Weakly nonlinear stochastic CGL equations

www.imstat.org/aihp 2013, Vol. 49, No. 4, 1033–1056

DOI:10.1214/11-AIHP482

© Association des Publications de l’Institut Henri Poincaré, 2013

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin

SNRS and CMLS at Ecole Polytechnique, Palaiseau, France. E-mail:kuksin@math.polytechnique.fr Received 22 June 2011; revised 8 December 2011; accepted 2 February 2012

Dedicated to Claude Bardos on his 70th birthday

Abstract. We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping:

d dtu+i

−+V (x) u=ν

u−γ_R|u|^2pu−iγ_I|u|^2qu +√

νη(t, x). (∗)

The forceηis white in time and smooth inx; the potentialV (x)is typical. We are concerned with the limiting, asν→0, behaviour of solutions on long time-intervals 0≤t≤ν⁻¹T, and with behaviour of these solutions under the double limitt→ ∞andν→0.

We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for(∗) which is a well posed semilinear stochastic heat equation with a non-local nonlinearity and a smooth additive noise, written in Fourier coefficients. The effective equations do not depend on the Hamiltonian part of the perturbation−iγ_I|u|^2qu(but depend on the dissipative part−γ_R|u|^2pu). Ifpis an integer, they may be written explicitly.

Résumé. Nous considérons l’équation de Schrödinger linéaire avec les conditions aux limites périodiques, perturbée par une force aléatoire et amortie par un terme quasi linéaire:

d dtu+i

−+V (x) u=ν

u−γ_R|u|^2pu−iγ_I|u|^2qu +√

νη(t, x). (∗)

La forceηest un processus aléatoire blanc en tempstet lisse enx; le potentielV (x)est typique. Nous étudions le comportement asymptotique des solutions sur de longs intervalles de temps 0≤t≤ν⁻¹T, quandν→0, et le comportement des solutions quand t→ ∞etν→0. Nous démontrons qu’on peut décrire ces deux comportements asymptotiques en termes des solutions dusystème d’équations effectives pour(∗). Ce dernier est une équation de la chaleur avec un terme quasi linéaire non local et une force aléatoire lisse additive, qui est écrite dans l’espace de Fourier. Les équations ne dépendent pas de la partie hamiltonienne de la perturbation−iγ_I|u|^2qu(mais elles dépendent de la partie dissipative−γ_R|u|^2pu). Sipest un entier, on peut écrire ces équations explicitement.

MSC:35Q56; 60H15

Keywords:Complex Ginzburg–Landau equation; Small nonlinearity; Stationary measures; Averaging; Effective equations

0. Introduction

In [9,10] we considered the KdV equation on a circle, perturbed by a random force and a viscous damping. There we suggested auxiliaryeffective equationswhich are well posed and describe long-time behaviour of solutions for the perturbed KdV through a kind of averaging.

1Supported by l’Agence Nationale de la Recherche through the grant ANR-10-BLAN 0102.

In this work we apply the method of [9,10] to a weakly nonlinear situation when the unperturbed equation is not an integrable nonlinear PDE (e.g. KdV), but a linear Hamiltonian PDE with a generic spectrum. Since analytic properties of the latter are easier and better understood then those of the former, in the weakly nonlinear situation we understand better properties of the effective system and its relation with the original equation. Accordingly we can go further in analysis of long time behaviour of solutions.

More precisely, we are concerned withν-small dissipative stochastic perturbations of the space-periodic linear Schrödinger equation

d dtu+i

−u+V (x)

u=0, x∈T^d, (0.1)

i.e. with equations d

dtu+iAu=ν

u−γ_Rf_p

|u|²

u−iγ_If_q

|u|² u

+√

νη(t, x), x∈T^d, (0.2)

whereη(t, x)=_dt^d_∞

j=1bjβ_j(t)ej(x). HereAu=AVu= −u+V (x)uand the potentialV (x)≥1 is sufficiently smooth; the real numbers p, q are nonnegative, the functions f_p(r) and f_q(r) are the monomials |r|^p and |r|^q, smoothed out near zero, and the constantsγ_R, γ_I satisfy

γ_R, γ_I≥0, γ_R+γ_I=1. (0.3)

Ifγ_R =0, then due to the usual difficulty with the zero-mode of a solution u, the term uin the r.h.s. should be modified to−u. The functions{e_j(x), j ≥1}in the definition of the random force form the real trigonometric base ofL₂(T^d), the real numbersb_jdecay sufficiently fast to zero whenj grows, and{βj(t), j≥1}, are the standard complex Wiener processes. So the noiseηis white in time and sufficiently smooth inx. It is convenient to pass to the slow timeτ=νtand write the equation as

˙

u+ν⁻¹iAu=u−γ_Rf_p

|u|²

u−iγIf_q

|u|²

u+η(τ, x), (0.4)

whereu˙=du/dτ. The equation is supplemented with the initial condition

u(0, x)=u₀(x). (0.5)

It is known that under certain restrictions onp, qandd the problem (0.4), (0.5) has a unique solutionu^ν(τ, x),τ≥0, and Eq. (0.4) has a unique stationary measureμ^ν. We review these results in Section1(there attention is given to the 1d case, while higher-dimensional equations are only briefly discussed).

