Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes
aa r X i v : . [ m a t h . P R ] J a n Generalized solutions of the Cauchy problem forthe Navier-Stokes system and diffusion processes
S. Albeverio
Institut f¨ur Angewandte Mathematik, Universit¨at Bonn,Wegelerstr. 6, D-53115 Bonn, Germany SFB 611, Bonn, BiBoS,Bielefeld - BonnCERFIM, Locarno and USI (Switzerland)
Ya. Belopolskaya,St.Petersburg State University for Architecture and CivilEngineering, 2-ja Krasnoarmejskaja 4,190005, St.Petersburg, Russia
Abstract
We reduce the construction of a weak solution of the Cauchy prob-lem for the Navier-Stokes system on R to the construction of a solu-tion to a stochastic problem. Namely, we construct diffusion processeswhich allow us to obtain a probabilistic representation of a weak (indistributional sense) solution to the Cauchy problem for the Navier-Stokes system on a small time interval. Strong solutions on a smalltime interval are constructed as well AMS Subject classification :
Key words:
Stochastic flows, diffusion process, nonlinear para-bolic equations, Cauchy problem.
Introduction
The main purpose of this article is to construct both strong and weaksolutions (in certain functional classes) of the Cauchy problem forthe Navier-Stokes (N-S) system in R . To this end we consider astochastic problem and show that the solution of the Cauchy problemfor the Navier-Stokes system can be constructed via the solution ofthis stochastic problem.The approach we develop in this article is based on the theoryof stochastic equations associated with nonlinear parabolic equationsstarted by McKean [1] and Freidlin [2],[3] and generalized by Belopol-skaya and Dalecky [4], [5] on one hand and on the theory of stochastic ows due to Kunita [6] on the other hand. In our previous paper [7]we have constructed a stochastic process that allows us to prove theexistence and uniqueness of a local in time classical ( C -smooth in thespatial variable) solution of the Cauchy problem for the Navier -Stokessystem. In the present paper we construct a process which allows us toobtain construction of solutions of the both weak and strong Cauchyproblem for this system. Later we plan to apply a similar approachfor the Navier-Stokes equation for compressible fluids extending theresults from [9], [10].A close but different approach is the Euler-Lagrange approach toincompressible fluids which was developed by Constantin [11] andConstantin and Iyer [12]. Shortly, the main differences in these ap-proaches are the following: we use a probabilistic representation forthe Euler pressure instead of the Leray projection and obtain differ-ent formulas for the stochastic representation of the velocity field. Wediscuss these differences with more details in the last section of thepresent work.Thestructure of the present article is as follows. In the first sectionwe give some preliminary information concerning different analyticalapproaches to the Navier-Stokes system. Here we recall some commonways to eliminate the pressure and to obtain a closed equation for thevelocity.The classical approaches here are based on the so called Leray(Leray-Hodge)-projection that is a projection of the space of squareintegrable vector fields to the space of divergence free square integrablevector fields. Applying such a projection to the velocity equationone can eliminate the pressure p and get the closed equation for thevelocity u . This operator is used both in numerous analytical papers(see [15] for references) and in papers where the N-S system is studiedfrom the probabilistic point of view [16],[17], [12]. Finally the pressureis reconstructed from the Poisson equation.One more possibility to eliminate the pressure appears when oneconsiders the equation for the vorticity of the velocity field u and usesthe Biot-Sawart law to obtain a closed system. From the probabilisticpoint of view this approach was investigated in [18].In our previous paper [7] we do not use the Leray projection but in-stead we start with consideration of a system consisting of the originalvelocity equation and the Poisson equation for the pressure and con-struct their probabilistic counterpart. The probabilistic counterpartof the N-S system was presented in the form of a system of stochas-tic equations. Furthermore we prove the existence and uniquenessof a solution to this stochastic system and show that in this way weconstruct a unique classical (strong) solution of the Cauchy problemfor the N-S system defined on a small time interval depending on theCauchy data.In the present paper we also reduce the N-S system to the sys-tem of equations consisting of the original velocity equation and thePoisson equation for the pressure but then an associated stochasticproblem considered here allows to construct a generalized (distribu-tional) solution to the Cauchy problem for the N-S system. The as- ociated stochastic problem is studied in section 5. In sections 1-4we expose auxiliary results used in section 5. Namely, in section 1 wegive analytical preliminaries and recall the notions of strong, weak andmild solutions to the Cauchy problem for the Navier-Stokes system.More detail can be found the recent book by Lemarie-Rieusset [15].In section 2 we give a short review of probabilistic approaches to theinvestigation of the Navier-Stokes system [7], [16] -[18]. In section 3we study a probabilistic representation of the solution to the Poissonequation, while in section 4 we recall some principal fact of the Ku-nita theory of stochastic flows and apply the results from [19], [20] toconstruct a solution of the Cauchy problem for a nonlinear parabolicequation (see also [21]). Finally all these preliminary results are usedto construct the probabilistic counterpart of the Navier-Stokes system,prove that there exists a unique local solution to the correspondingstochastic system and apply the results to construct both the strongand weak (and simultaneously mild) solutions to the Cauchy problemfor the Navier-Stokes system. As it was mentioned in the introduction the main purpose of thisarticle is to construct both strong and weak solutions (in certain func-tional classes) of the Cauchy problem for the Navier-Stokes system viadiffusion processes.Consider the Cauchy problem for the Navier-Stokes system ∂u∂t + ( u, ∇ ) u = ν ∆ u − ∇ p, u (0 , x ) = u ( x ) , x ∈ R , (1.1) div u = 0 . (1.2)Here u ( t, x ) ∈ R , x ∈ R , t ∈ [0 , ∞ ) is the velocity of the fluid at theposition x at time t and ν > p ( t, x )is a scalar field called the pressure which appears in the equation toenforce the incompressibility condition (1.2). Later we set ν = σ forreasons to be explained below.By eliminating the pressure from (1.1),(1.2) one gets a nonlinearpseudo-differential equation which is to be solved. There exist differentways to do it and we consider now some of them.Given a vector field f let P f be given by P f = f − ∇ ∆ − ∇ · f. (1.3)Here and below we denote by u · v the inner product of vectors u and v valued in R .The map P called the Leray projection is a projection of the space L ( R ) ≡ L ( R ) of square integrable vector fields to the space ofdivergence free vector fields and we discuss its properties below. Aquite direct definition of P is connected with the Riesz transformation R j . Recall that R k = ∇ k √− ∆ which means that for f ∈ L we have ( R j f ) = iξ j | ξ | ˆ f ( ξ ) where F ( f ) = ˆ f is the Fourier transform of f .Then P is defined on L ( R ) as P = Id + R ⊗ R or( P f ) j = f j + X k =1 R j R k f k . Since R k R j is a Calderon-Zygmund operator, P f may be defined onmany Banach spaces.Set γ ( t, x ) = X k,j =1 ∇ k u j ∇ j u k = Tr[ ∇ u ] (1.4)and note that γ can be presented as well in the form γ = ∇ · ∇ · u ⊗ u = X j,k ∇ k ∇ j ( u k u j ) . . By computing the divergence of both parts of (1.1) and taking intoaccount (1.2) we derive the equation − ∆ p ( t, x ) = γ ( t, x ) (1.5)thus arriving at the Poisson equation. The formal solution of thePoisson equation is given by p = ∆ − γ = ∆ − ∇ · ∇ · u ⊗ u (1.6)since div u = 0 and finally we present ∇ p in the form ∇ p = ∇ ∆ − ∇ · ∇ · u ⊗ u. Substituting this expression for ∇ p into (1.1) we obtain the followingCauchy problem ∂u∂t = ν ∆ u − P ∇ · ( u ⊗ u ) , u (0) = u . (1.7)There are a number of ways to define a notion of a solution forthe Cauchy problem (1.7). We will appeal mainly to the Leray weaksolution [13] or to the Kato mild solution [14]. Let D = D ( R ) = C ∞ c be the space of all infinitely