On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations

On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations

A. Agrachev

, S. Kuksin

, A. Sarychev

, A. Shirikyan

aSISSA-ISAS, via Beirut 2–4, Trieste 34014, Italy

bSteklov Institute of Mathematics, 8 Gubkina St., 117966 Moscow, Russia

cDepartment of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK dDiMaD, University of Florence, via C. Lombroso 6/17, Firenze 50134, Italy

eDépartement de Mathématiques, Université de Cergy-Pontoise, Site de Saint-Martin, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France

Received 1 February 2006; accepted 19 June 2006 Available online 13 December 2006

Abstract

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps.

We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier–Stokes equations perturbed by various random forces of low dimension.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

L’article est consacré à l’étude de l’image des mesures de probabilité sur un espace hilbertien par des applications analytiques de dimension finie. On établit des conditions suffisantes sous lesquelles l’image d’une mesure possède une densité par rapport à la mesure de Lebesgue et dépend continûment de l’application. Les résultats obtenus s’appliquent aux équations de Navier–Stokes 2D perturbées par diverses forces aléatoires de dimension peu élevée.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35Q30; 60H15; 93C20

Keywords:2D Navier–Stokes system; Analytic transformations; Random perturbations

0. Introduction

Let us consider the 2D Navier–Stokes equations on the torusT²⊂R²:

˙

u+(u,∇)u−νu+ ∇p=f (t, x), divu=0, x∈T². (0.1)

* Corresponding author.

E-mail addresses:[email protected] (A. Agrachev), [email protected] (S. Kuksin), [email protected] (A. Sarychev), [email protected] (A. Shirikyan).

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.06.001

Hereu=(u1, u2)andp are unknown velocity field and pressure,ν >0 is the viscosity, andf is an external force.

Eqs. (0.1) are supplemented with the initial condition

u(0)=u₀, (0.2)

whereu₀ is a given function belonging to the spaceH of divergence-free vector fields in L²(T²,R²). It is well known [20,5,21] that if the right-hand sidef satisfies some mild regularity assumptions, then problem (0.1), (0.2) has a unique solutionuin an appropriate functional class. Our aim is to study some qualitative properties of solutions in the situation whenfis a random process with a sufficiently non-degenerate distribution. More precisely, let us assume thatf is a stochastic process on the positive half-lineR₊ with range inL²(T²,R²)such that the distribution of its restriction to any interval[0, T]is a non-degenerate decomposable measure (see Condition (P) in Section 1.1). One of the main results of this paper says that for any finite-dimensional subspaceF ⊂H and anyt >0 the distribution of the projection ofu(t )toF has a density with respect to the Lebesgue measure. Similar properties are true in the case whenf is a white noise in time or a sum of independent identically distributed random forces. Furthermore, if the random dynamical system associated with (0.1) generates a Markov process, then the above-mentioned property is valid for any stationary distribution. These results are important since for a number of problems in pure and applied mathematics only finitely many Fourier components of a solutionumatter; these components correspond to a finite- dimensional subspace ofH.

The fact that a solution of a nonlinear SDE stirred by a degenerate noise has a continuous density against the Lebesgue measure is very well known. The proofs are usually based on the Malliavin calculus (see [17] and the references therein). There have been a few works in which various versions of the Malliavin calculus were developed for some stochastic PDE’s (for instance, see [18,4,6,15,8,16]). In particular, it was proved in [16] that iff is white noise in time and sufficiently non-degenerate in the space variables, then the distribution of the projection of solution for (0.1), (0.2) to any finite-dimensional subspace has a smooth density with respect to the Lebesgue measure.

The approach of this paper is completely different and is based on the controllability of (0.1) in finite-dimensional projections and an abstract result on the image of probability measures under analytic mappings. We emphasise that our method does not use the Gaussian structure of the noise, and the proof is simpler and shorter compared to the papers quoted above. At the same time, if the forcef is white in time, and the results of those works apply, then our information on the density of the distribution for the projection ofu(t )is weaker than, say, that obtained in [16].

The paper is organised as follows. In Section 1, we have compiled some preliminary results on decomposable measures on Hilbert spaces. Section 2 contains two abstract results on the transformation of probability measures under analytic mappings. In Section 3, we apply them to the 2D Navier–Stokes equations with different types of additive noise. Finally, in Appendix A, we prove some auxiliary results used in the main text.

0.1. Notation

LetX be a Polish space, i. e., separable complete metric space. We denote byB_X(a, R) the closed ball in X of radiusR centred ata. Ifa coincides with a selected point0∈X, then we writeB_X(R). Let B(X)be the Borel σ-algebra onXand letP(X)be the family of probability measures on(X,B(X)). The spaceP(X)is endowed with the total variation norm:

μ1−μ2var:= sup

Γ∈B(X)

μ1(Γ )−μ2(Γ ), μ1, μ2∈P(X).

Ifμ₁, μ₂∈P(X)andμ₁is absolutely continuous with respect toμ₂, then we writeμ₁μ₂. For a random variableξ, we denote byD(ξ )its distribution.