Let{ϕ_k, k≥1}, and{λ_k, k≥1}, be the eigenfunctions and eigenvalues ofA_V, 1≤λ₁≤λ₂≤ · · ·. We say that a potentialV isnonresonantif_∞

j=1λ_js_j =0 for every finite nonzero integer vector(s₁, s₂, . . .). In Sections1.4,1.5 we show that nonresonant potentials are typical both in the sense of Baire and in the sense of measure. Assuming that V is nonresonant we are interested in two questions:

Q1. What is the limiting behaviour asν→0 of solutionsu^ν(τ, x)on long time-intervals 0≤τ≤T? Q2. What is the limiting behaviour of the stationary measureμ^ν asν→0?

For any complex function u(x), x∈T^d, denote by Ψ (u)=v=(v₁, v₂, . . .)the complex vector of its Fourier coefficients with respect to the basis{ϕ_k}, i.e.u(x)=

v_jϕ_j. Denote I_j=1

2|v_j|², ϕ_j=Argv_j, j≥1. (0.6)

Then(I, ϕ)∈R^∞₊ ×T^∞are the action-angles for the linear Eq. (0.1). Thev- and(I, ϕ)-variables are convenient to study the two questions above. Writing (0.4) in the(I, ϕ)-variables we arrive at the following system:

d

dτI_j= · · ·, d

dτϕ_j=ν⁻¹λ_j+ · · ·, (0.7)

where the dots stand for terms of order one (stochastic and deterministic). We have got slow/fast stochastic equations to which the principle of averaging is formally applicable (e.g., see [2,14] for the classical deterministic averaging and [4,8] for the stochastic averaging). DenotingI_j^ν(τ )=I_j(u^ν(τ ))and averaging inϕtheI-equations in (0.7), using the rules of the stochastic calculus [4,8] and following the arguments in [10], we show in Section2that along sequences ν_j→0 we have the convergences

D I^νj(·)

D

I⁰(·)

, (0.8)

where the limiting processI⁰(τ ), 0≤τ ≤T, is a weak solution of the averagedI-equations. As in the KdV-case the averaged equations are singular and we do not know if their solution is unique. So we do not know if the convergence (0.8) holds asν→0. To continue the analysis we write Eq. (0.4) in thev-variables

˙

v_k+iν⁻¹λ_kv_k=P_k(v)+

j≥1

B_kjβ˙_j(τ ), (0.9)

where the driftP_k and the dispersionB_kj are constructed in terms of the r.h.s. of Eq. (0.4) and the transformationΨ. It turns out that the Hamiltonian term−iγIf_q(|u|²)ucontributes toP (v)a term which disappears in the averagedI- equations. We remove it fromP (v)and denote the restP (v). For any vector˜ θ=(θ₁, θ₂, . . .)∈T^∞denote byΦ_θthe linear transformation of the space of complex vectorsvwhich multiplies each componentv_jby e^iθj. Following [9] we average the vector fieldP˜by actions of the transformationsΦ_θ and get theeffective driftR(v)=

T^∞Φ_−θP (Φ˜ _θv)dθ.

In Section3.1we show that

Rk(v)= −λkvk+R_k⁰(v), (0.10)

whereR⁰(v)is a smooth locally Lipschitz nonlinearity.

Since the noise in (0.9) is additive (i.e., the matrixB is v-independent), then the construction of the effective dispersion, given in [9] for a non-additive noise, simplifies significantly and defines the effective noise for Eq. (0.9) whosekth component equals(

lb²_lΨ_kl²)^1/2dβ_k(τ ).Accordingly the effective equations for (0.4) become

˙

vk=Rk(v)dτ +

l

b²_lΨ_kl² 1/2

dβ_k(τ ), k≥1. (0.11)

By construction this system is invariant under rotations: if v(τ ) is its weak solution, then Φθv(τ ) also is a weak solution. Due to (0.10) this is the heat equationu˙= −Aufor a complex functionu(τ, x), perturbed by a non-local smooth nonlinearity and a nondegenerate smooth noise, written in terms of the complex Fourier coefficients v_j. It turns out to be a monotone equation, so its solution is unique (see Section3.2).