differentiablefunctions on R with compact support equipped with the Schwartztopology. Let D ′ be the topological dual of D and denote by h φ, ψ i = R R φ ( x ) ψ ( x ) dx the natural coupling between φ ∈ D and ψ ∈ D ′ . If itwill not lead to misunderstandings we will use the same notation forvector fields u and v as well, that is h h, u i = Z R X k =1 h k ( x ) u k ( x ) dx. e recall that a weak solution of the N-S system on [0 , T ] × R is a distribution vector field u ( t, x ) in ( D ′ ((0 , T ) × R )) where u islocally square integrable on (0 , T ) × R , div u = 0 and there exists p ∈ D ′ ((0 , T ) × R ) such that ∂u∂t = ν ∆ u − ∇ · ( u ⊗ u ) − ∇ p, lim t → u ( t ) = u (1.8)holds.The Leray solution to the N-S equations is constructed through alimiting procedure from the solutions to the mollified N-S equations ∂u∂t = ν ∆ u − ∇ · (( u ∗ q ε ) ⊗ u ) − ∇ p, ∇ · u = 0 , lim t → u ( t ) = u . (1.9)Namely it is proved that there exists a function u ε ∈ L ∞ ((0 , ∞ ) , L ) ∩ L ((0 , T ) , ( ˙ H ))such that (at least for a subsequence u ε k ) strongly converging in( L loc ((0 , T ) × R )) to u which satisfies (1.9).Here ˙ H is the homogenous Sobolev space ˙ H = { f ∈ S ′ : ∇ f ∈ L } with norm k f k H = k∇ f k L .On the other hand to construct the Kato solution means to con-struct a solution u to the following integral equation u ( t ) = e t ∆ u − Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( s ) ds. (1.10)Note that instead of looking for u ( t, x ) and p ( t, x ) one can preferto look for their Fourier images ˆ u ( t, λ ) = (2 π ) − R R e − iλ · x u ( t, x ) dx .The Leray and Kato approaches stated in terms of the Fouriertransformations of the Navier-Stokes system can be described as fol-lows.Applying the Fourier transformation to the relation (1.7) writtenin the form h h, u ( t ) i = h h, u (0) i + Z t h h, ∆ u ( s ) i − Z t h h, ∇ · ( u ⊗ u )( s ) i we derive the relation h ˆ h, ˆ u i = h ˆ h, ˆ u i − Z t h ˆ h, | λ | ˆ u ( s ) i ds − (1.11) i (2 π ) Z t Z R Z R X k,l =1 λ k ˆ h l ( λ ′ ) , ˆ u l ( s, λ ) u k ( s, λ − λ ′ ) dλdλ ′ ds. Here ˆ u corresponds to the Fourier transformation of u .On the other hand if we are interested in the Kato mild solutionof the N-S system then we may apply the Fourier transformation to(1.10) and derive the following equation χ ( t, λ ) = exp {− ν | λ | t } χ (0 , λ )+ (1.12) t ν | λ | e − ν | λ | ( t − s ) (cid:20)
12 ( χ ( s ) ◦ χ ( s )) (cid:21) ( λ ) ds for the function χ ( t, λ ) = 2 ν (cid:18) π (cid:19) | λ | ˆ u ( t, λ ) . Here χ ◦ χ ( λ ) = − iπ Z R ( χ ( λ ) · e λ )Π( λ ) χ ( λ − λ ′ ) | λ | dλ ′ | λ ′ | | λ − λ ′ | , (1.13) e λ = λ | λ | and Π( λ ) χ = χ − e λ ( χ · e λ ) , (1.14)Coming back to (1.7) we note that the Leray projection allowsto eliminate the pressure p ( t, x ) from the Navier-Stokes system, toconstruct u and finally to look for p defined by the solution of theauxiliary Poisson equation.Another way to eliminate p ( t, x ) from the system (1.1),(1.2) is toconsider the function v ( t, x ) = curl u ( t, x ) called the vorticity. Sincecurl ∇ p ( t, x ) = 0 one can derive a closed system for u and v . Namelyfor u and v we arrive at the system consisting of the equation ∂v∂t + ( u · ∇ ) v = ν ∆ v + ( v · ∇ ) u, (1.15)and the so called Biot-Savart law having the form u ( t, x ) = 14 π Z R ( x − y ) × v ( y ) | x − y | dy. (1.16)Here the cross-product u × v is given by u × v = det e e e u u u v v v =( u v − u v ) e + ( u v − u v ) e + ( u v − u v ) e , where ( e , e , e ) is the orthonormal basis in R .Note that the term ( v · ∇ ) u can be written as ( ∇ u ) v or even as D u v, where D u is the deformation tensor defined as the symmetricpart of ∇ u D u = 12 ( ∇ u + ∇ u T ) , since by direct computation we see that( ∇ u ) v − D u v = 12 ( ∇ u + ∇ u T ) v = 0 . To be able to present the precise statements concerning the ex-istence and uniqueness of solutions to the N-S equations we have tointroduce a number of functional spaces to be used in the sequel. .2 Functional spaces We describe here functional spaces which will be used in the sequel.Let D = D ( R ) be the space of all infinitely differentiable functionson R with compact supports equipped with the Schwartz topology.Let D ′ be the topological dual to D . The elements of D ′ are calledSchwartz distributions.The space of R -valued vector fields h with components h k ∈ D shall be denoted by D ( R ) and D ′ shall denote the space dual to D ( R ).Let L q ( R ) denote the Banach space of functions f which are ab-solutely integrable taken to the q -th power with the norm k f k q =( R R | f ( x ) | q dx ) q ;Let Z denote the set of all integers, and suppose that k ∈ Z ispositive and 1 < q < ∞ . Denote by W k,q = W k,q ( R ) the set of allreal functions h defined on R such that h and all its distributionalderivatives ∇ α of order | α | = P α j ≤ k belong to L q ( R ). It is aBanach space with norm k h k k,p = ( X | α |≤ k Z R | D α h ( x ) | q dx ) q . (1.17)We denote the dual space of W k,q by W − k,m where m + q = 1. El-ements of W − k,q can be identified with Schwartz distributions. Thespace W − k,q is also a Banach space with norm k φ k − k,q = sup k h k k,q ≤ |h φ, h i| , where h φ, h i = Z R φ ( x ) h ( x ) dx. The spaces W k,p for k ∈ Z and p > p = 2 we use the notation H k for the Hilbert spaces W k, . In anatural way one can define the spaces W k , q , H k of vector fields withcomponents in W k,p , and H k and so on.Set V = { v ∈ D : divv = 0 } and let H = { closure of V in L ( R ) } , V = { closure of V in H } . (1.18)Let C kb ( R , R ) denote the space of k-times differentiable fieldswith the norm k g k C kb = X | β |≤ k k D β g k ∞ and let C k,αb ( R , R ) be the space of vector fields whose k-th deriva-tives are H¨older continuous with exponent α, < α < k g k C k,αb = k g k C kb + [ g ] k + α here [ g ] k + α = X | β | = k sup x,y ∈ R | D β g ( x ) − D β g ( y ) || x − y | α . We denote by Lip( R ) the space of bounded Lipschitz continuousfunctions with the norm k g k Lip = sup x,y ∈ R | g ( x ) − g ( y ) || x − y | . Spaces of integrable functions on the whole R appear to be notsatisfactory to construct a solution to the N-S equations and one hasto consider spaces of locally integrable functions.Let f : R → R be a Lebesgue measurable function. A set offunctions { f : R K | f ( x ) | p dx < ∞} for all compact subsets K in R isdenoted by L ploc and called a space of locally integrable functions. Notethat L ( R ) ⊂ L loc ( R ) . Although L ploc ( R ) are not normed spacesthey are readily topologized. Namely a sequence { u n } converges to u in L ploc ( R ) if { u n } → u in L p ( K ) for each open K ⊂ G having compactclosure in R . Local spaces W k,ploc ( R ) can be defined to consist offunctions belonging to W k,p ( K ) for all compact K ⊂ R .A local space W k,ploc ( G ) is defined as a space of functions belongingto W k,p ( G ′ ) for all G ′ ⊂ G with compact closure in G . A function f ∈ W k,ploc ( G ) with compact support will in fact belong to W k,p ( G ) . Alsofunctions in W ,p ( G ) which vanish continuously on the boundary ∂G will belong to W ,p ( G ) since they can be approximated by functionswith compact support.In the whole space R and with p , q satisfying 1 ≤ q ≤ p < ∞ de-note by M pq a nonhomogenous Morrey space and by M pq a homogenousMorrey space with norms given respectively by M pq = ( f ∈ L qloc : k f k M pq = sup x ∈ R sup Definition 1.1.(Weak solutions) A weak solution of the Navier-Stokes system on (0 , T ) × R is a distribution vector field u ( t, x ) , u ∈ ( D ′ ((0 , T ) × R ) d such thata) u is locally square integrable on (0 , T ) × R ,b) ∇ · u = 0 , c) there exists p ∈ D ′ ((0 , T ) × R ) such that ∂ t u = ∆ u − ∇ · ( u ⊗ u ) − ∇ p. The classical results concerning the existence of square integrableweak solutions are due to Leray [13]. Theorem 1.1. (Leray’s theorem) Let u ∈ ( L ( R )) so that ∇ · u = 0 . Then there exists a weak solution u ∈ L ∞ ((0 , ∞ ) , ( L ) d ) ∩ L ((0 , ∞ ) , ( H ) ) for the Navier -Stokes equation on (0 , ∞ ) × R so hat lim t → k u ( t ) − u k = 0 . Moreover, the solution u satisfies theenergy inequality k u ( t ) k + 2 Z t Z R k∇ ⊗ u k dxds ≤ k u k , (1.22) where k u ( t ) k = d X k =1 Z R | u k ( t, x ) | dx, ∇ ⊗ u = d X k =1 d X j =1 | ∂ k u j | . Definition 1.2. (Mild solution) The Kato mild solution of(1.1), (1.2) is a solution of (1.7) constructed as a fixed point of thetransform v e t ∆ u ( x ) − Z t e ( t − s )∆ P ∇ · ( v ⊗ v )( θ, x ) dθ = e t ∆ u − B ( v, v ) . (1.23) . Note that the right hand side of (1.7) e t ∆ u ( x ) − Z t e ( t − s )∆ P ∇ · ( u ⊗ u )( θ, x ) dθ = Φ( t, x, u ) (1.24)is a nonlinear map in the corresponding space and the solution u isobtained by the iterative procedure u = e t ∆ u , u n +1 = e t ∆ u − B ( u n , u n ) . (1.25)Hence to construct a mild solution to (1.1), (1.2) means to finda suitable functional space for which Φ( t, x, u ) given by (1.20) is acontraction.To this end one has to find a subspace E T of L uloc,x L t ((0 , T ) × R ) so that the bilinear transformation B ( u, v ) of the form (1.11) isbounded as a map E T × E T → E T . Then one may consider the space E T ⊂ S ′ defined by f ∈ E iff f ∈ S ′ and ( e t ∆ f ) Let E T ⊂ L uloc,x L t ([0 , T ) × R ) be such that the bilinear map B isbounded on E T Then:(a) If u ∈ E T is a weak solution for the Navier-Stokes equation(1.1) (1.2) then the associated initial value belongs to E T .