For any Banach spaceX, we denote by · Xthe norm inX. IfY is another Banach space, thenL(X, Y )stands for the space of bounded linear operators fromXtoY. In the caseX=Y, we shall writeL(X). IfXis finite-dimensional, then_Xdenotes the Lebesgue measure onX.

LetJ⊂Rbe a closed interval and letR₊= [0,+∞). We use the following functional spaces.

• Cb(X)is the space of bounded continuous functionsf:X→Rendowed with the norm f∞=sup

x∈X

f (x).

• C(J, X)is the space of continuous functionsu:J→X.

• L^p(J, X)is the space of Bochner-measurable functionsu:J→Xsuch that uL^p(J,X)=

J

u(t )^p

Xdt 1/p

<∞.

• L^p_loc(R₊, X) is the space of functionsu:R₊→X whose restriction to any finite interval J ⊂R₊ belongs to L^p(J, X).

IfXis a Hilbert space andF ⊂Xis a closed subspace, then^PF:X→F denotes the orthogonal projection inX ontoF.

1. Decomposable measures on Hilbert spaces

1.1. Definitions and examples

Let X be a separable Hilbert space with a scalar product (·,·) and the corresponding norm · X. We denote byB(X)the Borelσ-algebra onXand byP(X)the family of probability measures on(X,B(X)).

Definition 1.1.We shall say that a measureμ∈P(X)isdecomposableif there is an orthonormal basis{g_j} ⊂Xsuch that

μ= ∞ j=1

μj, (1.1)

whereμ_j is the projection ofμto the one-dimensional spaceX_j generated byg_j and⊗denotes the tensor product of measures.

Example 1.2.Letμ∈P(X)be a Gaussian measure (for instance, see [3]). It is well known that there is a vectora∈X and a self-adjoint nuclear operatorK∈L(X)such that the characteristic function ofμhas the form

ˆ

μ(z)=exp

i(a, z)−1

2(Kz, z) , z∈X. (1.2)

In this case, the vectorais the mean value ofμ, a=

X

xμ(dx),

andKis the covariance operator forμ, (Kz, z)=

X

(z, x−a)²μ(dx);

we refer the reader to Chapter 2 in [3] for more details. Let{g_j}be an orthonormal basis inX formed of the eigen- vectors ofKand letλ_j be the eigenvalue ofKcorresponding tog_j. If a vectorz∈Xis written in the form

z=^∞

j=1

z_jg_j,

then relation (1.2) takes the form ˆ

μ(z)= ∞ j=1

exp

i(a, gj)zj−1

2λjz²_j . (1.3)

It follows that μ admits decomposition (1.1) in which μj is a one-dimensional Gaussian measure with mean value(a, g_j)and variance λ_j. Note also that if λ_j =0 for somej 1, then the measureμ is degenerate in the sense that its support is contained in a proper affine subspace ofX.

In what follows, we shall deal with decomposable measures possessing some additional properties. Namely, we consider a measureμ∈P(X)satisfying the following condition.

(P) The measureμis decomposable and has a finite second moment

X

x²_Xμ(dx) <∞. (1.4)

Moreover, every measureμ_j in (1.1) possesses a continuous density ρ_j with respect to the Lebesgue measure onXj.

The following lemma describes the random variables whose distribution satisfies property (P); its proof is obvious.

Lemma 1.3.The distribution of anX-valued random variableξ satisfies condition(P)if and only ifξ has the form ξ=^∞

j=1

b_jξ_jg_j, (1.5)

where {gj} is an orthonormal basis in X,{ξj} is a sequence of scalar independent random variables such that Eξ_j²=1 and the law ofξ_j has a continuous density with respect to the Lebesgue measure, and b_j >0 are some constants satisfying the condition

∞ j=1

b_j²<∞. (1.6)

Example 1.4. Letμ∈P(X)be a Gaussian measure with a mean value a∈X and a covariance operatorK. We claim thatμsatisfies condition (P) if and only if it is non-degenerate, i.e., all eigenvalues ofKare positive. Indeed, Fernique’s theorem (see [3]) implies that condition (1.4) is satisfied for any Gaussian measure. Furthermore, it fol- lows from (1.3) that the projectionμj ofμtoXj is a one-dimensional Gaussian measure with varianceλj. Thus, μ_j possesses a continuous density with respect to the Lebesgue measure if and only ifλ_j>0.

For the sequel, we note that ifμ∈P(X)is a non-degenerate Gaussian measure, then

μ(B) >0 for any ballB⊂X; (1.7)

see Section 3.5 in [3] for a proof.

1.2. Support of decomposable measures

Let X be a separable Hilbert space and letμ∈P(X) be a measure possessing property (P). Denote by {g_j} an orthonormal basis in X for which representation (1.1) holds and, for any integer N 1, define X_{(N )} as the N-dimensional space spanned bygj, 1jN. Let

μ_{(N )}= N

j=1

μ_j, ν_{(N )}= ∞ j=N+1

μ_j. (1.8)

Ifa∈X_{(N )}andR >0, then we denote byB_N(a, R)the closed ball inX_{(N )}of radiusRcentred ata. In the casea=0, we writeB_N(R).