In particular, if in (0.4)p=1, then the system of effective equations takes the form

˙

v_k= −v_k (λ_k−M_k)+γ_R

|v_l|²L_kl

dτ+

l

b²_lΨ_kl² 1/2

dβ_k(τ ), k≥1, (0.12)

whereM_k=

V (x)ϕ_k²(x)dxandL_kl=(2−δ_kl)

ϕ_k²(x)ϕ_l²(x)dx. See Example3.1(the calculations, made there for d=1, remain the same ford≥2).

It follows directly from the construction of effective equations that actions{I (v_k(τ ))=¹₂|v_k(τ )|², k≥1}of any solutionv(τ )of (0.11) is a solution of the system of averagedI-equations. On the contrary, every solutionI⁰(τ )of the averagedI-equations, obtained as a limit (0.8), can be lifted to a weak solution of (0.11). Using the uniqueness we get

Theorem 0.1. LetI^ν(τ )=I (u^ν(τ )),whereu^ν(τ ), 0≤τ ≤T,is a solution of(0.4), (0.5).Thenlimν→0D(I^ν(·))= D(I⁰(·)),whereI⁰(τ ), 0≤τ ≤T,is a weak solution of the averagedI-equations.Moreover,there exists a unique solutionv(τ )of(0.11)such thatv(0)=v0=Ψ (u0)andD(I (v(·))=D(I⁰(·)),whereI (v(τ ))j=¹₂|vj(τ )|².

The solutionsI⁰(τ )andv(τ )satisfy some apriori estimates, see Theorem3.5. Concerning distribution of the angles ϕ(u^ν(τ ))and their joint distribution with the actions see Section2.4.

Now let μ^ν be the unique stationary measure for Eq. (0.4) and u^ν be a corresponding stationary solution, D(u^ν(τ ))≡μ^ν. As above, along sequencesν_j →0 the actionsI^νj(τ )=I (u^νj(τ ))converge in distribution to a stationary solutionI(τ )of the averagedI-equations. This solution can be lifted to a stationary weak solutionsv(τ ) of effective Eqs (0.11). Since that system is monotone, then its stationary measuremis unique. So the limit above holds asν→0. As the effective system is rotation invariant, then in the(I, ϕ)-variables its unique stationary measure has the form dm=m_I(dI )×dϕ, where dϕ=dϕ1dϕ2· · ·is the Haar measure onT^∞. It turns out that the measure limν→0μ^νalso has the rotation-invariant form and we arrive at the following result (see Theorem4.3,4.4for a precise statement):

Theorem 0.2. When ν→0 we have the convergences D(I (u^ν(·)) DI (v(·)) and Ψ ◦μ^ν m,where dm= m_I(dI )×dϕ.

Accordingly every solutionu^ν(τ )of (0.2) obeys the following double limit

νlim→0 lim

t→∞D u^ν(t)

=Ψ⁻¹◦m. (0.13)

By Theorems0.1and0.2, the actionsI (u^ν(τ ))of a solutionu^ν of (0.4), (0.5) converge in distribution to those of a solutionv(τ )of the effective system (0.11) withv(0)=Ψ (u₀), both for 0≤τ ≤T and whenτ→ ∞. We conjecture that this convergence hold for eachτ ≥0, uniformly in τ (the space of measures is equipped with the Wasserstein distance).

In Example4.6we discuss Theorem0.2for equations withp=1, when the effective equations become (0.12). In particular, we show that Theorem0.2implies that in Eqs (0.2) with smallν there is no direct or inverse cascade of energy.

In Example4.5we discuss Theorem0.2for the caseγ_R=0 (when the nonlinear part of the perturbation is Hamil- tonian) and its relation to the theory of weak turbulence.

We note that the effective Eqs (0.11) depend on the potentialV (x)in a regular way and are well defined without assuming thatV (x)is nonresonant (cf. Eqs (0.12)). In particular, ifV^M(x)→1 asM→ ∞, where eachV^M(x)≥1 is a nonresonant potential, then in (0.13)m^M m(1), wherem(1)is a unique stationary measure for Eq. (0.12) with V (x)≡1. In this equationΨ_kl=δ_k,l,M_k≡1 and the constantsL_klcan be written down explicitly.

In Section5we show that Theorems0.1,0.2remain true for 1d equations with non-viscous damping (whenuin the l.h.s. of (0.2) is removed, butγ_R>0).