(b) There exists a positive constant C such that for all u ∈ E T satisfying ∇ · u = 0 and k e t ∆ u k E T < ∞ there exists a weak solution u ∈ E T of (1.1) (1.2) associated with the initial value u u = e t ∆ u − Z t e ( t − s )∆ P ∇ · ( u ⊗ u ) ds. (1.26)The classical results assert that for sufficiently smooth initial datafor example for u in the Sobolev space H k , k > d + 1 , there existsa short time strong unique solution to (1.1), (1.2). On the otherhand Leray proved the existence of a global weak solution of finiteenergy, i.e. u ∈ L called the Leray-Hopf solution. Although theuniqueness and full regularity of this solution are still an open problemnevertheless one knows that if a strong solution exists then a weaksolution coincides with it. Probabilistic approaches to the so-lution of the N-S equations Along with above functional analytical approaches recently a numberof probabilistic approaches to the problems of hydrodynamics wasdeveloped (see [18]-[17], [7]). In this section we give a short survey ofseveral different probabilistic approaches.Let (Ω , F , P ) be a complete probability space, w ( t ) , B ( t ) be a cou-ple of independent Wiener processes valued in R .Assume that ( u ( t, x ) , p ( t, x )) is a unique strong solution to (1.1),(1.2) or (to be more precise) to (1.1), (1.5) and set ν = σ to simplifynotations in stochastic equations. Since in this case u is C -smoothone can check that the stochastic equation dξ ( τ ) = − u ( t − τ, ξ ( τ )) dτ + σdw ( τ ) , ξ (0) = x, (2.1)has a unique solution and the function v ( t, x ) = E [ u ( ξ ( t )) − Z t ∇ p ( t − τ, ξ ( τ )) dτ ] (2.2)satisfies (1.1) and hence equals to u by the uniqueness of the strongsolution to (1.1). The relation − p ( t, x ) = Z ∞ Eγ ( t, x + B ( θ )) dθ (2.3)with γ given by (1.4) allows us to verify that div u = 0. Thus (2.2),(2.3)give the probabilistic representation of the solution to the N-S system.If u ( t, x ) is a C -smooth solution to (2.1)-(2.3), then Ito’s formulayields that for v ( θ, x ) = u ( t − θ, x ) v ( t, ξ ( t )) = v ( θ, x ) + Z tθ [ ∂v∂τ − ( v, ∇ ) v + σ v ]( τ, ξ ( τ )) dτ + Z tθ σ ∇ v ( τ, ξ ( τ )) dw ( τ ) . Then (2.2) and the relation u ′ τ ( t − τ, x ) = − v ′ τ ( τ, x ) yield Eu (0 , ξ ( t )) = v ( t, x ) − E [ Z t [ ∂u∂τ + ( u, ∇ ) u − σ u ]( t − τ, ξ x ( τ ))+ ∇ p ( t − τ, ξ x ( τ ))] dτ ] + E [ Z t ∇ p ( t − τ, ξ x ( τ ))] dτ ] . Finally it results from (2.2) that E [ Z t [ ∂u∂τ + ( u, ∇ ) u − σ u + ∇ p ]( t − τ, ξ x ( τ ))] dτ ] = 0 . Since the latter equality holds for all t and x we deduce that (1.1)alsoholds. The relation u (0 , x ) = u ( x ) immediately follows from (2.2).The system (2.1)-(2.3) is a closed system of equations and we cantry to look for its solution. Then at a second step we will look for theconnection between this solution and a solution of the N-S system. his approach was realized in paper [7]. It appears that to provethe existence of smooth solutions to (2.1)-(2.3) we have to considerthe stochastic representations for ∇ u and ∇ p along with this system.Using general results of diffusion process theory and in particularthe Bismut-Elworthy formula [8] we note that heuristic differentiationof (2.1)- (2.3) leads to the relations ∇ k u i ( t, x ) = E [ ∇ j u i ( ξ ( t )) η jk ( t ) − Z t στ ( ∇ i p ( t − τ, ξ ( τ )) Z τ η kl ( θ ) dw l ( θ )) dτ ] (2.4)and dη ik = −∇ j u i ( t − τ, ξ ( τ )) η jk ( τ ) dτ, η ik (0) = δ ik . (2.5)In addition by Bismut-Elworthy’s formula (integration by parts)we can derive from (1.17) the probabilistic representation for ∇ p ( t, x )2 ∇ p ( t, x ) = − Z ∞ s E [ γ ( t, x + B ( s )) B ( s )] ds. (2.6)The main results in [7] can be stated in the following way.Let V = ( u, ∇ u ), V = { V ( t, x ) : k V ( t ) k L rloc < ∞} , if < r < V = { V ( t, x ) : k V ( t ) k L rloc < ∞} , if r > V = V ∩ V ∩ C α and let M = C ([0 , T ] , V ) denote the Banach space with the norm k V k r,α = sup t ∈ [0 ,T ] [ k V ( t ) k V + [ ∇ u ( t )] α ] . Theorem 2.1. ([7]) Assume that V (0) = V ∈ V . Then there ex-ist a bounded interval [0 , T ] depending on V and a unique solution ( ξ ( t ) , u ( t, x ) , p ( t, x ) , η ( t ) , ∇ u ( t, x )) to the system (2.1)-(2.5) belongingto M for each τ ∈ [0 , T ] . Theorem 2.2. ([7]) Assume that the conditions of theorem 2.1 holdand u ∈ C α . Then there exists an interval [0 , T ] , T ≤ T , suchthat for all t ∈ [0 , T ] there exists a unique solution to (1.1), (1.4) in ˜ M ⊂ M where ˜ M = M ∩ C and this solution is given by (2.2),(2.3). A close approach based on a similar diffusion process was devel-oped by Busnello, Flandoli, Romito [18], though their starting pointwas the system that governs the vorticity v = curl u and velocity u .The corresponding probabilistic counterpart of (1.15),(1.16) can bepresented in the form of the following stochastic system dξ ( τ ) = − u ( t − τ, ξ ( τ )) dτ + σdw ( t ) , ξ ( s ) = x, (2.7)and the following two relations v ( t, x ) = E [ U ( t, s ) u ( ξ ( t )) , (2.8)2 u ( t, x ) = Z ∞ θ E [ v ( t, x + B ( θ )) × B ( θ )] dθ, (2.9)where U ( t, s ) = exp ( R ts ∇ u ( t − τ, ξ ( τ ) dτ ) is a solution to the linearequation dU t,xs = ∇ u ( t − s, ξ ( s )) U t,xs ds, U t,x = Id. (2.10) he main results in [18] are as follows. Denote by U α ( T ) = { u ∈ C ([0 , T ] , C b ( R , R )) ∩ L ∞ ([0 , T ] , C ,αb ( R , R )) | div u = 0 } the Banach space endowed with the norm k u k U α = sup ess ≤ t ≤ T k u ( t ) k C ,αb and by V α,q ( T ) = { v ∈ C ([0 , T ] , C b ( R , R )) ∩ L ∞ ([0 , T ] , C αb ( R , R )) } the Banach space endowed with the norm k v k V α,q = sup ess ≤ t ≤ T k v ( t ) k L q ∩ C αb . Theorem 2.3. ([18]) Given p ∈ [1 , ) , α ∈ [0 , and T > let ξ ∈ C αb ( R , R ) ∩ L p ( R , R ) and set ε = k v k C αb ∩ L p . Then there exists τ ∈ [0 , T ] depending only on ε , such that there is aunique solution of (2.7)-(2.10). The diffusion process ξ ( t ) plays a roleof a Lagrangian path, vector field u belongs to U α , and vector field v belongs to V α,p ( τ ) . In addition the deformation matrix U x,ts satisfies(2.7). After these developments P.Constantin kindly attracted our at-tention to his papers [11] [12] where the Lagrangian approach wassuccessfully applied to the investigation of the Navier-Stokes system.The presentation of this approach and the discussion of their similarityand difference will be given in the last section of the present article.A probabilistic representation of the solution to the Fourier trans-formed Navier-Stokes equation (1.11) was constructed by Le Jan andSznitman [16].To describe their approach recall the definition of the solution ofthe Fourier representation (FNS) of the Navier-Stokes system.First for a solution u ( t, x ) of (1.26) and its Fourier transform ˆ u ( t, λ )one can introduce a function χ ( t, λ ) defined on [0 , T ] × R such that χ t ( λ ) = 2 ν ( π | λ | ˆ u t ( λ ) , a.e. for t ∈ [0 , T ] , and χ t ( λ ) · λ = 0 , χ t ( − λ ) = ¯ χ t ( λ ) . In addition, for Lebesgue a.e. λ , χ t ( λ ) solves the equation χ t ( λ ) = exp ( − ν | λ | t ) χ ( λ ) + Z t ν | λ | e − ν | λ | ( t − s ) 12 [ χ s ◦ χ s ( λ )] ds, (2.11)where χ ◦ χ ( λ ) = − iπ Z ( χ ( λ ) · e λ )Π( λ ) χ ( λ − λ ) | λ | dλ | λ | | λ − λ | , f the initial function χ is a measurable function and for a.e. λ ∈ R \{ } χ : R \{ } → C , χ ( λ ) · λ = 0 , χ ( − λ ) = χ ( λ ) . Introducing the kernel K from R \ R \ Z h ( λ , λ ) K λ ( dλ , dλ ) = 1 π Z h ( λ , λ − λ ) | λ | dλ | λ | | λ − λ | for h ≥ R \ one gets χ ◦ χ ( λ ) = − i Z ( χ ( λ ) · e λ )Π( λ ) χ ( λ ) K λ ( dλ , dλ ) . It turns out that K is a Markovian kernel with some