Proposition 1.5.Letμ∈P(X)be a measure satisfying condition(P). Then there is a point A=

∞ j=1

ajgj∈X (1.9)

such that¹

ρj(aj) >0 for allj1. (1.10)

Furthermore, ifA∈Xis a point satisfying(1.10), then for any integerN1there isrN>0such that suppμ⊃B_N(A_N, r_N), A_N=

N

j=1

a_jg_j. (1.11)

Proof. To prove the existence ofA, denote byA⁰=

ja_j⁰gjany point in the support ofμ. Thena⁰_j∈suppμjfor any j 1. SinceμjX_j (whereX_j denotes the Lebesgue measure onXj), there isaj∈Rsuch that|aj−a_j⁰|j⁻¹ and (1.10) holds. DefiningA∈Xby relation (1.9), we obtain the required result.

When proving the second part of the proposition, we shall assume, without loss of generality, thatA=0. Let us fix any integerN1. It follows from (1.8) and (1.10) that

suppμ(N )⊃BN(rN), (1.12)

whererN>0 is sufficiently small. We claim that (1.11) holds for the same constantrN>0. Indeed, letx∈BN(rN) andε >0. Consider anX-valued random variableξ with distributionμ. By Lemma 1.3, it has the form (1.5). We wish to show that

P

ξ−xXε

>0. (1.13)

To this end, define ϕN=

N

j=1

bjξjgj, ψN= ∞ j=N+1

bjξjgj.

It is clear that (1.13) will be established if we prove that P

ϕ_N−xXε/2

>0, P

ψ_NXε/2

>0. (1.14)

The first inequality follows immediately from (1.12). To prove the second, choose an integerM > N and write ψN²X=

M

j=N+1

b_j²ξ_j²+ ∞ j=M+1

b_j²ξ_j²=SM+RM. (1.15)

In view of (1.6) and the relationEξ_j²=1, we have ER_M=

∞ j=M+1

b²_j→0 asM→ ∞. By Chebyshev’s inequality, it follows that

P{RMδ} →1 asM→ ∞, (1.16)

whereδ >0 is an arbitrary constant. Furthermore, inequalities (1.10) with aj=0 imply that, for any fixedM > N andδ >0, we have

P{SMδ}>0. (1.17)

Combining (1.15)–(1.17) and recalling thatRM andSM are independent, we arrive at the second inequality in (1.14).

This completes the proof of the proposition. 2

1 In what follows, we identifyX_j with the real line and regardρ_jas a function of a real variable.

1.3. A zero-one law for analytic functions

LetXbe a separable Hilbert space and letf:X→Rbe a continuous function. Recall thatf is said to beanalytic if for anyx₀∈Xthere isδ >0 such that

f (x)=f (x₀)+ ∞ m=1

L_m(x−x₀) forx∈B_X(x₀, δ), (1.18)

whereL_m:X→Ris anm-linear functional and the series in (1.18) converges regularly. The latter means that ∞

m=1

Lmδ^m<∞,

where · stands for the norm of multilinear functionals:

L_m = sup

xX1

|L_mx|.

We refer the reader to [13] or [21] for more details.

For any continuous functionf:X→R, we set Nf =

x∈X: f (x)=0 .

It is well known that if dimX <∞andf is analytic, then either_X(Nf)=0 orf≡0. The following theorem shows that a similar result is true in the infinite-dimensional case.

Theorem 1.6.Letf:X→Rbe an analytic function and letμ∈P(X)be a measure possessing property(P). Then

μ(Nf)=0or1. (1.19)

Furthermore, iff is not identically zero, thenμ(Nf)=0.

Proof. We shall need a result from [7]. To formulate it, let us introduce some definitions.

Recall that the one-dimensional spacesXj are defined in condition (P) and thatX(N )stands for the vector space spanned byXj, 1jN. Denote byX_{(N )}^⊥ the orthogonal complement ofX(N )inXand setX(∞)=

NX(N ). We shall say thatA∈B(X)is a finite zero-oneμ-set if

ν_{(N )}

y∈X^⊥_{(N )}: μ_{(N )} A_N(y)

=0 or 1

=1 for any integerN1,

whereA_N(y)= {x∈X_{(N )}: x+y∈A}, and the measuresμ_{(N )}andν_{(N )}are defined by (1.8).

According to Theorem 4 in [7], ifμis a decomposable measure onX, then for any finite zero-oneμ-setA∈B(X) there isA∈B(X)such that

A+X(∞)=A, μ(A)=μ(A).

Applying the Kolmogorov zero-one law (see [9]), we see thatμ(A)=0 or 1. Thus, the measure of any finite zero-one μ-set is either zero or one.

To prove (1.19), note that if f is analytic, then for any integer N 1 and any y ∈X_{(N )}^⊥ , we have either _N(Nf(y))=0 or_N(X_{(N )}\Nf(y))=0, where _N denotes the Lebesgue measure on X_{(N )}. Since μ_{(N )N} (see condition (P)), we see that

μ_{(N )} Nf(y)

=0 or 1 for anyy∈X^⊥_{(N )}.

Thus,Nf is a finite zero-oneμ-set, and (1.19) follows from what has been said above.