Inviscid limit

A stationary measureμ^ν for Eq. (0.4) also is stationary for the fast-time Eq. (0.2). LetU^ν(t)be a corresponding stationary solution,DU^ν(t)≡μ^ν. It is not hard to see that the system of solutionsU^ν(t)is tight on any finite time- interval[0,T˜]. Let{U^νj, ν_j→0}, be a converging subsequence, i.e.

D(U^νj) Q^∗, μ^νj μ^∗.

Thenμ^∗is an invariant measure for the linear Eq. (0.1) andQ^∗=D(U^∗(·)), whereU^∗(t),0≤t≤ ˜T, is a stationary process such thatD(U^∗(t))≡μ^∗and every trajectory ofU^∗is a solution of (0.1). The limitD(U^νj) D(U^∗)is the inviscid limit for Eq.(0.2). Equation (0.1) has plenty of invariant measures: if we write it in the action-angle variables (0.6), then every measure of the formm(dI )×dϕis invariant (see [12] for the more complicated inviscid limit for nonlinear Schrödinger equation). Theorem0.2explains which one is chosen by Eq. (0.2) for the limit limν→0μ^ν.

The inviscid limit for the damped/driven KdV equation, studied in [9,10] is similar: the limit of the stationary measures for the perturbed equations is a stationary measure of the corresponding effective equations. Due to a com- plicated structure of the nonlinear Fourier transform which integrates KdV, uniqueness of their invariant measure is not proved yet. So the final results concerning the damped/driven KdV are less complete than those for the weakly perturbed CGL equation in Theorem0.2.

Finally consider the damped/driven 2d Navier–Stokes equations with a small viscosity ν and a random force, similar to the forces above and proportional to√

ν:

v_t−νv+(v· ∇)v+ ∇p=√

νη(t, x); divv=0, v∈R², x∈T². (0.14) It is known that (0.14) has a unique stationary measureμ^ν, the family of measures{μ^ν,0< ν≤1}is tight, and every limiting measure limνj→0μ^νjis a non-trivial invariant measure for the 2d Euler Eq. (0.14)ν=0, see Section 5.2 of [13].

Hovewer it is nonclear if the limiting measure is unique and how to single it out among all invariant measures of the Euler equation. The research [9,10] was motivated by the belief that the damped/driven KdV is a model for (0.14).

Unfortunately, we still do not know up to what extend the description of the inviscid limit for the damped/driven KdV and for weakly nonlinear CGL in terms of the effective equations is relevant for the inviscid limit of the 2d hydrodynamics.

Agreements

Analyticity of mapsB₁→B₂ between Banach spacesB₁ andB₂, which are the real parts of complex spaces B₁^c andB₂^c, is understood in the sense of Fréchet. All analytic maps which we consider possess the following additional property: for anyRa map analytically extends to a complex(δ_R>0)-neighbourhood of the ball{|u|B₁< R}inB₁^c. Notations

χ_A stands for the indicator function of a setA (equal 1 inAand equal 0 outside A). By κ(t)we denote various functions oft such thatκ(t)→0 whent→ ∞, and byκ_∞(t)denote functionsκ(t)such thatκ(t)=o(t^−N)for eachN. We writeκ(t)=κ(t;R)to indicate thatκ(t)depends on a parameterR.

1. Preliminaries 1.1. Apriori estimates

We consider the 1d CGL equation on a segment [0,π] with a conservative linear part of order one and a small nonlinearity. The equation is supplemented with Dirichlet boundary conditions which we interpret as odd 2π-periodic boundary conditions. Introducing the slow timeτ=νt(cf. theIntroduction) we write the equation as follows:

˙

u+iν⁻¹

−u_xx+V (x)u

=κu_xx−γ_R|u|^2pu−iγI|u|^2qu+ d dτ

∞ i=1

b_jβ_j(τ )e_j(x),

(1.1) u(x)≡u(x+2π)≡ −u(−x).

Hereu˙=_dτ^du,p, q∈Z+:=N∪ {0}(only for simplicity, see next section),κ>0, constantsγRandγI satisfy (0.3) andRV (x)≥0 is a sufficiently smooth even 2π-periodic function,{e_j, j≥1}is the sine-basis,

ej(x)= 1

√πsinj x,

andβ_j, j≥1, are standard independent complex Wiener processes. That is,β_j(τ )=βj(τ )+iβ₋j(τ ), whereβ_±j(τ ) are standard independent real Wiener processes. Finally, the real numbersb_jall are nonzero and decay whenj grows in such a way thatB₁<∞, where

B_r :=2 ∞ j=1

j^2rb²_j≤ ∞ forr≥0.