remarkablefeatures that allow to study existence and uniqueness problems for(2.11) with the help of a critical branching process on R \{ } calledthe stochastic cascade. Namely, LeJan and Sznitman have describeda particle located in λ such that after an exponentially holding timewith parameter ν | λ | with equal probability the particle either diesor gives birth to two descendants, distributed according to K λ . Arepresentation formula for the solution of (2.11) is constructed as theexpectation of the result of a certain operation performed along thebranching tree generated by the stochastic cascade.One more probabilistic model was recently developed by M. Os-siander [17]. A binary branching process with jumps that correspondsto the formulation of solutions to N-S in physical space was con-structed in [17].Once again the N-S system is reformulated incorporating incom-pressibility via the Leray projection P and then the Duhamel principleis applied to derive u = e νt ∆ u − Z t e − ν ( t − s )∆ P ∇ · ( u ⊗ u )( s ) ds + Z t e − ν ( t − s )∆ P f ( s ) ds, (2.12) ∇ · u = 0 . (2.13)Let K ( y, t ) = (2 πt ) − e − | y | t be the transition density of the Brow-nian motion w ( t ) ∈ R ( P y ) ij = δ ij − ( e y ) i ( e y ) j b ( y ; u, v ) = ( u · e y ) P y v + ( v · e y ) P y u,b ( y ; u, v ) = b ( y ; u, v ) + u · ( I − e y e Ty ) ve y . Then (2.12) can be rewritten in the form u ( x, t ) = Z R u ( x − y ) K ( y, νt ) dy + Z t Z R { | z | νs K ( z, νs ) b ( z ; u ( x − z, t − s ) , u ( x − z, t − s ))+ | z | K ( z, νs ) − π | z | Z { y : | y |≤ z } K ( z, νs ) dy ) b ( z ; u ( x − z, t − s ) , u ( x − z, t − s )) + ( K ( z, νs ) P z − π | z | ( I − e z e Tz ) Z { y : | y |≤ z } K ( y, νs ) dy ) g ( x − z, t − s ) } dzds. (2.14) Theorem 2.4. ([17]) Let h : R → [0 , ∞ ] and ˜ h : R → [0 , ∞ ] with h locally integrable and h, ˜ h jointly satisfying Z R h ( x − y ) | y | − dy ≤ h ( x ) and Z R ˜ h ( x − y ) | y | − dy ≤ h ( x ) for all x ∈ R . If for all x ∈ R and t > πνt ) − | Z R u ( x − y ) e − | y | νt dy | ≤ πν h ( x )11 and | g ( x, t ) | < ( πν ) ˜ h ( x )11 then there exists a collection of probabilistic measures { P x : x ∈ R } defined on a common measurable space (Ω , F ) and a measurable func-tion Σ : (0 , ∞ ) × Ω → R such that P x ( { ω : | Σ( t, ω ) | < πν for all t > 0) = 1 for which a weak solution u ( t, x ) to the N-S can be presented in theform u ( x, t ) = h ( x ) Z Ω Σ( t, ω ) dP x ( ω ) for all x ∈ R , t > . Furthermore the solution u is unique in the class { v ∈ ( S ′ ( R × (0 , ∞ ))) : | v ( t, x ) | < πνh ( x )11 } for all x ∈ R , t > . Our survey is still far from being exhaustive. As already men-tioned the discussion of the Euler-Lagrangian approach developed byConstantin and Iyer will be postponed to of the present paper the lastsection. Within the framework of the approach developed in this paper weintend to construct diffusion processes associated with the system (1.1)(1.5). First we will start with (1.5) and recall some results concerningthe solution of the Poisson equation in an open domain G ⊆ R . irst we recall that by the divergence theorem a C ( G ) solution of − ∆ p = γ satisfies the integral identity Z G ∇ p · ∇ φ dx = − Z G γφ dx for all φ ∈ C ( G ) . In the space W , ( G ) which is the completion of C ( G ) under the inner product h p, φ i = Z G ∇ p · ∇ φ dx the linear functional F ( φ ) = − Z G γφ dx may be extended to a bounded linear functional on the space W , ( G ).Hence by the Riesz theorem there exists an element p ∈ W , ( G )satisfying h p, φ i = F ( φ ) for all φ ∈ C ( G ) . Then the existence of ageneralized solution to the Dirichlet problem − ∆ p = γ and p = 0on ∂G is readily established. The question of classical existence isaccordingly transformed into the question of regularity of generalizedsolution under the appropriately smooth bounded conditions.We give in this section a brief summary of a probabilistic approachto the solution of the Poisson equation. We will try to give the prob-abilistic proofs of the necessary facts inasmuch as they are known.Proofs of similar statements can be found in [18]. The source foranalytical results is the book by Gilbarg and Trudinger [22].Consider the Poisson equation − ∆ p ( x ) = γ ( x ) (3.1)where p and γ are scalar integrable functions defined on G . A Newtonpotential with density γ is defined by N γ ( x ) = 14 π Z G k x − y k γ ( y ) dy. (3.2)If γ is regular and has a compact support then N γ is known to be asolution of the Poisson equation (2.1).To derive a probabilistic interpretation of the relation (3.2) we con-sider the generator A = ∆ of a Wiener process B ( t ) ∈ R defined ona given probability space (Ω , F , P ). It is well known that on the space C ( R ) of all continuous functions vanishing at infinity the Wienerprocess generates the strongly continuous semigroup T t γ ( x ) = Eγ ( x + B ( t )) , x ∈ R , t ≥ , γ ∈ C ( R ) . Given a function with a compact support in G we extend it to thewhole space R by zero.By a direct computation we can check that Z ∞ E [ γ ( x + B ( t ))] dt = Z R γ ( x + y ) Z ∞ πt ) e − t k y k dtdy = R π k y k γ ( x + y ) dy = 2 N γ. (3.3)To prove that p = 2 N γ solves the Poisson equation − ∆ p = γ we need some additional regularity properties of N γ . Lemma 3.1. Let γ ∈ L m ( R ) ∩ L q ( R ) with ≤ m < < q < ∞ .Then N γ ∈ C ( R ) and k N γ k ∞ ≤ C m,q ( k γ k m + k γ k q ) . Proof. First we note that for every l, m such that l + r = 1 byH¨older inequality we have E | γ ( x + B ( t )) | = 1(2 πt ) Z R | γ ( x + y ) | e − k y k t dy ≤ (3.4) C r t − + l k γ k r ≤ C r t − r k γ k m , since − + l = − r . Finally we rewrite the left hand side of (3.1) as Z ∞ E [ γ ( x + B ( t ))] dt = Z E [ γ ( x + B ( t ))] dt + Z ∞ E [ γ ( x + B ( t ))] dt and applying the estimate (3.4) for r = q and r = m we derive Z ∞ E [ γ ( x + B ( t ))] dt ≤ C ( k γ k m + k γ k q ) , with C = max ( C m , C q ).By Sobolev embeddings it is known [22] that if γ ∈ L ( R ) then N γ ∈ C ( R ). To check that N γ ∈ C ( R ) we note that for any R > Z ∞ E [ γ ( x + B ( t ))] dt = Z ∞ E [ γ ( x + B ( t )) I k B ( t ) k >R ] dt + (3.5) Z ∞ E [ γ ( x + B ( t )) I k B ( t ) k≤ R ] dt. Let us prove that the first term on the right hand side of (3.5)converges to 0 uniformly in x as R → ∞ and the second term convergesto 0 as k x k → ∞ for each R . For the first term we apply the estimate(3.4) to derive sup x ∈ R E [ | γ ( x + B ( t )) | I {k B ( t ) k >R } ] ≤ C ( k γ k p + k γ k q )( t − m I { [1 , ∞ ) } ( t ) + t − q I { [0 , } ( t ))andsup x ∈ R E [ | γ ( x + B ( t )) | I {k B ( t ) k >R } ] ≤ Ct − k γ k m ( Z {k y k >R } e − k y k t dy ) q → s R → ∞ . To obtain the estimate for the second term we apply (3.4)once again and obtain E [ | γ ( x + B ( t )) | I k B ( t ) k≤ R ] ≤ Ct − m ( Z R | γ ( y ) | p I {k y − x k≤ R } dy ) m I { [1 , ∞ ) } ( t )+ Ct − q ( Z R | γ ( y ) | q I {k y − x k≤ R } dy ) q I [0 , ( t )) , that yields after the integration in time that the second term on theright hand side of (3.5) converges to 0, since γ ∈ L m ( G ) ∩ L q ( G ) andis zero outside G .To study derivatives of N γ we apply the Bismut-Elworthy-Li for-mula ∇ x i E [ γ ( x + B ( t ))] = 1 t E [ γ ( x + B ( t )) B i ( t )]that holds for a regular γ . Lemma 3.2. Let γ ∈ L m ( R ) ∩ L q ( ) for some ≤ m < < N γ one has to apply the Schauder estimates and the Bismut-Elworthy-Liformula.Let us recall two more useful results (see [22] theorem 4.5) con-cerning the Newton potential. Lemma 3.3. Let γ ∈ L q ( R ) ∩ C αb ( R ) with ≤ q ≤ . Then N γ ∈ C ,αb ( R ) ∩ C ( R ) , k N γ k C ,αb ( R ) ≤ C ( k γ k L q ( R ) + k γ k C αb ( G ) ) and p = 2 N γ is the unique solution of the Poisson equation − ∆ p = γ in C ( R ) ∩ C ( R ) . Theorem 3.4. Let N γ ∈ C ( R ) , γ ∈ C ( R ) satisfy the Poissonequation ∆ N γ = γ in R . Then N γ ∈ R and if B = B R ( x ) is anyball containing the support of N γ then k∇ N γ k ,α ; B ≤ C α k γ k ,α ; B , k N γ k ′ ,B ≤ CR k γ k ,B . (3.9) n the sequel we will need as well L q type estimates for the Newtonianpotential. Lemma 3.5. The operator N maps L q ( R ) into L q ( R ) and thereexists a positive constant C such that k N γ k L q ( R ) ≤ C k γ k L q ( R ) (3.10)Proof. By H¨older inequality we have | N γ | ( x ) = | Z G γ ( y )(Γ( x − y )) q (Γ( x − y )) − q dy | ≤{ Z G | γ ( y ) | q Γ( x − y ) dy } q { Z G Γ( x − y ) dy } − q ≤ C { Z G | γ ( y ) | q Γ( x − y ) dy } q . Next we obtain by Fubini’s theorem Z R | N γ | p ( x ) dx ≤ Z R C p { Z R | γ ( y ) | p Γ( x − y ) dy } dx = C p Z G Z R | γ ( y ) | p Γ( x − y ) dydx = C p Z R | γ ( y ) | p ( Z R Γ( x − y ) dx ) dy ≤ C Z R | γ ( y ) | p dy. Note that all above results in this section are valid if we considera bounded domain G ⊂ R instead of R . To get further regularityproperties of the Newton potential we need more auxiliary results.Define the distribution ν γ ( λ ) of the function γ : G → R by ν γ ( λ ) = |{ x ∈ G : | γ ( x ) | > λ }| (3.11)where | G | denotes the Lebesgue volume of the domain G . Lemma 3.6. Assume that γ ∈ L q ( G ) for some q > . Then ν γ ( λ ) ≤ λ − q Z G | γ ( x ) | dx, Z G | γ ( x ) | q dx = p Z ∞ λ q − ν γ ( λ ) dλ. Proof. It is easy to check that Z G | γ ( x ) | p dx ≥ Z { γ>λ } | γ ( x ) | p dx ≥ λ p |{ x : γ ( x ) > λ }| = λ p ν γ ( λ ) . If p = 1 we can apply the Fubini theorem to change the order ofintegration Z G | γ ( x ) | dx = Z G Z | γ ( x ) | dtdx = Z ∞ Z G I { x ∈ G : γ ( x ) >λ } dxdλ = Z ∞ ν f ( λ ) dλ. or arbitrary q we have ν γ q ( λ ) = |{ x : γ ( x ) > λ q }| = ν γ ( λ q )and hence p Z ∞ λ q − ν γ ( λ ) dλ = Z ∞ ν γ q ( λ q ) d ( λ q ) = Z G | γ ( x ) | q dx. Lemma 3.7. Let γ ∈ L q ( G ) for some < q < ∞ . Then N γ ∈ W ,q ( G ) and k∇ N γ k L q ( G ) ≤ C ( q, G ) k γ k L q ( G ) (3.12) Moreover for q = 2 the equality Z R k∇ N γ k ( x ) dx = Z G γ ( x ) dx (3.13) holds. Proof. The proof of this fact is based on the Calderon-Zygmundtechnique of cube decomposition and estimates of the function ν γ ( λ )of the form (2.5).Let ˜ K be a cube in R , γ ≥ κ > | ˜ K | Z ˜ K γ ( x ) dx ≤ κ. Bisect ˜ K into 2 equal ( in volume) subcubes. Let Q be a set of thosesubcubes K for which | K | R K γ ( x ) dx > κ . For each of the remainingsubcubes ( which do not belong to Q ) we repeat the same procedure,that is bisect each one into 2 sub-cubes and add those smaller ones,where f is highly concentrated to Q . Now repeating the procedureagain and again we obtain a partition of ˜ K . For any K in Q denoteby ˆ K its immediate predecessor. Since K ∈ Q, while ˆ K / ∈ Q , we have λ < | K | Z K γ ( x ) dx < | K | Z ˆ K γ ( x ) dx = | ˆ K || K | | ˆ K | Z ˆ K γ ( x ) dx < λ. Set F = ∪ K ∈ Q K, J = ˜ K \ F = ∩ K ∈ Q K C . Note that each point in J belongs to infinitely many nested cubes with bounded concentration of γ with diameters converging to zero, that is | K i | R ˆ K i γ ( x ) dx ≤ κ , with | K i | → . By the Lebesgue theorem we deduce that | K i | R ˆ K i γ ( x ) dx → γ a.e. with respect to the Lebesgue measure, that is γ ≤ κ a.e. on J .Then we have an average estimate on F and a point-wise estimate on J . At the second step we need the Marcinkiewicz interpolation theo-rem. Marcinkiewicz interpolation theorem. Let ≤ q < r < ∞ and let T : L q ( G ) ∩ L r ( G ) → L q ( G ) ∩ L r ( G ) be a linear map. Supposethere exist constants C , C such that ∀ γ ∈ L q ( G ) ∩ L r ( G ) and for any λ > ν T γ ( λ ) ≤ C k γ k L q ( G ) λ ! q , ν Qγ ( λ ) ≤ C k γ k L r ( G ) λ ! r . hen for any exponent m such that q < m < r the map T can beextended to a map from L m ( G ) to L m ( G ) and kT γ k L m ( G ) ≤ KC α C − α k γ k L m ( G ) . all γ ∈ L q ( G ) ∩ L p ( G ) where m = αq + − αr and the constant K dependsonly on m, q and r . At the end we define an operator T : L ( G ) → L ( G ) by T γ = ∇ i ∇ j N γ to obtain the necessary result. Theorem 3.8.(Calderon-Zygmund inequality) Let γ ∈ L p ( G ) , < q < ∞ . Then the Newton potential N γ = p ∈ W ,q ( G ) , solvesthe Poisson equation ∆ p = γ a.e. and k∇ p k L q ( G ) ≤ C k γ k L q ( G ) , (3.14) where C depends only on d and q . Furthermore , when q = 2 we have Z R k∇ N γ ( x ) k dx = Z G γ ( x ) dx. For the proof of the above interpolation theorem and theorem 3.8see, e.g., [22]. In this section we adapt the results of the Kunita theory of stochas-tic flows acting on Schwartz distributions [19],[20] to the case underconsideration. The considerations in this section are similar to [21].Unlike the Kunita case we assume here that the coefficients ofSDEs under consideration are at most C α -smooth with 0 < α < H k for k = 1 , − ξ ( t )having the Ito differential of the form dξ ( t ) = [ a ( ξ ( t )) + 12 T r ∇ σ ( ξ ( t )) σ ( ξ ( t ))] dt + σ ( ξ ( t )) dw has the Stratonovich differential of the form d S ξ ( t ) = a ( ξ ( t )) dt + σ ( ξ ( t )) ◦ dw. We say that condition C 4.1 holds if for all t ∈ [0 , T ] the functions g ( t ) and σ belongs respectively to C αb and C αb .Throughout this section we assume that C4.1 holds. We shall firstgive a brief review of the results which will be needed in the sequel.Consider a stochastic differential equation in the Stratonovich form dξ ( τ ) = − g ( t − τ, ξ ( τ )) dτ − σ ( ξ ( τ )) ◦ dw ( τ ) , ξ ( s ) = x ∈ R , (4.1) ≤ s ≤ τ ≤ t. Here g ( t, x ) ∈ R , σ ( t, x ) ∈ R × R and w ( t ) ∈ R isa Wiener process.Assuming that g ( t ) ∈ C ( R ) and σ ( t ) is a C -smooth matrix weare in the framework of the Kunita theory [6] and know that there ex-ists a local C -diffeomorphism of R generated by the solution ξ s,x ( τ )of (4.1).Namely, by general results on the SDE theory the existence anduniqueness of the solution ξ s,x ( τ ) to (4.1) are granted for a C - smoothbounded function g . Moreover, in this case, one can prove that thesolution ξ gs,x ( τ ) of (4.1) has a modification φ gs,τ ( x, ω ) such that for all ω outside a null set N ⊂ Ω1) φ gs,τ ( x, ω ) is continuous in ( s, τ, x ), and differentiable in x ;2) φ gτ,τ ( φ gs,τ ( x, ω ) , ω ) = φ gs,τ ( x, ω ), if 0 < s < τ < t ;3) the mapping φ gs,τ ( ω ) : R → R is a C - diffeomorphism in R .The map φ gs,τ ( ω ) is called a stochastic flow of C - diffeomorphismsin R .We will denote by ( φ gs,τ ) − ( ω ) = ψ gτ,s ( ω ) the map inverse to thestochastic flow φ gs,τ ( ω ) and will write simply ψ gτ,s ( x ) for ψ gτ,s ( x, ω ). Wecheck a simple property of an inverse stochastic flow. Lemma 4.1. Consider the σ -algebras F ws = σ { w ( θ ) : θ ∈ [0 , s ] } F ˆ wt,s = σ { ˆ w ( τ ) − ˆ w ( τ ) : s ≤ τ ≤ τ ≤ t } and a continuous bounded process m ( s ) adapted to F ws . Then theprocess f ( s ) = g ( t − s ) for s ∈ [0 , t ] is F ˆ wt,s adapted and for all α, β such that ≤ α ≤ β ≤ t we have Z βα f ( τ ) dw ( τ ) = Z t − βt − α g ( s ) d ˆ w ( s ) . Proof. Note that since ˆ w ( s ) = w ( t − s ) − w ( t ) we have ˆ w ( β ) − ˆ w ( α ) = w ( t − β ) − w ( t − α ) , that yields F wt − s = F ˆ wt,s .Now we consider a partition of the interval [0 , t ] { t ≤ t ≤ . . . ≤ t k ≤ t k +1 ≤ . . . ≤ t N = t } such that | t k +1 − t k | → N → ∞ . Set θ k = t − t k for k = 1 , . . . , N, then Z βα f ( s ) dw ( s ) = lim n →∞ N X k =1 f ( t k )[ w ( t k +1 ) − w ( t k )] =lim n →∞ N X k =1 g ( t − θ k )[ w ( t − θ k +1 ) − w ( t − θ k )] = − lim n →∞ N X k =1 g ( θ k )[ ˆ w ( θ k +1 ) − ˆ w ( θ k )] = − Z t − αt − β g ( s ) d ˆ w ( s ) . The main point of Kunita’s theory is that the stochastic flow isa bijection and that the inverse stochastic flow satisfies a couple ofSDEs which will be used for different purposes. One of these SDEsis given by the following lemma due to Malliavin ( see ([23], lemma5.2.2) or [5]). emma 4.2. Let ξ g ( τ, x, w ) be a solution of the stochastic equation(4.1) with s = 0 or equivalently of the SDE dξ ( τ ) = − g ( t − τ, ξ ( τ )) dτ + m ( ξ ( τ )) dτ − σ ( ξ ( τ )) dw ( τ ) , ξ ( s ) = x (4.2) where m ( x ) = T r ∇ σ ( x ) σ ( x ) . Then, for every fixed T > we have ξ ( t − θ, x, w ) = ˆ ξ ( θ, ξ ( t, x, w ) , ˆ w ) for every ≤ θ ≤ t , and x , a.s. ( P w ) . In what follows we need as well some generalizations of the Itˆ o formula. The first one called the Itˆ o -Wentzel formula reads as follows. Lemma 4.3. (It ˆ o -Wentzel formula) Assume that the process ξ ( t ) ∈ R has a stochastic differential of the form dξ ( t ) = g ( t, ξ ( t )) dt + σ ( ξ ( t )) dw ( t ) and the process f ( t, x ) ∈ R has a stochastic differential df ( t, x ) = Ψ( t, x ) dt + Φ( t, x ) dw ( t ) with the same Wiener process w ( t ) . Let the vector field Ψ( t, x ) ∈ R and the operator field Φ( t, x ) ∈ R × R be C smooth in x andcontinuous in t . Then the process η ( t ) = f ( t, ξ ( t )) has a stochasticdifferential df m ( t, ξ ( t )) = Ψ m ( t, ξ ( t )) dt + Φ mk ( t, ξ ( t )) dw k + ∇ i f m ( t, ξ ( t )) dξ i ( t )+(4.3)12 ∇ i ∇ j f m ( t, ξ ( t )) σ ik ( ξ ( t )) σ jk ( ξ ( t )) dt + ∇ i Φ mk ( t, ξ ( t )) σ ik ( ξ ( t )) dt. Remark 4.4. Note that (4.3) can be rewritten in the Stratonovichform as follows df m ( t, ξ ( t )) = Ψ m ( t, ξ ( t )) dt + Φ mk ( t, ξ ( t )) dw k + ∇ i f m ( t, ξ ( t )) ◦ dξ i ( t )+(4.4)+ ∇ i Φ mk ( t, ξ ( t )) σ ik ( ξ ( t )) dt. We apply lemma 4.3 to check that the inverse flow ψ gt, to the flow φ g (0 , t ) (generated by the solution of the equation in (4.1)) can alsobe represented as a solution of the following stochastic equation dψ gt, ( x ) = ∇ φ g ,t ( ψ gt, ) − g ( t, x ) dt + ∇ φ g ,t ( ψ gt, ) − σ ( x ) ◦ dw, (4.5)where ( ∇ φ g ,t ) − is the inverse matrix of the Jacobian matrix ∇ φ g ,t ofthe map φ g ,t .Namely, we have the following statement proved by Kunita (see [6]Theorem 4.2.2) in a slightly different context. Theorem 4.5. Let C4.1 hold and φ g ,t be the solution of the equa-tion (4.1). Then the inverse flow [ φ g ,t ] − = ψ gt, satisfies (4.5). Proof. To verify the statement of the theorem note first that theJacobian matrix κ g ( t ) = ∇ φ g ,t solves the Cauchy problem for thestochastic equation dκ ( τ ) = −∇ g ( τ, φ g ,τ ( y )) κ ( τ ) dτ −∇ σ ( φ g ,τ ( y )) κ ( τ ) ◦ dw ( τ ) , κ (0) = I. (4.6) hen, consider the stochastic process G ( y, t ) = Z t ∇ φ g ,τ ( y ) − g ( τ, φ g ,τ ( y )) dτ + Z t ∇ φ g ,τ ( y ) − σ ( φ g ,τ ( y )) ◦ dw ( τ ) , and compute φ g ,t ( ψ gt, ( x )), where ψ gt, has the stochastic differential dψ gt, = dG ( ψ gt, ( x ) , t ) . Set φ g ,t ( y ) = φ g ( y, t ) . By the Itˆ o -Wentzell formula we have φ g ,t ( ψ gt, ( x )) = x + Z t d S φ g ( ψ gθ, ( x ) , θ )+ Z t ∇ φ g ( ψ gθ, ( x ) , θ ) ◦ dψ gθ, ( x ) = x − Z t g ( θ, φ g ,θ ( ψ gθ, ( x ) , θ )) dθ − Z t σ ( φ g ,θ ( ψ gθ, ( x ) , θ )) ◦ dw ( θ )+ Z t ∇ φ g ( ψ gθ, ( x ) , θ )[ ∇ φ g ( ψ gθ, ( x ) , θ )] − g ( θ, φ g ,θ ( ψ gθ, ( x ))) dθ + Z t ∇ φ g ( ψ gθ, ( x ) , θ )[ ∇ φ g ( ψ gθ, ( x ) , θ )] − σ ( φ g ,θ ( ψ gθ, ( x ) , θ )) ◦ dw ( θ ) = x. Hence, φ g ,t ( ψ gt, ( x )) = x and thus ψ gt, is the inverse to φ g ,t . Remark 4.6. Recall that by lemma 4.2 the process ψ t, ( x ) = ˆ ξ ( t )along with (4.5) satisfies the SDE d ˆ ξ ( θ ) = g ( θ, ˆ ξ ( θ ))) dθ + σ ( ˆ ξ ( θ )) ◦ d ˆ w ( θ ) . (4.7)Denote by J g ,t ( ω ) the Jacobian of the map φ g ,t ( ω ). Given h ∈ H and f ∈ H − one can define the composition of f with the stochasticflow ψ gt, ( x ) as a random variable with values in H − defined by therelation h S gt, ( ω ) , h i = h f, h ◦ φ g ,t ( ω ) J g ,t ( ω ) i , h ∈ H , (4.8)for any t and ω / ∈ N . Note that if ˜ f is a distribution of the form˜ f = f ( x ) dx where f is a continuous function then ˜ f ◦ ψ t, is just thecomposition of the function f with the map ψ t, ( ω ) and Z R f ( ψ t, ( y, ω )) h ( y ) dy = Z R f ( x ) h ( φ ,t ( x, ω )) J ,t ( x, ω ) dx by the formula of the change of variables. Remark 4.7. Consider the case of constant diffusion coefficient σ ( x ) ≡ σ and assume that the drift possessed the property divg = 0.Then (4.8) has the form h S gt, ( ω ) , h i = h f, h ◦ φ g ,t ( ω ) i , h ∈ H , (4.9)since in this case J g ,t ( ω ) = Id is the identity map.Consider a linear PDE dfdt = L g f − γ ( t ) , f (0) = f , (4.10) here L g = ( g, ∇ ) + L ,L f = 12 F ij ∇ i ∇ j f + m j ∇ j f, and F ij = σ ik σ jk , m j = ∇ j σ ik σ jk , and g ∈ C is a given bounded smooth function and f ∈ C (or moregenerally f ∈ D ′ ).To construct a probabilistic representation of a weak solution to(4.8) in the case when the initial data f is a C function (or even adistribution f ∈ D ′ ) we consider the stochastic process λ ( t ) = f − Z t γ ( τ ) ◦ φ g ,τ dτ (4.11)and define its composition with a stochastic flow ψ gt, ( x ) solving (4.7).Recall that ψ gt, ( x ) is inverse to the stochastic flow φ g ,t ( x ) generatedby the solution ξ ( t ) of (4.2).It is proved in [19] that the generalized solution of (4.8) is givenby the generalized expectation of λ ( t ) ◦ ψ gt, ( x ).To define the generalized expectation we consider the Sobolevspaces W k,q or the weighted Sobolev spaces S k,q defined in section1 and check that Eλ ( t ) ◦ ψ gt, ( x ) is well defined. Lemma 4.8. For each integer k and q > , T > there are existpositive constants c k,,q,T , c ′ k,,q,T depending only on the flow ψ t, suchthat for any t ∈ [0 , T ] E k λ ( t ) ◦ ψ t, k qk ≤ c k,,q,T k f k qk,q + c ′ k,,q,T Z t k γ ( τ ) k qk,q dτ, (4.12) for all f ∈ W k,q and γ ( t ) ∈ W k,q . If f ∈ W k,q then by this lemma for any h ∈ W − k,q there exists h S ,t , h i = E h λ ( t ) ◦ ψ t, , h i , and S ,t can be considered as an element from W k,q . This elementwill be called the generalized expectation of λ ( t ) ◦ ψ t, and denoted by E [ λ ( t ) ◦ ψ t, ].For k = 1 , q = 2 we consider h ¯ S gt, , h i = E h f ◦ ψ gt, , h i which is acontinuous linear functional on H and can be regarded as an elementof H − . Set U t, f = E [ f ◦ ψ t, ] (4.13)and call it the generalized expectation of f ◦ ψ t, . It is easy to seethat U t, is a linear map form H − into itself. Moreover the family U t,s f = E [ f ◦ ψ t,s ] possesses the evolution property U t,τ U τ,s = U t,s forany 0 ≤ s ≤ t ≤ T . It can be immediately deduced from the evolutionproperties of φ gs,t and J gs,t .Finally we compute the infinitesimal operator of the evolution fam-ily U t,s . To this end we need a version of the Itˆ o formula. heorem 4.9. (The generalized Itˆ o formula) Let f ( t ) ∈ H be acontinuous in t nonrandom function. Then, given stochastic flows φ ,t , ψ t, generated by (4.1), (4.6) the following relations hold f ( t ) ◦ φ g ,t = f (0) + Z t [ ∂f ( θ ) ∂θ ◦ φ g ,θ + L f ( θ ) ◦ φ g ,θ ] dθ + Z t ∇ i f ( θ ) ◦ φ g ,θ d [ φ g ] i ,θ and f ( t ) ◦ ψ gt, = f (0) + Z t [ ∂∂θ [ f ( θ )] ◦ ψ gθ, + L [ f ( θ ) ◦ ψ gθ, ]] dθ + (4.14) Z t ∇ i [ f ( θ ) ◦ ψ gθ, ] σdw + Z t ∇ i [ f ( θ ) ◦ ψ gθ, ] g ( θ ) dθ. Here we understand the action of the operator L in the sense of gen-eralized functions. The proof of theorem 4.9 employs the classical Itˆ o formula for C - smooth functions f ε that approximate the C function f , usesequations (4.2) and (4.6) for the flows φ ,t and ψ t, , respectively, andthen justifies the passage to the limit under the integral sign in theintegral identity. The details can be found in [19] for a much moregeneral case.Let us come back to the parabolic equation (4.10) and set γ = 0.We can show that the stochastic flow ψ t, gives rise to an evolutionfamily acting in spaces of of distributions and the function f ( t ) = E [ f ◦ ψ t, ] is a weak solution of (4.10) with γ = 0. Theorem 4.10. Assume that the coefficients of the stochasticequation (4.2) satisfy C 4.1 and ψ gt, is generated by the solution of(4.6). Then, for any functions f , g ( t ) ∈ H the relation f ( t ) = E [ f ◦ ψ gt, ] defines the unique generalized solution to the problem (4.10) with γ =0 . The restriction of U gt, to H defines a strongly continuous family ofevolution mappings acting on the space H . The domain of definitionof its infinitesimal operator A g (in a weak sense) contains the subspace H and A g f = L g f for any u ∈ H . Proof. From the relation (4.13) and the properties of stochasticflows we deduce that the relation h U g ( t ) f , h i = h E [ f ◦ ψ gt, ] , h i defines a continuous linear functional on H . Thus, we can treat U g ( t ) f as an element from H . It follows from the representation U gt, f ( x ) = E [ f ◦ ψ gt, ( x )] that U g ( t ) is a linear mapping from thespace H into itself. Note that the above definition of the family U gt,s through the integral identity allows to check that it possesses theevolution property h U gt,τ U gτ,s f , h i = h U gt,s f , h i . ndeed, by the Markov property of the process ψ gt, ( x ) we deducethat h U gt,s f , h i = h U gt,τ U gτ,s f , h i = E [ h U gτ,s f , h ◦ φ gτ,t J gτ,t i ] = E [ h f , [ v ◦ φ gs,τ ] J gs,τ i| v = h ◦ φ gτ,t J gτ,t ] = E h f , [ h ◦ φ gs,τ ◦ φ gτ,t ] J gs,τ ] ◦ φ gτ,s J gτ,t i = E [ h f , [ h ◦ φ gs,t ] J gs,t i ] = h U gs,t f, h i . Now we apply the generalized Itˆ o formula to obtain the relation E [ f ◦ ψ gt, ] = f + E [ Z t L g ( f ◦ ψ gθ, ) dθ ] . Note that in the latter expression each summand belongs to H . Inaddition, E [ Z t h L g ( f ◦ ψ gθ, ) , h i dθ ] = Z t h E [ f ◦ ψ gθ, ] , ( L g ) ∗ h i dθ = Z t h L g ( E [ f ◦ ψ gθ, ]) , h i dθ, that yields E [ f ◦ ψ gt, ] = f + Z t L g ( E [ f ◦ ψ gθ, ]) dθ. In other words U gt, f = f + Z t L g U gθ, f dθ. As the result we get that f ( t ) = E [ f ◦ ψ t, ] satisfies (4.8) and f (0) = f . One can prove the corresponding result in the case γ ( t ) = 0 ina similar way applying the above reasons to λ ( t ) of the form (4.11)instead of f . Theorem 4.11. Given tempered distributions f and γ ( t ) define λ ( t ) by (4.11). Then U ( t ) = E [ λ ( t ) ◦ ψ gt, ] defines the unique solutionof equation (4.10) if ψ gt, satisfies (4.6) and φ ,t is its inverse. Proof. By the generalized Ito formula we get λ ( t ) ◦ ψ gt, = f − Z t γ ( τ ) dτ + Z t ∇ i ( λ ( τ ) ◦ ψ gτ, ) g ( τ ) dτ + Z t ∇ i ( λ ( τ ) ◦ ψ gτ, ) σ ( τ ) dw ( τ ) + Z t L g ( λ ( τ ) ◦ ψ gτ, ) dτ. As a consequence we get λ ( t ) ◦ ψ gt, = f + Z t ∇ i ( λ ( τ ) ◦ ψ gτ, ) σdw ( τ )+ Z t L g ( λ ( τ ) ◦ ψ gτ, ) dτ − Z t γ ( τ ) dτ. (4.15) ach term in (4.15) has a generalized expectation as an element of S ′ .The generalized expectation of the second term in the right hand sideof (4.15) is equal to zero. For the third term we have h E (cid:20)Z t L g ( λ ( τ ) ◦ ψ gτ, ) dτ (cid:21) , h i = E (cid:20)Z t h λ ( τ ) ◦ ψ gτ, , [ L g ] ∗ h i dτ (cid:21) = Z t h E [ λ ( τ ) ◦ ψ gτ, ] , [ L g ] ∗ h i dτ = Z t h L g ( E [ λ ( τ ) ◦ ψ gτ, ]) , h i dτ. (4.16)Hence E [ λ ( t ) ◦ ψ gt, ] = f + Z t L g ( E [ λ ( τ ) ◦ ψ gτ, ]) dτ − Z t γ ( τ ) dτ. (4.17)Differentiating each term with respect to t we check that U ( t ) = E [ λ ( t ) ◦ ψ gt, ] satisfies (4.10). In addition lim t → h U g ( t ) , h i = h f , h i ,that is lim t → h U g ( t ) = f and we proved that U g ( t ) solves the Cauchyproblem (4.10).To prove the uniqueness of the solution to (4.8) suppose to the con-trary that there exist two solutions f ( t ) and ˜ f ( t ) to (4.10). Then thefunction u ( t ) = f ( t ) − ˜ f ( t ) satisfies du ( t ) dt = L g u ( t ) and lim t → u ( t ) = 0.Fix t and choose a function h ( t, · ) ∈ D . Then there exists a solution h ( τ, x ) , ≤ s ≤ τ ≤ t, x ∈ R , to the Cauchy problem ∂h ( τ, x ) ∂τ + [ L g ] ∗ h ( τ, x ) = 0 , lim τ → t h ( τ, x ) = h ( t, x ) . If the coefficients a g and σ g are C -smooth , then there exists a uniqueclassical solution to this Cauchy problem. As a result, h u ( t ) , h i = Z t h ddθ u ( θ ) , h ( θ ) i dθ + Z t h u ( θ ) , ddθ h ( θ i dθ = Z t h L g u ( θ ) , h ( θ ) i dθ − Z t h u ( θ ) , [ L g ] ∗ h ( θ ) i dθ = 0 . Let us come back to the Navier-Stokes system ∂u∂t + ( u, ∇ ) u = σ u − ∇ p, u (0 , x ) = u ( x ) , x ∈ R (5.1) − ∆ p = γ, (5.2)with γ defined by (1.4).Our main purpose in this section is to construct a diffusion pro-cess that allows us to obtain a probabilistic representation of a weaksolution to (5.1), (5.2). To be more precise we intend to reduce thesolution of this system to solution of a certain stochastic problem, to olve it and then to verify that in this way we have constructed a weaksolution of (5.1), (5.2).Let as above w ( t ) , B ( t ) be standard R -valued independent Wienerprocesses defined on a probability space (Ω , F , P ). Given a boundedmeasurable function f ( x ) and a stochastic process ξ ( t ) we denote E s,x f ( ξ ( t )) ≡ Ef ( ξ s,x ( t )) a conditional expectation under the con-dition ξ ( s ) = x .In section 2 we recalled the probabilistic approach developed in ourprevious paper [7] that allows to construct a probabilistic representa-tion of a C -smooth (classical) solution to (5.1)-(5.2) via the solutionof the stochastic problem dξ ( τ ) = − u ( t − τ, ξ ( τ )) dτ + σdw ( τ ) , (5.3) u ( t, x ) = E ,x [ u ( ξ ( t )) + Z t ∇ p ( t − τ, ξ ( τ )) dτ ] (5.4) − p ( t, x ) = E [ Z ∞ γ ( t, x + B ( t )) dt ] = E [ Z ∞ tr[ ∇ u ] ( t, x + B ( t )) dt ] . (5.5)In this section we consider a similar stochastic system but nowwe choose to invert the time direction of the stochastic process itselfrather then of the function u to obtain the possibility to reduce aconstruction of a generalized solution to the Navier-Stokes system tothe construction of a solution of a stochastic problem.Our considerations will be based on the result of sections 3 and4. Note that since we consider the case where the diffusion coefficient σ is constant the Ito form and the Stratonovich form of a stochasticequation coincide.Let as above w ( t ) , B ( t ) be standard R -valued independent Wienerprocesses defined on a probability space (Ω , F , P ).Let φ ,t ( y ) be a stochastic process satisfying the stochastic equa-tion dφ ,t ( y ) = u ( t, φ ,t ( y )) dt − σdw ( t ) , φ , = y and the stochastic process λ ( t ) be of the form λ ( t ) = u − Z t ∇ p ( τ, φ ,τ ) dτ. (5.6)Consider the system dψ t,θ, ( x ) = − u ( θ, ψ t,θ ( x )) dθ + σd ˆ w ( θ ) , ψ t,t ( x ) = x, (5.7) u ( t, x ) = E [ u ( ψ t, ( x )) − Z t ∇ p ( τ, ψ t,τ ( x )) dτ ] . (5.8) − ∇ p ( t, x ) = E [ Z ∞ τ γ ( t, x + B ( τ )) B ( τ ) dτ ] , (5.9)where γ is given by (1.4) and prove the existence and uniqueness ofits solution.To this end we apply the Picard principle to the solution of thestochastic system and construct a solution to (5.7)-(5.9) by the suc-cessive approximation technique. et u ( t, x ) = u ( x ) , ψ t, ( x ) = x, p ( t, x ) = 0 (5.10)and consider a family of stochastic processes ψ kt,θ ( x ) and families ofvector fields u k ( t, x ) and scalar functions p k ( t, x ) given by the followingrelations dψ kt,θ = − u k ( θ, ψ kt,θ ) dθ + σd ˆ w ( θ ) , ψ kt,t = x, (5.11) u k +1 ( t, x ) = E [ u ( ψ kt, ( x )) − Z t ∇ p k +1 ( τ, ψ kt,τ ( x )) dτ ] , (5.12) − p k +1 ( t, x ) = Z ∞ E [ γ k +1 ( t, x + B ( τ ))] dτ, (5.13)where γ k +1 ( t, x ) = T r [ ∇ u k ( t, x ) ∇ u k +1 ( t, x )] . (5.14)Note that for a fixed k the first stochastic equation (5.11) that de-termines the family of stochastic processes ψ kt, ( x ) may be solved inde-pendently on equations (5.12)-(5.14). Then given the process ψ kt, ( x )and keeping in mind the properties of the function p k that satisfiesthe Poisson equation − ∆ p k +1 ( t, x ) = γ k +1 ( t, x ) , (5.15)one has to compute ∇ p k ( t, x ), u k +1 ( t, x ) by (5.12), (5.13).To investigate the convergence of the stochastic processes ψ kt, ( x )and functions u k ( t, x ) , p k ( t, x ) defined above we need some auxiliaryresults concerning the behavior of solutions of stochastic equations.Let g ∈ V be a given function. Consider the stochastic equation dψ gt,θ = − g ( θ ) ◦ ψ gt,θ dθ + σd ˆ w ( θ ) , ψ gt,t ( x ) = x (5.16)and define vector fields u g ( t, x ) and ∇ p g ( t, x ) by u g ( t, x ) = E [ u ( ψ gt, ( x )) − Z t ∇ p g ( τ, ψ gt,τ ( x )) dτ ] , (5.17) − p g ( t, x ) = Z ∞ E [ γ g ( t, x + B ( τ ))] dτ, (5.18) γ g ( t, x ) = T r [ ∇ g ∇ u g ]( t, x ) . (5.19)Recall that p g solves the Poisson equation − ∆ p g = T r [ ∇ g ∇ u g ] . (5.20)To investigate the convergence of the stochastic processes ψ kt, ( x ) andfunctions u k ( t, x ) , p k ( t, x ) defined above we need some auxiliary resultsconcerning the behavior of solutions of stochastic equations. Moreover long with the system (5.16) – (5.18) we will need the system to de-scribe the process η k ( τ ) = ∇ ψ kt,τ ( x ) and the functions ∇ u k ( t, x ) and ∇ p k ( t, x ).To derive the necessary apriori estimates we start with the consid-eration of a linearized system.Let g ( t ) be a given vector field. Consider the stochastic equation dψ gt,θ = − g ( θ ) ◦ ψ gt,θ dθ + σd ˆ w ( θ ) , ψ gt,t ( x ) = x (5.21)and define the vector fields u g ( t, x ) and ∇ p g ( t, x ) by u g ( t, x ) = E [ u ( ψ gt, ( x )) − Z t ∇ p g ( τ, ψ gt,τ ( x )) dτ ] , (5.22) − p g ( t, x ) = Z ∞ E [ γ g ( t, x + B ( τ ))] dτ, (5.23) γ g ( t, x ) = T r [ ∇ g ∇ u g ]( t, x ) . (5.24)Finally we derive from (5.23) the relation − ∇ p g ( t, x ) = Z ∞ E [ 1 τ γ g ( t, x + B ( τ )) B ( τ )] dτ (5.25)by applying the Bismut – Elworthy – Li formula first checking theconditions that validate such an application are satisfied. Below wewill need some estimates of a solution of the Poisson equation fromsection 3. For convenience of references we formulate them in thefollowing statement. Lemma 5.1. 1. Let γ g ∈ L q ( R ) ∩ L m ( R ) for some ≤ q < < < m < ∞ .Then k∇ p g k ∞ ≤ C qm ( k γ g k q + k γ g k m ) k∇ i ∇ j p g k ∞ ≤ C ( k γ g k q + [ γ g ] α ) . 