We now suppose thatf ≡0. In this case, there isx₀∈Xsuch thatf (x₀)=0, and since{g_j}is a basis inXandf is continuous, there is no loss of generality in assuming thatx₀∈X_{(N )}for some integerN1. In view of (1.19), the required assertion will be established if we show that

μ

x∈X: f (x)=0

=0. (1.20)

The restriction off toX(N )is an analytic function on a finite-dimensional space. It follows that f (x)=0 for_N-almost everyx∈X_{(N )}.

Recalling Proposition 1.5, we can find a pointx₀∈suppμsuch thatf (x₀)=0. By continuity, there isδ >0 such that f (x)=0 forx∈B(x0, δ).

This implies that{f (x)=0} ⊃B(x₀, δ), and therefore (1.20) holds. 2 2. Image of measures under analytic maps

2.1. Formulations of the results

Let(U, d) be a metric space and letX andF be finite-dimensional vector spaces. Consider a continuous oper- ator f:U×X→F. For any probability measure μ∈P(X) and any u∈U, denote byf_∗(u, μ)the image ofμ underf (u,·).

Theorem 2.1.Suppose that, for anyu∈U, the functionf (u,·)is analytic and the interior of the setf (u, X)is non- empty. Letμ∈P(X)be a measure possessing a continuous densityρ(x)with respect to the Lebesgue measure onX.

Then the following assertions hold.

(i) For anyu₀∈U, the measuref_∗(u₀, μ)is absolutely continuous with respect to_F.

(ii) The functionf_∗(·, μ)fromUto the spaceP(F )endowed with the total variation norm is continuous.

Our next goal is to study the case in whichX is an infinite-dimensional space. More precisely, suppose that X is a separable Hilbert space and F is a finite-dimensional vector space. Recall that condition (P) is introduced in Section 1.1.

Theorem 2.2. Let f:U×X→F be a continuous function such thatf (u,·)is analytic for any u∈U and the derivativeD_xf (u, x)is continuous with respect to(u, x). Suppose that, for anyu∈U, there is a ballB_u in a finite- dimensional subspaceX_u⊂Xsuch that the interior of the setf (u, B_u)is non-empty. Then for any measureμ∈P(X) satisfying condition(P)statements(i)and(ii)of Theorem2.1take place.

2.2. Proof of Theorem 2.1

Let us fix any pointu₀∈Uand denote byD_xf (u₀, x)the derivative (Jacobian) of the mapf (u₀,·):X→F at the pointx∈X. We first show that the matrixD_xf (u₀, x)has a minorm(x)of the size dimF such that

m(x)=0 forμ-almost everyx∈X. (2.1)

To this end, recall that a pointx∈Xis said to beregularforf (u₀,·)if the rank ofD_xf (u₀, x)is maximal. Any point that is not regular is said to be singular. In view of Sard’s theorem (see [19]), the image under the smooth functionf (u₀,·)of the set of its singular points has zero Lebesgue measure. Since the interior off (u₀, X)is non- empty, we conclude thatf (u₀,·)has a regular pointx₀∈X.

Let m(x)be the minor of D_xf (u₀, x)that is non-zero at x₀. Sincem(x) is analytic, we see thatm(x)=0 al- most everywhere with respect to the Lebesgue measureX. Sinceμis absolutely continuous with respect toX, we conclude that (2.1) holds.

For anyε >0, we denote X^ε=

x∈X: |x|ε⁻¹,m(x)ε . Then

νε:=μ X\X^ε

→0 asε→0.

Let us take anyx∈X^εand write it asx=(x₁, x₂), wherex₁denotes the variables entering the minorm(x). Accord- ingly, the spaceXcan be represented as a direct productX=X₁×X₂. Applying the implicit function theorem, we

can find open ballsV1⊂X1 andV2⊂X2 such that for anyx2∈V2 andu∈B^ε=BX(u0, rε),the mapf (u,·, x2) is a diffeomorphism of the domainV1onto its imageW (u, x2). Hererε>0 is a constant that goes to zero withε.

Accordingly, we can writex1in terms ofu∈B^ε,x2∈V2andy=f (u, x1, x2)∈W (u, x2):

x₁=g(u, y, x₂).

The setsV =V1×V2corresponding to variousx∈X^εform an open cover of the compact setX^ε. Let us find a finite sub-cover{V^j}. We denote by{ϕj(x)}a continuous partition of unity onX^ε subordinate to{V^j}. That is, ϕj0, suppϕj⊂V^j, and(

ϕj)(x)=1 forx∈X^ε. Letμj=ϕjμ. Then

μ_j=ϕ_j(x)ρ(x)dx1dx2=ϕ_j(x₁, x₂)ρ(x₁, x₂)m(u, x₁, x₂)⁻¹dydx2, (2.2) wherem(u, x)is the minor ofD_xf (u, x)corresponding tox1, andx1=g(u, y, x2)on the right-hand side of (2.2).