By H^r,r∈Rwe denote the Sobolev space of orderr of complex odd periodic functions and provide it with the homogeneous norm · r,

u²r = ∞ l=1

|ul|²l^2r foru(x)= ∞ l=1

ulel(x), u0= u

(ifr∈N, thenur= |^∂_∂x^rur|L2).

Letu(t, x)be a solution of (1.1) such thatu(0, x)=u0.Applying Ito’s formula to¹₂u²we get that d

1 2u²

=

−γ_R|u|^2p_2p⁺₊²₂−κu²₁+1 2B₀

dτ+dM(τ ), (1.2)

whereM(τ )is the martingale_τ

0

b_ju_j·dβ_j(τ ). Here |u|r stands for theL_r-norm, 1≤r≤ ∞, and for complex numbersz1,z2we denote byz1·z2their real scalar product,

z₁·z₂=Rez1z₂.

So(uj+iu₋j)·(dβj+i dβ₋j)=ujdβj+u₋jdβ₋j. From (1.2) we get in the usual way (e.g., see Section 2.2.3 in [13]) that

Ee^{ρκu(τ )2}≤C

κ, B0,u0

∀τ≥0 (1.3)

for a suitableρ_κ>0, uniformly inν >0.

Denoting E(τ )=1

2u(τ )²+γ_{R τ}

0

|u|^2p_2p⁺₊²₂ds+κ 2

_τ

0

u²₁ds

and noting that the characteristic of the martingaleMisM(τ )=

b_j²|u_j|²τ ≤b²_Mu²τ, whereb_M=max|b_j|, we get from (1.2) that

E(τ )≤ 1

2u₀²+1

2B₀τ+M(τ )−κ 2

_τ

0 u²1ds

≤ 1

2u₀²+1

2B₀τ+κ⁻¹b²_M

κb⁻_M²M(τ )

−1 2

κb_M⁻²M (τ )

.

Applying in a standard way the exponential supermartingale estimate to the term in the square bracket in the r.h.s.

(e.g., see [13], Section 2.2.3 ), we get that P

sup

τ≥0

E(τ )−1 2B₀τ

≥1

2u₀²+ρ

≤e^{−2κρb−M2} (1.4)

for anyρ >0.

Now let us re-write Eq. (1.1) as follows:

˙

u+iν⁻¹

−u_xx+V (x)u+νγ_I|u|^2qu

=κu_xx−γ_R|u|^2pu+ d dτ

b_jβ_j(τ )e_j. (1.5)

The l.h.s. is a Hamiltonian system with the hamiltonian−ν⁻¹H (u), H (u)=1

2Au, u +γI

ν 2q+2

|u|^2q+2dx, A= − ∂²

∂x²+V (x).

For anyj ∈Nwe denote ur

2= A^ru, u

.

Then dH (u)(v)= Au, v +γ_Iν|u|^2qu, vand 1

2 ·2 ∞ j=1

b²_jd²H (u)(ej, ej)=1

2B₁+γIνX(τ ),

where

B_r =2

b²_je_j²_r=2

b_j²λ^r_j ∀r, and

X(τ )=2qRe |u|^2q−2u²

j

b²_je_j(x)²

dx+

|u|^2q

j

b²_je_j(x)²dx

≤CB₀u(τ )^2q

2q.

Therefore applying Ito’s formula we get that dH

u(τ )

=

−γR

Au,|u|^2pu

+κAu, uxx −γIνγR

|u|^2p+2q+2dx +κγ_Iν

|u|^2qu, u_xx

+B₁ +γ_IνX(τ )

dτ+dM(τ ), (1.6)

where dM(τ )=

b_jAu+γ_Iν|u|^2qu, e_j ·dβ_j(τ ).

DenotingU_q(x)=_q+¹₁u^q+1andU_p(x)=_p+¹₁u^p+1,we have |u|^2qu, u_xx

≤ −

|u_x|²|u|^2qdx= − ∂

∂xU_q ², and a similar relation holds forq replaced byp. Accordingly,

dH u(τ )

≤ −1 2

κu²2+γR

∂

∂xUp

²+κγIν ∂

∂xUq

² +νγ_Iγ_R

|u|^2p+2q+2dx−C_κu²−2B₁

dτ+dM(τ ), (1.7)

whereC_κmay be chosen independent fromκifγR>0. Considering relations onH (u)^m,m≥1, which follow from (1.7) and (1.6), using (1.4) and arguing by induction we get that

E

sup

0≤t≤T

H u(t )m

+κ 2

T 0

H^m−1(u)u²₂ds

≤H (u₀)^m+C_m(κ, T , B₁)

1+ u₀^cm

, (1.8)

EH u(t )m

≤C_m(κ, B₁)

1+H (u₀)^m+ u₀^cm

∀t >0, (1.9)

for anym. Estimates (1.8) in a traditional way (cf. [5,12,16,18]) imply that Eq. (1.1) isregular in spaceH¹in the sense that for anyu₀∈H¹it has a unique strong solution, satisfying (1.4), (1.8).

