titi, Young measure approach to computing slowly advansing fast oscillations, Multiscale Model. Simul. 6 (2007), no. 4, 1085––1097. [5] A. Babin, A. Ilyin, and E.S. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equa- tion, Commun. Pure Appl. Math. 64 (2011), no. 5, 591–648. [6] A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Appl. Math. Lett. 13 (2000), no. 4, 51–57. [7] A. Babin and M. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl. (9) 62 (1983), 441–491. [8] A. Babin and M. Vishik, Attractors of Evolution Equations, Amsterdam etc.: North-Holland, 1992. [9] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications. Vol. 5, North-Holland Publ. Amsterdam - New York - Oxford, 1978. [10] N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Kiev: Izd. AN USSR, 1945. [11] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209–262. [12] J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115–159. [13] J. Bourgain, Global Solutions of Nonlinear Schr¨odinger Equations. American Mathematical Society Collo- quium Publications, Vol. 46, Providence, RI: AMS, 1999. [14] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: American Mathematical Society (AMS), 2002. ##LTLine## ##LTLine##

text: LARGE

text: DISPERSION

text: and

text: the

text: dynamics

text: on

text: this

text: manifold

text: is

text: given

text: by

text: the

text: equations

text: y˙

text: βy

text: iω)[y(

text: y

text: v

text: v˙

text: (β

text: iγ)v

text: y

text: v

text: Substituting

text: y

text: reiϕ,

text: v

text: ρeiψ,

text: η

text: ϕ)

text: and

text: scaling

text: time

text: by

text: the

text: factor

text: of

text: we

text: arrive

text: at

text: the

text: following

text: system:

text: r˙

text: r[β

text: r

text: 4ρ

text: η

text: ω

text: sin

text: η)],

text: ρ˙

text: ρ[β

text: 3ρ

text: r(cos

text: η

text: ω

text: sin

text: η)],

text: η˙

text: γ

text: ω(ρ

text: r)

text: r(sin

text: η

text: ω

text: cos

text: η)

text: η

text: ω

text: cos

text: η)],

text: see

text: for

text: the

text: details.

text: As

text: shown

text: there,

text: there

text: are

text: values

text: of

text: parameters

text: (γ

text: β

text: ω

text: for

text: which

text: equations

text: possess

text: an

text: equilibrium

text: (r

text: ρ

text: η

text: with

text: zero

text: eigenvalues.

text: Then,

text: the

text: standard

text: bifurcation

text: analysis

text: (also

text: performed

text: in

text: shows

text: that

text: this

text: system

text: possesses

text: the

text: Shilnikov

text: saddle-focus

text: homoclinic

text: loop

text: at

text: certain

text: values

text: of

text: the

text: parameters

text: and,

text: as

text: a

text: result

text: there

text: is

text: an

text: open

text: region

text: in

text: the

text: space

text: of

text: parameters

text: (β,

text: γ,

text: ω)

text: for

text: which

text: the

text: corresponding

text: dynamics

text: generated

text: by

text: is

text: chaotic.

text: Thus,

text: in

text: contrast

text: to

text: the

text: averaged

text: equations

text: cannot

text: be

text: transformed

text: to

text: a

text: gradient

text: system.

text: Moreover,

text: the

text: standard

text: perturbation

text: arguments

text: show

text: that

text: the

text: chaotic

text: dynamics

text: of

text: the

text: limit

text: averaged

text: equations

text: persists

text: at

text: the

text: initial

text: equations

text: if

text: L

text: is

text: large

text: enough.

text: The

text: work

text: of

text: S.Z.

text: was

text: partially

text: supported

text: by

text: the

text: Russian

text: Foundation

text: of

text: Basic

text: Researches

text: (projects

text: and

text: and

text: by

text: the

text: grant

text: of

text: RSF.

text: The

text: work

text: of

text: E.S.T.

text: was

text: supported

text: in

text: part

text: by

text: the

text: ONR

text: grant

text: and

text: the

text: NSF

text: grants

text: DMS-

text: and

text: References

text: M.

text: Al-Jaboori

text: and

text: D.

text: Wirosoetisno,

text: Navier-Stokes

text: equations

text: on

text: the

text: β-plane,

text: Discrete

text: Contin.

text: Dyn.

text: Syst.

text: Ser.

text: B

text: no.

text: V.

text: Arnold,

text: Mathematical

text: Methods

text: of

text: Classical

text: Mechanics

text: (Graduate

text: Texts

text: in

text: Mathematics,

text: Vol.

text: New

text: York,

text: NY:

text: Springer,

text: Z.

text: Artstein,

text: I.