Hence,f_∗(u, μ_j)=g_j(u, y)dy, where gj(u, y)=

˜

ϕj(y, x2)ρ(y, x˜ 2)m(u, y, x2)⁻¹dx2

withϕ˜_j(y, x₂)=ϕ_j(g(u, y, x₂), x₂), etc. Let us denoteμ_ε=(

ϕ_j)μ. Thenμ−μ_εvarν_ε. We have

f_∗(u, μ_ε)=g_ε(u, y)dy, (2.3)

whereg_ε(u, y)=

g_j(u, y)is a continuous function ofu∈B^εandy. Clearly, f_∗(u, μ_ε)−f_∗(u, μ)

varν_ε for anyu∈B^ε. (2.4)

Relations (2.3) and (2.4) withu=u₀andε→0 imply assertion (i). Indeed, for any measurable setQ⊂F with zero Lebesgue measure, we have

f_∗(u, μ)(Q)=

f_∗(u, μ)−f_∗(u, με)

(Q)+f_∗(u, μ^ε)(Q), sof_∗(u, μ)(Q)ν_εfor eachε. Hence,f_∗(u, μ)(Q)=0.

To prove (ii), we fix anyγ >0 and chooseε >0 such thatν_ε<¹₃γ. Due to (2.4), foru∈B^εwe have f_∗(u, μ)−f_∗(u₀, μ)

var2νε+f_∗(u, μ_ε)−f_∗(u₀, μ_ε)

var

2νε+1

2 gε(u, y)−gε(u0, y)dyγ ,

ifuis sufficiently close tou0. Sinceu0is an arbitrary point, what has been said implies (ii).

Remark 2.3.Analysing the proof given above, one easily sees that, instead of assuming the existence of interior points for the setf (u, X), we could require that the functionf (u,·)should have at least one regular point for anyu∈U(cf.

the beginning of the proof).

2.3. Proof of Theorem 2.2

Step 1. Let us fix anyu₀∈Uand show that there is finite-dimensional subspaceX₁⊂Xspanned by some vectors of the basis{g_j}such that dimX₁=dimF and (2.1) holds, where m(x)denotes the determinant of the matrix for the restriction ofD_xf (u₀, x)toX₁. Indeed, by the hypothesis, there is a ballB_u0 in a finite-dimensional subspace X_u0 ⊂Xsuch that the interior off (u₀, B_u0)is non-empty. Since{g_j}is a basis inX, for anyδ >0 there is a finite- dimensional subspaceY ⊂Xspanned by some vectors of{g_j}such that

^PYy−yXδ for anyy∈B_u0, (2.5)

whereP_Y:X→X is the orthogonal projection in X onto the subspace Y. Now note thatf (u0,·) is continuous andBu₀ is compact. Therefore for anyε >0 we can findδ >0 such that

f (u₀, z)−f (u₀, y)

F ε fory∈B_u0,z−yXδ. (2.6)

Combining (2.5) and (2.6), we see that for anyε >0 there is a finite-dimensional subspaceY ⊂Xspanned by some vectors of{g_j}such that

f (u0,P_Yy)−f (u0, y)

Fε fory∈Bu₀.

Choosing ε >0 sufficiently small and applying Proposition A.1 of the Appendix (see Appendix A), we conclude that the interior of the set f (u0, Y ) is non-empty. Since dimY <∞, Sard’s theorem implies that the function f (u₀,·):Y →F has at least one regular pointx₀∈Y. Let us denote byX₁ a subspace spanned by some vectors of the basis{g_j}such that the restriction ofD_xf (u₀, x₀)toX₁is an isomorphism fromX₁ontoF. Thenm(x₀)=0.

Sincem(x)is an analytic function, the required result follows from Theorem 1.6.

Step 2. We now repeat the argument used in the proof of Theorem 2.1. Let us representX as the direct product X=X₁×X₂, whereX₁ is constructed in Step 1 andX₂denotes the orthogonal complement of X₁inX. Denote by λ₁andλ₂the projections of μ to the subspacesX₁ andX₂, respectively, and byρ(x₁)the density ofλ₁ with respect toX₁. For anyε >0, let us choose a compact setX^ε⊂Xsuch that

xXε⁻¹, m(x)ε forx∈X^ε, ν_ε:=μ X\X^ε

ε.

As in the proof of Theorem 2.1, we can find a constantr_ε>0 going to zero withεand a finite cover{V^j=V₁^j×V₂^j} of the compact set X^ε such that V₁^j⊂X1 andV₂^j⊂X2 are balls, and for any x2∈V2 andu∈B^ε=BX(u0, rε), the map f (u,·, x2)is a diffeomorphism of the domainV1 onto its imageW (u, x2). We denote by x1=g(u,·, x2) the inverse function off (u,·, x₂). Let{ϕ_j(x)}be a continuous partition of unity onX^εsubordinate to{V^j}and let μ_j=ϕ_jμ. Then we have (cf. (2.2))

μj=ϕj(x)ρ(x1)dx1λ(dx2)=ϕj(x1, x2)ρ(x1)m(u, x1, x2)⁻¹dyλ(dx2),

wherem(u, x)denotes the determinant of the restriction ofDxf (u, x)toX1, andx1=g(u, y, x2)on the right-hand side of the formula. The rest of the proof is literally the same as that of Theorem 2.1, and therefore we omit it.