1.2. Stationary measures

The a-priori estimates on solutions of (1.1) and the Bogolyubov–Krylov argument (e.g., see in [13]) imply that Eq. (1.1) has a stationary measureμ^ν, supported by spaceH². Now assume that

bj =0 ∀j. (1.10)

Then the approaches, developed in the last decade to study the 2d stochastic Navier–Stokes equations, apply to (1.1) and allow to prove that under certain restrictions on the equation the stationary measureμ^ν is unique. In particular

this is true ifγ_I =0 (the easiest case), or ifp≥q andγ_R =0 (see [16]), or ifγ_R=0 andp=1 (see [18]). In this case any solutionu(t )of (1.1) withu(0)=u₀∈H¹satisfies

Du(t ) μ^ν ast→ ∞. (1.11)

This convergence and (1.3), (1.9) imply that

e^ρκu2μ^ν(du)≤C(κ, B), (1.12)

u^2m₁ μ^ν(du)≤C_m(κ, B₁) ∀m. (1.13)

1.3. Multidimensional case

In this section we briefly discuss a multidimensional analogy of Eq. (1.1):

˙

u+iν⁻¹Au=u−γRfp

|u|²

u−iγIfq

|u|² u + d

dτ ∞ j=1

b_jβ_j(τ )e_j(x), u=u(τ, x), x∈T^d. (1.14)

HereAu= −u+V (x)u,V ∈C^N(T^d,R)andV (x)≥1. The numbersγI, γR satisfy (0.3). Functionsfp≥0 and fq≥0 are real-valued smooth and

f_p(t)=t^p fort≥1, f_q(t)=t^q fort≥1,

wherep, q≥0. Ifγ_R=0, then the termuin the r.h.s. should be modified to−u. By{e_k, k≥1}, we denote the usual trigonometric basis of the spaceL₂(T^d)(formed by all functionsπ^−d/2f_s1(x₁)· · ·f_sd(x_d), where eachf_s(x)is sinsx or cossx), parameterised by natural numbers. These are eigen-functions of the Laplacian,−e_r =λ_re_r. We assume that

B_N

1=2

k

λ^N_k¹b²_k<∞, (1.15)

whereN1=N1(d)is sufficiently large. In this section we denote by(H^r, · r)the Sobolev spaceH^r =H^r(T^d,C), regarded as a real Hilbert space, and·,·stands for the realL2-scalar product.

Noting that(fp(|u|²)u− |u|^2pu)and(fq(|u|²)u− |u|^2qu)are bounded Lipschitz functions with compact support we immediately see that the a-priori estimates from Section1.1remain true for solutions of (1.14). Accordingly, for anyu0∈H¹∩L2q+2Eq. (1.1) has a solutionu(t, x)such thatu(0, x)=u0, satisfying (1.3), (1.8), (1.9).

Now assume that

p, q <∞ ifd=1,2, p, q < 2

d−2 ifd≥3. (1.16)

Applying Ito’s formula to the processesA^mu(τ ), u(τ )ⁿ,m, n≥1, using (1.3), (1.8), (1.9) and arguing by induction (first innand next inm) we get that

E

sup

0≤τ≤T

u(τ )²ⁿ

2m+ _T

0

u(s)²

2m+1u(s)²ⁿ⁻²

2m ds

≤ u0²ⁿ2m+C(m, n, T )

1+ u0^cm,n

, (1.17)

Eu(τ )²ⁿ

2m≤C(m, n) ∀τ ≥0, (1.18)

for each m andn, whereC(m, n, T ) andC(m, n) also depends on |V|C^N andB_N1 (see (1.15)), and N =N (m), N₁=N₁(m).

Relations (1.17) withm=m₀≥1 in the usual way (cf. [5,12,16,18]) imply that Eq. (1.14) isregular in the space H^m0∩L_2q+2in the sense that for anyu₀∈H^m0 ∩L_2q+2 it has a unique strong solutionu(t, x), equalu₀att=0, and satisfying estimates (1.3), (1.17) withm=m₀ for anyn. By the Bogolyubov–Krylov argument this equation has a stationary measureμ^ν, supported by the spaceH^m0 ∩L2q+2, and a corresponding stationary solutionu^ν(τ ), Du^ν(τ )≡μ^ν, also satisfies (1.3) and (1.18) withm=m0.