text: Kevrikidis,

text: M.

text: Slemrod,

text: and

text: E.S.

text: Titi,

text: Slow

text: observables

text: of

text: singularly

text: perturbed

text: diﬀerential

text: equations,

text: Nonlinearity

text: Z.

text: Artstein,

text: J.

text: Linshiz,

text: and

text: E.S.

text: Titi,

text: Young

text: measure

text: approach

text: to

text: computing

text: slowly

text: advansing

text: fast

text: oscillations,

text: Multiscale

text: Model.

text: Simul.

text: no.

text: A.

text: Babin,

text: A.

text: Ilyin,

text: and

text: E.S.

text: Titi,

text: On

text: the

text: regularization

text: mechanism

text: for

text: the

text: periodic

text: Korteweg-de

text: Vries

text: equa-

text: tion,

text: Commun.

text: Pure

text: Appl.

text: Math.

text: no.

text: A.

text: Babin,

text: A.

text: Mahalov,

text: and

text: B.

text: Nicolaenko,

text: Global

text: regularity

text: of

text: rotating

text: Navier-Stokes

text: equations

text: for

text: resonant

text: domains,

text: Appl.

text: Math.

text: Lett.

text: no.

text: A.

text: Babin

text: and

text: M.

text: Vishik,

text: Regular

text: attractors

text: of

text: semigroups

text: and

text: evolution

text: equations,

text: J.

text: Math.

text: Pures

text: Appl.

text: A.

text: Babin

text: and

text: M.

text: Vishik,

text: Attractors

text: of

text: Evolution

text: Equations,

text: Amsterdam

text: etc.:

text: North-Holland,

text: A.

text: Bensoussan,

text: J.-L.

text: Lions,

text: and

text: G.

text: Papanicolaou,

text: Asymptotic

text: Analysis

text: for

text: Periodic

text: Structures.

text: Studies

text: in

text: Mathematics

text: and

text: Its

text: Applications.

text: Vol.

text: North-Holland

text: Publ.

text: Amsterdam

text: New

text: York

text: Oxford,

text: N.

text: Bogolyubov,

text: On

text: Some

text: Statistical

text: Methods

text: in

text: Mathematical

text: Physics,

text: Kiev:

text: Izd.

text: AN

text: USSR,

text: J.

text: Bourgain,

text: Fourier

text: transform

text: restriction

text: phenomena

text: for

text: certain

text: lattice

text: subsets

text: and

text: applications

text: to

text: nonlinear

text: evolution

text: equations.

text: II.

text: The

text: KdV-equation,

text: Geom.

text: Funct.

text: Anal.

text: no.

text: J.

text: Bourgain,

text: Periodic

text: Korteweg

text: de

text: Vries

text: equation

text: with

text: measures

text: as

text: initial

text: data,

text: Selecta

text: Math.

text: (N.S.)

text: no.

text: J.

text: Bourgain,

text: Global

text: Solutions

text: of

text: Nonlinear

text: Schr¨odinger

text: Equations.

text: American

text: Mathematical

text: Society

text: Collo-

text: quium

text: Publications,

text: Vol.

text: Providence,

text: RI:

text: AMS,

text: V.

text: Chepyzhov

text: and

text: M.

text: Vishik,

text: Attractors

text: for

text: Equations

text: of

text: Mathematical

text: Physics,

text: Providence,

text: RI:

text: American

text: Mathematical

text: Society

text: (AMS),

text: ##LTLine##

text: ##LTLine##


text: 4 A. KOSTIANKO, E. TITI AND S. ZELIK and using the transformation F (2.6) u(t) = (t)w(t) L for equation (1.1). Then we will average the obtained rapidly oscillating in time terms. To this end, we need to compute the resounant terms appearing from the nonlinearities. We will do that in the following several lemmas. We start with the case of equation (1.2). ∈ Lemma 2.2. Let w Hs, s > 1/2, and let H − ◦ H · |H | (2.7) F (τ, w) := ( τ) ( (τ)w (τ)w 2). 1 1 1 → Then, for every ﬁxed τ, the map w F (τ, w) is a bounded smooth map from Hs to itself: (cid:107) (cid:107) ≤ (cid:107) (cid:107) (2.8) F (τ, w) C w 3 , Hs s Hs with a constant C that is independent of(cid:90)τ. Moreover, the operator F is 2π-periodic with s respect to τ, and its time averaging has the form: (cid:104) · (cid:105) 2π F ( , w) := 1/(2π) F (τ, w) dτ =: N(w), 0 where the operator N has the following explicit form: (cid:88) (cid:107) (cid:107) − | | − | | | | (2.9) N(w) = 2w w 2 + w¯[w, w] 2w w 2e w ( w 2 + 2 w− 2)e . (cid:80) H (cid:80) 0 0 0 n n n n (cid:54) n=0 (cid:0) (cid:1) with [w, v] := n∈Z wnv−n and w = n∈Z wnen. In particular, (cid:88) (cid:0) (cid:1) (cid:107) (cid:107) − | | (2.10) N(w) = 2w ( w 2 w 2) + w¯ [w, w] e + 0 H 0 0 0 (cid:107) (cid:107) − | | − | | + w (2 w 2 w 2 2 w− 2) + w¯− [w, w] e . n H n n n n (cid:54) n=0 Proof. Indeed, estimate (2.8) follows immediately from the fact that Hs is an algebra, for s > H 1/2, and that (τ) are isometries. So, we only need to compute the average of F (τ, w). Indeed, L inserting (cid:88) H − (2.11) u = (τ)w = e in3τw e 1 n n ∈Z n to (2.7) and using that e e(cid:88)= e , we have k l k+l − − (2.12) F (τ, w) = e i(n3+m3+k3 (n+m+k)3)τw w− w e . n m k n+m+k ∈Z n,m,k Thus, the resonance condition reads n3 + m3 + k3 = (m + n + k)3, ##LTLine##

text: A.

text: KOSTIANKO,

text: E.

text: TITI

text: AND

text: S.

text: ZELIK

text: and

text: using

text: the

text: transformation

text: F

text: u(t)

text: (t)w(t)

text: L

text: for

text: equation

text: Then

text: we

text: will

text: average

text: the

text: obtained

text: rapidly

text: oscillating

text: in

text: time

text: terms.

text: To

text: this

text: end,

text: we

text: need

text: to

text: compute

text: the

text: resounant

text: terms

text: appearing

text: from

text: the

text: nonlinearities.

text: We

text: will

text: do

text: that

text: in

text: the

text: following

text: several

text: lemmas.

text: We

text: start

text: with

text: the

text: case

text: of

text: equation

text: Lemma

text: Let

text: w

text: Hs,

text: s

text: and

text: let

text: H

text: H

text: |H

text: F

text: (τ,

text: w)

text: τ)

text: (τ)w

text: (τ)w

text: Then,

text: for

text: every

text: ﬁxed

text: τ,

text: the

text: map

text: w

text: F

text: (τ,

text: w)

text: is

text: a

text: bounded

text: smooth

text: map

text: from

text: Hs

text: to

text: itself:

text: F

text: (τ,

text: w)

text: C

text: w

text: Hs

text: s

text: Hs

text: with

text: a

text: constant

text: C

text: that

text: is

text: independent

text: Moreover,

text: the

text: operator

text: F

text: is

text: with

text: s

text: respect

text: to

text: τ,

text: and

text: its

text: time

text: averaging

text: has

text: the

text: form:

text: 2π

text: F

text: w)

text: F

text: (τ,

text: w)

text: dτ

text: N(w),

text: where

text: the

text: operator

text: N

text: has

text: the

text: following

text: explicit

text: form:

text: N(w)

text: w

text: w¯[w,

text: w]

text: w

text: w

text: w

text: w−

text: H

text: n

text: n

text: n

text: n

text: with

text: [w,

text: v]

text: n∈Z

text: wnv−n

text: and

text: w

text: n∈Z

text: wnen.

text: In

text: particular,

text: N(w)

text: w

text: w

text: w¯

text: [w,

text: w]

text: e

text: H

text: w

text: w

text: w

text: w−

text: w¯−

text: [w,

text: w]

text: e

text: n

text: H

text: n

text: n

text: n

text: n

text: Proof.

text: Indeed,

text: estimate

text: follows

text: immediately

text: from

text: the

text: fact

text: that

text: Hs

text: is

text: an

text: algebra,

text: for

text: s

text: H

text: and

text: that

text: (τ)

text: are

text: isometries.

text: So,

text: we

text: only

text: need

text: to

text: compute

text: the

text: average

text: of

text: F

text: (τ,

text: w).

text: Indeed,

text: L

text: inserting

text: H

text: u

text: (τ)w

text: e

text: e

text: n

text: n

text: ∈Z

text: n

text: to

text: and

text: using

text: that

text: e

text: e

text: we

text: have

text: k

text: l

text: k+l

text: F

text: (τ,

text: w)

text: e

text: w−

text: w

text: e

text: n

text: m

text: k

text: n+m+k

text: ∈Z

text: n,m,k

text: Thus,

text: the

text: resonance

text: condition

text: reads

text: (m

text: n

text: ##LTLine##