Remark 2.4.The proof given above implies that the claim of Theorem 2.2 remains true if we replace the condition of existence of interior points for the setf (u, B_u)by the following one: for anyu∈U, there is a pointx_u∈Xand a finite-dimensional subspaceX_u⊂Xsuch that the restriction ofD_xf (u, x_u)toX_uis an isomorphism fromX_uontoF (cf. Step 1 of the proof).

3. Applications

Throughout this section, we use the standard functional spaces H andV arising in the theory of Navier–Stokes equations; they are defined in Subsection 3.1. We shall also use the spaces

X=C(R₊, H )∩L²_loc(R₊, V ), XT =C(JT, H )∩L²(JT, V ), whereT >0 andJ_T = [0, T].

3.1. Navier–Stokes equations perturbed by a time-discrete random force

Let us consider the 2D Navier–Stokes (NS) system on the torusT²=R²/2πZ². Define the spaces H=

u∈L² T²,R²

: divu=0 onT²

, V =H¹ T²,R²

∩H,

endowed with natural norms. Here H^s(T²,R²)denotes the space of vector functions (u₁, u₂)whose components belong to the Sobolev space of orders. LetΠ:L²(T²,R²)→Hbe the orthogonal projection inL²(T²,R²)ontoH. After applying the projectionΠ, the NS system reduces to the following evolution equation inH:

˙

u+νLu+B(u, u)=g(t, x), (3.1)

whereν >0 is the viscosity,L= −Π , andB(u, v)=Π ((u,∇)v). In this subsection, we assume that g(t, x)=

∞ k=1

Ik,T(t )ηk(x), (3.2)

whereT >0 is a parameter,Ik,T(t )is the indicator function of the time interval[(k−1)T , kT ), and{ηk}is a sequence ofH-valued i.i.d. random variables defined on a probability space(Ω,F,P). Standard theorems on well-posedness of the 2D NS system (e.g., see [5]) imply that, for almost everyω∈Ω, problem (3.1), (3.2) has a unique solution u∈Xthat satisfies the initial condition

u(0)=u0, (3.3)

whereu₀∈H is an arbitrary function. We shall denote byS_t:H→Hthe random operator that takesu₀tou(t ). Our aim is to study the distribution for projections of the random variablesS_kT(u0)to finite-dimensional subspaces ofH.

Let{ej}be a complete set of eigenfunctions forLindexed in an increasing order of the corresponding eigenval- uesαj and letHNbe the vector space spanned byej,j=1, . . . , N. We shall assume that the i.i.d. random variablesηk

satisfy the following condition.

(D) The random variablesηkhave the form η_k=^∞

j=1

b_jξ_{j k}e_j, (3.4)

whereb_j0 are some constants such that ∞

j=1

b²_j<∞, (3.5)

andξ_{j k}are independent scalar random variables whose distributionπ_jpossesses a continuous density with respect to the Lebesgue measure, and suppπ_j0 for anyj0.

The following theorem shows that, under the above condition, all finite-dimensional projections of the distribution forS_kT(u)have continuous densities with respect to the Lebesgue measure, provided thatkis sufficiently large andT does not belong to a discrete exceptional set.

Theorem 3.1.Suppose that condition(D)is fulfilled. Then there is an integerN1 not depending onν and{η_k} such that the following two statements hold, provided that

b_j=0 forj =1, . . . , N. (3.6)

(i) For any constantν >0 and any finite-dimensional subspaceF ⊂H, there is a discrete subsetT=T(ν, F )⊂ R₊\ {0}such that ifT /∈TandR >0, then for anyu∈B_H(R)and an appropriate integerk=k(ν, F, R)1 the distribution ofP_FS_kT(u)possesses a density with respect to_F.

(ii) Let us setλ_u(t )=D(P_FS_t(u)). Thenλ_u(kT )continuously depends onu∈B_H(R)in the total variation norm.

Remark 3.2.It is possible to give a more precise description of the integerN in (3.6). Namely, it is the minimal integerN 1 such that the vectorse1, . . . , eN form a saturating set (see Section A.3 in Appendix A for a definition of a saturating set). In particular, if the eigenfunctions ofLare indexed in a suitable way, then one can takeN=6.

Proof of Theorem 3.1. Step 1. We wish to apply Theorem 2.2 and Remark 2.4. LetH₀⊂H be the subspace spanned by those vectorse_j for whichb_j=0 and letXk be the direct product ofkcopies ofH₀. We fix a finite-dimensional subspaceF⊂Hand consider the operator

f_k:R^∗₊×H×Xk→F, (T , u₀, η₁, . . . , η_k)→^PFu(kT ),

whereR^∗₊=R₊\ {0}andu(t )denotes the solution of problem (3.1)–(3.3) in which{η_k}is regarded as a sequence of deterministic functions inH₀. In view of Proposition A.2, the operatorf_k is analytic onR^∗₊×H ×Xk. Fur- thermore, if condition (D) is fulfilled, then for any integerk1 the distribution of the Xk-valued random variable η_k=(η₁, . . . , η_k)satisfies property (P). Suppose we have established the existence of a discrete subsetT⊂R^∗₊pos- sessing the following property:

(C) for anyR >0 there is an integerk1 such that ifT /∈Tandu0∈BH(R), then the derivative

(D_ηkf_k)(T , u₀,η_k):Xk→F (3.7)

is surjective for at least one pointη_k=(η₁, . . . , η_k)∈Xk.