If (1.10) holds and (1.16) is replaced by a stronger assumption, then a stationary measure is unique. IfγI=0, the uniqueness readily follows, for example, from the abstract theorem in [13]. In [18] this assertion is proved if

γ_R=0 and q≤1 ifd=1, q <1 ifd=2, q≤2/d ifd=3. (1.19) In [16] it is established if

p=q, γR, γI>0 ifd=1,2, and p=q < 2

d−2, γR, γI>0 ifd≥3; (1.20) the argument of that work also applies ifp > q.

Note that whenγ_R=0 or whenp < q (i.e., when the nonlinear damping is weaker than the conservative term), the assumptions (1.19), (1.20), needed for the uniqueness of the stationary measure, are much stronger than the assumptions (1.16), needed for the regularity. This gap does not exist (at least it shrinks a lot) if the random force in Eq. (1.14) is not white in time, but is a kick-force. See in [11] the abstract theorem and its application to the CGL equations.

1.4. Spectral properties ofA_V:One-dimensional case

As in Section1.1we denoteA_V =A= −∂²/∂x²+V (x), where the potentialV (x)≥0 belongs to the spaceC_e^N ofC^N-smooth even and 2π-periodic functions,N≥1. Letφ₁, φ₂, . . .be theL₂-normalised complete system of real eigenfunctions ofA_V with the eigenvalues 1≤λ₁< λ₂<· · ·. Consider the linear mapping

Ψ:Hu(x)→v=(v₁, v₂, . . .)∈C^∞, defined by the relationu(x)=

vkφk(x). In the space of complex sequencesvwe introduce the norms

|v|²h^m=

k≥1

|vk|²λ^m_k, m∈R,

and denoteh^m= {v| |v|h^m<∞}. Due to the Parseval identity,Ψ:H→h⁰is a unitary isomorphism. By{Ψ_km, k, m≥1}we denote the matrix ofΨ with respect to the basis{e_j}inHand the standard basis inh⁰. SinceΨ maps real vectors to real, its matrix has real entries.

For anym∈Nwe have|v|²_h^m= A^mu(x), u(x). So the norms|v|h^m andumare equivalent form=0, . . . , N.

SinceΨ^∗=Ψ⁻¹, then the norms are equivalent for integer|m| ≤N. By interpolation they are equivalent for all real

|m| ≤N. So

the maps Ψ:H^m→h^m, |m| ≤N, are isomorphisms. (1.21)

DenoteG=Ψ⁻¹:h^m→H^m. Then Ψ ◦A◦G=diag{λ_k, k≥1} =:A.

Consider the operator

L:=Ψ◦(−)◦G=Ψ◦(A−V )◦G=A−Ψ◦V◦G=:A−L⁰. (1.22)

By (1.21)L⁰=Ψ◦V◦Gdefines bounded linear maps

L⁰:h^m→h^m ∀|m| ≤N, (1.23)

and in the spaceh⁰it is selfadjoint.

For any finiteMconsider the mapping

Λ^M:C_e^N→R^M, V (x)→(λ₁, . . . , λ_M).

Since the eigenvaluesλj are different, this mapping is analytic. As∇λj(V )=φj(x)²and the functionsφ²₁, φ₂², . . . are linearly independent by the classical result of G. Borg (1946), then for anyV ∈C_e^Nthe linear mapping

dΛ^M(V ):C^N_e →R^M is surjective (1.24)

(all this result may be found in [17]; e.g. see there p. 46 for Borg’s theorem). In the spaceC_e^N consider a Gaussian measureμ_Kwith a nondegenerate correlation operatorK(so for the quadratic functionf (V )= V , ξL₂V , ηL₂ we have

f (V )μ_K(dV )= Kξ, η). Relation (1.24) easily implies

Lemma 1.1. For anyM≥1the measureΛ^M◦μ_Kis absolutely continuous with respect to the Lebesgue measure on R^M.

We will call a vectorΛ∈R^∞nonresonantif for any nonzero integer vectorsof finite length we have

Λ·s =0. (1.25)

A potentialV (x)is called nonresonant if its spectrumΛ(V )=(λ₁, λ₂, . . .)is nonresonant. The nonresonant potentials are defined inC_e^N by a countable family of open dense relations (1.25). So

the nonresonant potentials form a subset ofC_e^Nof the second Baire category. (1.26) Applying Lemma1.1we also get

the nonresonant potentials form a subset ofC_e^Nof fullμ_Kmeasure (1.27) for any Gaussian measureμ_Kas above.