In this case, statements (i) and (ii) of the theorem are straightforward consequences of Theorem 2.1 and Remark 2.3 in whichX=Xk andU=B_H(R).

Step 2. To prove (C), we first assume thatu₀=0. Let us denote byR1:L²(J₁, H )→H the operator that takes each functiong∈L²(J₁, H )tou(1, x), whereu∈X1is the solution of (3.1), (3.3) withu₀=0. By Proposition A.5, there is an integerN 1 such that the Navier–Stokes system (3.1) with g∈L²(J₁, H_N)is solidly controllable in time 1 for the projection toF. (See Definition A.4 for the concept of solid controllability.) In particular, there is a compact subsetK⊂L²(J₁, H )and a constantε >0 such thatΦ(K)⊃B_F(1)for any continuous map satisfying the inequality

sup

g∈K

Φ(g)−^PFR1(g)

F ε. (3.8)

For any integerm1, denote byYm⊂L²(J1, HN)the subspace of functions that are constant on any interval of the form[^l−_m¹,_m^l), 1lm. It is clear that

mY_mis dense inL²(J₁, H_N). SinceKis compact, for anyδ >0 we can find an integerm1 such that

sup

g∈K

g−^PYmgV δ.

It follows from the continuity of R1 that for anyε >0 there is an integerm1 such that (3.8) is satisfied for Φ(g)=^PFR1(P_Ymg). This implies that

P_FR1(Y_m)⊃B_F(1). (3.9)

Now note that if we denote byI_l,m(t )the indicator function of the interval[^l−_m¹,_m^l)and identify the function g(t, x)=

m

l=1

I_l,m(t )η_l(x)∈Y_m

with the vectorη=(η₁, . . . , η_m)∈Xm, then we can write f_m

m⁻¹,0,η_m

=^PFR1(g). (3.10)

Combining (3.9) and (3.10), we conclude that there is a finite-dimensional subspaceX⊂X_msuch that fm

m⁻¹,0, X

⊃BF(1).

Sard’s theorem now implies that the derivative (D_ηmf_m)

m⁻¹,0,η_m

:Xm→F

is surjective for at least one pointη⁰_m∈Xm. Since(Dη_mf_m)(T ,0,η⁰_m)is an analytic function with respect toT >0, we see that there is a discrete setT⊂R^∗₊such that

(D_ηmf_m)

T ,0,η⁰_m

:Xm→F is surjective for anyT /∈T. (3.11)

Step 3. We can now verify property (C). Let us fix anyT /∈T andR >0. We claim that if an integerl1 is sufficiently large andk=l+m, then the linear operator (3.7) is surjective forη_k=(0, . . . ,0,η⁰_m)and anyu0∈BH(R).

Indeed, the definition offkimplies that fk

T , u0,η⁰_k

=fm

T , u(lT ),η⁰_m ,

whereu(t )is the solution of (3.1), (3.3) withg≡0. It follows that Im

(D_ηkf_k)

T , u₀,η⁰_k

⊃Im

(D_ηmf_m)

T , u(lT ),η⁰_m

, (3.12)

where Im{A} denotes the image of a linear operator A. In view of (3.11) and the continuity of Dη_mfm, we can findr >0 such that

Im

(Dη_mfm)

T , v,η⁰_m

=F for anyv∈BH(r). (3.13)

Furthermore, the dissipation property of the homogeneous Navier–Stokes system implies that

u(lT )∈BH(r) for anyu0∈BH(R), (3.14)

wherel=l(R, T )1 is sufficiently large. Combining (3.12)–(3.14), we arrive at the required result. 2

Remark 3.3.Analysing the proof given above, it is possible to establish the following property, which shows that the

“bad” subsetTconstructed in Theorem 3.1 cannot accumulate to zero if we allow the integerkto depend onT:

• For any constant ν >0 and any finite-dimensional subspace F ⊂H there is T₀>0 such that if T ∈(0, T0] andR >0, then for anyu∈B_H(R)and an appropriate integerk=k(ν, F, R, T )1 the distribution ofP_FS_kT(u) possesses a density with respect to_F and continuously depends onuin the total variation norm.

We now study stationary solutions of (3.1), (3.2). Since{η_k}are i.i.d. random variables inH, for any deterministic initial function u₀ the sequence {u(kT ), k0}is a Markov chain. Thus, the set of all solutions restricted to the timeskT form a Markov family inH. LetP_k(u, Γ )be the corresponding transition function and letP^∗_kbe the Markov operator associated withP_k. Using standard a priori estimates for solutions of the Navier–Stokes system and applying the Bogolyubov–Krylov argument (e.g., see [12]), one can show thatP^∗_k has at least one stationary distributionμ:

P^∗_kμ=μ for allk1. (3.15)

A simple consequence of Theorem 3.1 is the following result:

Corollary 3.4.Suppose that the conditions of Theorem3.1are fulfilled. LetF ⊂H be a finite-dimensional subspace and letT /∈T(ν, F ). Then for any stationary measureμthe projectionP_Fμpossesses a density with respect to the Lebesgue measure onF.

Proof. LetΓ ⊂F be a Borel set of zero Lebesgue measure and letε >0. Choose a constantR >0 so large that μ

B_H(R)

1−ε. (3.16)

By assertion (i) of Theorem 3.1, we can find an integerk1 such that

(P_FP_k)(u, Γ )=0 for allu∈B_H(R). (3.17)

Combining (3.15)–(3.17), we derive P_Fμ(Γ )=

H

(P_FP_k)(u, Γ )μ(du)=

B_H^c(R)

(P_FP_k)(u, Γ )μ(du)μ B_H^c(R)

ε,

whereB_H^c(R)denotes the complement ofB_H(R). Sinceε >0 was arbitrary, we conclude thatP_Fμ(Γ )=0. This completes the proof of the corollary. 2

3.2. Navier–Stokes equations with a non-degenerate random perturbation

In this subsection, we consider the NS system (3.1) with a right-hand side of the form

g(t, x)=h(t, x)+η(t, x). (3.18)

We assume that h∈L²_loc(R+, H ) is a deterministic function andηis a random process whose trajectories belong toL²_loc(R₊, H₀), whereH₀⊂His a closed subspace. Denote byμ_T,T >0, the distribution of the restriction ofηto the intervalJ_T and byS_t:H→H the random operator that takes eachu₀∈H tou(t ), whereu∈X is the solution of (3.1), (3.3), (3.18).

Theorem 3.5.There is an integerN1 not depending onν >0such that ifh∈L²_loc(R+, H )is a given function, T >0 is a constant, H₀⊂H is a subspace containing H_N, and μ_T is a decomposable measure on L²(J_T, H₀) satisfying property(P), then for anyu₀∈Hand any positivet∈J_T the following statements take place.²

(i) LetF⊂H be a finite-dimensional subspace and letλ_u(t )be the distribution ofP_FS_t(u). Thenλ_u(t )_F. (ii) The measureλ_u(t )continuously depends onu∈H in the total variation norm.

Proof. Both assertions are straightforward consequences of Theorem 2.2 and Propositions A.2 and A.5. Indeed, let f_t:H×L²(J_T, H )→F be the operator that takes each pair(u₀, η)toP_Fu(t ), whereu∈XT is the solution of prob- lem (3.1), (3.3), (3.18) with deterministic functionshandη. By assumption, the distributionμ_T of the restriction ofη toJ_T satisfies condition (P), and by Proposition A.2, the operatorf_t is analytic with respect to(u₀, η). Furthermore, taking into account Proposition A.5 and repeating the argument used in Section 3.1, for anyu∈H and any positive t∈JT we can find a ballBu in a finite-dimensional subspaceXu⊂L²(JT, H )such thatft(u, Bu)⊂F has at least one interior point. Thus, the conditions of Theorem 2.2 are fulfilled, and we can conclude that assertions (i) and (ii) hold. 2

Example 3.6.Let us consider an example of a random forceη(t, x)for which the hypotheses of Theorem 3.5 are satisfied for any T >0. LetH₀⊂H be any finite-dimensional subspace and let {η(t ), t0}be a homogeneous Gaussian process inH₀with a correlation functionK(t ). (For the existence of such a process, see [11, Chapter 3].) Suppose thatK(t )is a positive-definite operator for anyt0. Then, for anyT >0, the distribution of the restriction ofηtoJ_T is a non-degenerate Gaussian measure. As is explained in Example 1.4, such a measure satisfies property (P).

3.3. Navier–Stokes equations perturbed by a white noise

This subsection is devoted to studying the NS system (3.1), (3.18), in which h∈L²_loc(R+, H )is a deterministic function andη(t, x) is a random process white in time andH²-regular in the space variables. More precisely, we define the spaceU=V ∩H²(T²,R²)endowed with theH²-norm and assume that there is a non-negative nuclear operatorQ∈L(U )such that

η(t, x)= ∂

∂tζ (t, x), (3.19)

whereζ is a Gaussian process inUwith continuous trajectories and the covariance operator K(t, s)=(t∧s)Q, t, s0.

It follows that ζ (t, x)=

∞ j=1

bjβj(t )gj(x),

where{g_j}is an orthonormal basis inU formed of the eigenvectors ofQ,b²_j is the eigenvalue ofQcorresponding tog_j, and{β_j}is a sequence of independent standard Brownian motions. It is well known (see [21,10]) that for almost every value of the random parameter the Cauchy problem for (3.1), (3.18), (3.19) has a unique solution in the spaceX, and we denote byS_t:H→Hits resolving (random) operator.

Theorem 3.7.There is an integer³N1 not depending onν >0such that ifh∈L²_loc(R+, H )is a given function and the image ofQcontainsH_N, then for anyu₀∈H andt >0assertions(i)and(ii)of Theorem3.5hold, and

suppλu(t )=F. (3.20)

2 See Remark 3.2 for a more precise description ofN.

3 A more precise description ofNcan be found in Remark 3.2.