The nonresonant vectorsΛare important because of the following version of the Kronecker–Weyl theorem:

Lemma 1.2. Letf ∈Cⁿ⁺¹(Tⁿ)for somen∈N.Then for any nonresonant vectorΛwe have

Tlim→∞

1 T

T 0

f

q₀+t Λⁿ

dt=(2π)⁻ⁿ

fdx, Λⁿ=(Λ₁, . . . , Λ_n), uniformly inq₀∈Tⁿ.The rate of convergence depends onn,Λand|f|_Cⁿ+1. Proof. Let us write f (q) as the Fourier series f (q) =

f_se^is·q. Then for each nonzero s we have |f_s| ≤ Cn+1|f|Cⁿ⁺¹|s|⁻ⁿ⁻¹.So for anyε >0 we may findR=Rεsuch that|

|s|>Rfse^is·q| ≤^ε₂for eachq. Now it suffices to show that

1 T

_T

0

f_R

q₀+t Λⁿ

dt−f₀ ≤ε

2 ∀T ≥T_ε (1.28)

for a suitableTε, wherefR(q)=

|s|≤Rfse^is·q. But 1

T T

0

e^{is·(q0+t Λn)}dt ≤ 2

T|s·Λⁿ|

for each nonzeros. Therefor the l.h.s. of (1.28) is

≤ 2 T inf

|s|≤R

s·Λⁿ⁻¹

|f_s| ≤T⁻¹|f|C⁰C(R, Λ).

Now the assertion follows.

1.5. Spectral properties ofA_V:Multi-dimensional case

Now let, as in Section 1.3,A=AV be the operator A= −+V (x), x∈T^d, where 1≤V (x)∈C^N(T^d). Let {φ_k(x), k≥1} be its L₂-normalised eigenfunctions and {λ_k, k≥1}, be the corresponding eigenvalues, 1≤λ₁≤ λ₂≤ · · ·. For anyM≥1 denote byF_M⊂C^N(T^d)the open domain

F_M= {V |λ₁< λ₂<· · ·< λ_M}.

Its complementF_M^c is a real analytic variety inC^N(T^d)of codimension≥2, soF_M is connected (see [6] and refer- ences therein). The functionsλ1, . . . , λ_M are analytic inF_M. Let us fix any nonzero vectors∈Z^∞such thats_l=0 forl > M. The set

Q_s=

V ∈F_M|Λ(V )·s=0

clearly is closed inF_M. Since the functionΛ(V )·sis analytic inF_M, then eitherQ_s=F_M, orQ_s is nowhere dense inF_M. Theorem 1 from [6] immediately implies thatQ_s =F_M, so (1.26) also holds true in the case we consider now.

Letμ_K be a Gaussian measure with a nondegenerate correlation operator, supported by the spaceC^N(T^d). As Λ(V )·s is a non-trivial analytic function on F_M and F_M^c is an analytic variety of positive codimension, then μ_K(Q_s)=0 (e.g., see Theorem 1.6 in [1]). Since this is true for anyM and any s as above, then the assertion (1.27) also is true.

2. Averaging theorem

The approach and the results of this section apply both to Eqs (1.1) and (1.14). We present it for Eq. (1.1) and at Section2.5discuss small changes, needed to treat (1.14). Everywhere belowT is an arbitrary fixed positive number.

2.1. Preliminaries

In Eq. (1.1) withu∈H¹we pass to thev-variables,v=Ψ (u)∈h¹:

˙

v_k+iν⁻¹λ_kv_k=P_k(v)dτ+

j≥1

B_kjdβ_j(τ ), k≥1. (2.1)

HereB_kj =Ψ_kjb_j (a matrix with real entries, operating on complex vectors), and

Pk=P_k¹+P_k²+P_k³, (2.2)

whereP¹, P²andP³are, correspondingly, the linear, dissipative and Hamiltonian parts of the perturbation:

P¹(v)=κΨ◦ ∂²

∂x²u, P²(v)= −γ_RΨ

|u|^2pu

, P³(v)= −iγIΨ

|u|^2qu , whereu=G(v). We will refer to Eqs (2.1) as to thev-equations.

Fork≥1 let us denoteI_k=I (v_k)=¹₂|v_k|²andϕ_k =ϕ(v_k)=Argv_k∈S¹, whereϕ(0)=0∈S¹. Consider the mappings

ΠI:h^rv→I=(I1, I2, . . .)∈h^r_I+, Πϕ:h^rv→ϕ=(ϕ1, ϕ2, . . .)∈T^∞.