2. Let γ g ∈ L r ( R ) for < r < ∞ . Then p g ∈ W ,r ( R ) and theCalderon- Zygmund inequality k∇ i ∇ j p g k r,loc ≤ C k γ g k r,loc holds. Let Lip be the subspace of the space C ( R × R , R ) of continuous( in t ∈ [0 , T ] , x ∈ R ), bounded functions which consists of Lipschitz-continuous (in x ) functions g such that k g ( t, x ) − g ( t, y ) k ≤ L g ( t ) k x − y k , t ∈ [0 , T ] x, y ∈ R , where k · k is the norm in R .Condition C 5.1 Let g ( t, x ) ∈ R be a vector field defined on [0 , T ] × R that belongsto C ,α ( R , R ) , < α ≤ t ∈ [0 , T ] and satisfies thefollowing estimates: . k g ( t ) k L qloc ≤ N g ( t ) for some q to be specified below, k g ( t ) k ∞ ≤ K g ( t ) and k g ( t, x ) − g ( t, y ) k ≤ L g ( t ) k x − y k , k∇ g ( t, x ) −∇ g ( t, y ) k ≤ L g ( t ) k x − y k . . k∇ g ( t ) k ∞ ≤ K g ( t ) , k∇ g ( t ) k r,loc ≤ N g ( t ) , where K g ( t ) , L g ( t ), N g ( t ) and K g ( t ) , L g ( t ) , N g ( t ) are positive functions bounded on an in-terval [0 , T ], r = m and r = q for 1 < q < < < m < ∞ .Set ψ ( τ ) = ψ t,τ ( x ) and consider the stochastic equation ψ ( τ ) = x − Z tτ g ( τ , ψ ( τ )) dτ + Z tτ σd ˆ w ( τ ) , (5.26)with 0 ≤ τ ≤ t < T for a certain constant T . If we are interested inthe particular dependence of the process ψ ( τ ) on the parameters t, x and g , we write ψ ( τ ) = ψ gt,x ( τ ). Lemma 5.2. Assume that C 5.1 holds. Then there exists a uniquesolution ψ gx ( τ ) of (5.21) that satisfies the following estimates: E k ψ gx ( τ ) k ≤ k x k + σ ( t − τ ) + ( t − τ ) Z tτ [ K g ( τ )] dτ ] , (5.27) E k ψ gx ( τ ) − ψ gy ( τ ) k ≤ k x − y k e R tτ L g ( θ ) dθ , (5.28) E k ψ gx ( τ ) − ψ g x ( τ ) k ≤ Z tτ k g ( τ ) − g ( τ ) k ∞ dτ e R tτ L g ( θ ) dθ . (5.29)Proof. The proof of the estimates of this lemma is standard. Weonly show the proof of (5.28). In view of C 5.1 we have E k ψ gx ( τ ) − ψ gy ( τ ) k ≤ k x − y k + Z tτ L g ( τ ) k ψ gx ( τ ) − ψ gy ( τ ) k dτ , where 0 ≤ τ ≤ t ≤ T with some constant T to be chosen later. Finally,by Gronwall’s lemma, we get E k ψ gx ( τ ) − ψ gy ( τ ) k ≤ k x − y k e R tτ L g ( θ ) dθ . Along with the equations (5.21)-(5.23) we will need below the equa-tions for the mean square derivative η ( t ) = ∇ ψ t. ( x ) of the diffusionprocess ψ t, ( x ) that satisfies (5.21) and the gradient v ( t, x ) = ∇ u ( t, x )of the function u ( t, x ) of the form (5.22). Lemma 5.3 Assume that C 5.1 holds. Then the process η g ( τ ) = ∇ ψ gt,τ satisfies the stochastic equation dη g ( τ ) = −∇ g ( τ, ψ gt,τ ) η g ( τ ) dτ, η g ( t ) = I, (5.30) where I is the identity map. Furthermore the process η g ( τ ) possessesthe following properties.The determinant det η ( τ ) is equal to 1, i. e. det η g ( τ ) = J t,τ = 1 , and the estimate k η g ( τ ) k ≤ e R tτ K g ( θ ) dθ (5.31) olds.In addition the following integration by parts formula is valid Z R f ( ψ gx ( τ )) dx = Z R f ( x ) dx, f ∈ L ( R ) . (5.32)Proof. Under the above assumptions the first statement immedi-ately follows from the results of the stochastic differential equationtheory. By a direct computation one can check that J t,τ satisfies thelinear equation dJ t,τ = − div g ( ψ gt,τ ) J t,τ dτ, J t,t = I and since div g = 0 we get the second statement that yields the in-tegration by parts formula (5.32). Finally (5.31) is deduced from theinequality E k η ( τ ) k ≤ Z tτ K g ( θ ) E k η ( θ ) k dθ by the Gronwall lemma.In the sequel we denote by η x,g ( t ) the solution of the equation dη x,g ( τ ) = ∇ g ( t, ψ t,τ ( x )) η x,g ( τ ) dτ, η x,g (0) = I if we will be interested in the properties of the process η x,g ( t ). Onecan easily check that k η x,g ( τ ) − η y,g ( τ ) k ≤ Z tτ k∇ g ( θ, ψ t,θ ( x )) − ∇ g ( θ, ψ t,θ ( y )) k dθe R tτ K g ( θ ) dθ ≤ Z tτ L g ( θ ) k ψ t,θ ( x )) − ψ t,θ ( y )) k dθ and by (5.28) we have k η x,g ( τ ) − η y,g ( τ ) k ≤ C ( τ ) k x − y k where C ( τ ) is a bounded function over a certain interval [0 , T ] de-pending on g .Let us state conditions on initial data u of the N-S system.We say that C 5.2 holds if for 0 < α ≤ u ∈ C ,α satisfies the following estimates k u k ∞ ≤ K , k∇ u k ∞ ≤ K , k u k r,loc ≤ M , k∇ u k r,loc ≤ M with r to be specified below and let L , L be Lipschitz constants forthe functions u and ∇ u respectively. Lemma 5.4. Assume that g ( t, x ) satisfies C 5.1 and u satisfies C 5.2 with r = q and r = m for 1 < q < < < m < ∞ . Then thevector field u g ( t, x ) given by u g ( t, x ) = E [ u ( ψ gt,x (0)) − Z t ∇ p g ( τ, ψ gt,τ ( x )) dτ ] (5.33) atisfies the following estimate k u g ( t ) k ∞ ≤ K + Z t C qm [ k∇ g ( τ ) ∇ u g ( τ ) k q,loc + k∇ g ( τ ) ∇ u g ( τ ) k m,loc ] dτ. (5.34)The proof of the estimate can easily be obtained by direct compu-tation from (5.33) using the estimates of the Newton potential givenin lemma 5.1. Lemma 5.5. Assume that conditions of lemma 5.4 hold. Thengiven the function u g ( t, x ) of the form (5.33) the function ∇ u g ( t, x ) admits a representation of the form ∇ u g ( t, x ) = E [ ∇ u ( ψ gt, ( x )) η x,g ( t ) − Z t σ ( t − τ ) ∇ p g ( τ, ψ gt,τ ( x )) Z tτ η x,g ( θ ) d ˆ w ( θ ) dτ ] (5.35) and the estimate k∇ u g ( t ) k ∞ ≤ e R t K g ( θ ) dθ K + Z t C qm σ √ t − τ e R tτ K g ( θ ) dθ K g ( τ )[ k∇ u g ( τ ) k q,loc + k∇ u g ( τ ) k m,loc ] dτ (5.36) holds for < q < < < m < ∞ . Proof. To derive (5.35) we compute directly the gradient of thefirst term in (5.34) and apply the Bismut-Elworthy -Li formula [8] tocompute the gradient of the second term in this relation. To verifythe estimate (5.36) we use the above estimates for the process η ( t )and the estimates of the Newton potential derivative from lemma 5.1.Then we obtain k∇ u g ( t ) k ∞ ≤ e R t K g ( θ ) dθ K + Z t C m,q σ √ t − τ e R tτ K g ( θ ) dθ [ k∇ g ( τ ) ∇ u g ( τ ) k m,loc + (5.37) k∇ g ( τ ) ∇ u g ( τ ) k q,loc ] dτ ] ≤ e R t K g ( θ ) dθ K + Z t C qm σ √ t − τ e R tτ K g ( θ ) dθ K g ( τ )[ k∇ u g ( τ ) k q,loc + k∇ u g ( τ ) k m,loc ] dτ ] . Now we have to derive the estimate for the function k∇ u ( t ) k r,loc . Lemma 5.6. Assume that the conditions of lemma 2.4 hold.Then for < r < ∞ the function u g ( t, x ) given by (5.33) satisfies theestimate k∇ u g ( t ) k r,loc ≤ e R t K g ( θ ) dθ k∇ u k r,loc + (5.38) C qm Z t e R τ K g ( θ ) dθ K g ( τ ) k∇ u g ( τ ) k r,loc , dτ with a constant C depending on r and a certain constant T to bespecified later. roof. Recall that along with (5.35) ∇ u g ( t, x ) admits the repre-sentation ∇ u g ( t, x ) = E [ ∇ u ( ψ gt, ( x )) η x,g ( t ) − Z t ∇ p g ( τ, ψ gt,τ ( x )) η x,g ( τ ) dτ ] . To derive the estimate for k∇ u ( t ) k rr,loc = R G k∇ u ( t, x ) k r dx (where G is an arbitrary compact in R ) by the triangle inequality we get k∇ u g ( t ) k r,loc ≤ α + α , where α = (cid:18)Z G E [ k∇ u ( ψ gt, ( x )) η x,g ( t ) k r ] dx (cid:19) r ,α = (cid:18)Z G Z t k∇ p g ( τ, ψ gt,τ ( x ))) η x,g ( τ ) k r dτ dx (cid:19) r . To estimate α we apply the H¨older inequality and recall that ψ t,τ ( x ) preserves the volume. As a result we have α ≤ (cid:18)Z G ( E [ k∇ u ( ψ gt, ( x )) k ] E [ k η g ( t ) k ]) r dx (cid:19) ) r ≤k∇ u k r,loc e R t K g ( θ ) dθ . To estimate α we apply the Calderon-Zygmund inequality andthe above property of ψ t,τ ( x ) to obtain α r ≤ C r Z t e R τ K g ( θ ) dθ K g ( τ ) Z G k∇ u ( τ, x ) k r dxdτ. Combining the above estimates for α and α we obtain the requiredestimate k∇ u g ( t ) k r,loc ≤ e R t K g ( θ ) dθ [ k∇ u k r,loc + C r Z t e R τ K g ( θ ) dθ K g ( τ ) k∇ u g ( τ ) k r,loc dτ ] . Theorem 5.7. Assume that conditions C 5.1 and C 5.2 hold.Then there exists an interval ∆ = [0 , T ] and functions α ( t ) , β ( t ) , κ bounded for t ∈ ∆ , such that, if for all t ∈ ∆ , k g ( t ) k ∞ ≤ κ ( t ) and k∇ g ( t ) k ∞ ≤ α ( t ) , k∇ g ( t ) k r ≤ β r ( t ) then the function k∇ u g ( t, x ) k (where u g ( t, x ) is given by (5.21)) satisfies the estimates k u g ( t ) k ∞ ≤ κ ( t ) , k∇ u g ( t ) k ∞ ≤ α ( t ) , k∇ u g ( t ) k r,loc ≤ β r ( t )(5.39) for r = q and r = m and < m < < < q < ∞ . Proof. Analyzing the above estimates for the functions u g ( t, x )and ∇ u g ( t, x ) we get the following estimates k∇ u g ( t ) k ∞ ≤ e R t K g ( θ ) dθ K + (5.40) Z t C qm e R τ K g ( θ ) dθ K g ( τ )[ k∇ u g ( τ ) k q,loc + k∇ u g ( τ ) k m,loc ] dτ ] , ∇ u ( t ) k r,loc ≤ e R t K g ( θ ) dθ [ k∇ u k r,loc + (5.41) C r Z t e R τ K g ( θ ) dθ K g ( τ ) k∇ u g ( τ ) k r,loc dτ ] . To derive the required estimates consider the integral equations α ( s ) = e R ts α ( θ ) dθ K + C qm Z ts e R τs α ( θ ) dθ α ( τ )[ n q ( τ ) + n m ( τ )] dτ, (5.42) n r ( s ) = e R ts α ( θ ) dθ k∇ u k r + C r Z ts e R τs α ( θ ) dθ n r ( τ ) α ( τ ) dτ for r = q and r = m and C qm = max ( C q , C m ). Finally we considerthe equation β ( s ) = e R ts α ( θ ) dθ β + C qm Z ts e R τs α ( θ ) dθ α ( τ ) β ( τ ) dτ, (5.43)where β ( τ ) = n q ( τ ) + n m ( τ ), and k∇ u k q,loc + k∇ u k m,loc = n q (0) + n m (0) = β . Next instead of the above system of integral equations we consider thesystem of ODEs dαds = − α ( s ) − C qm α ( s ) β ( s ) , α ( t ) = K , (5.44) dβds = − α ( s ) β ( s ) − C qm α ( s ) β ( s ) , β ( t ) = β . (5.45)By classical results of the ODE theory we know that there exists aninterval [0 , T ] depending on K , N and C, C qm such that the system(5.44), (5.45) has a bounded solution defined on this interval.To prove the convergence of functions u k ( t, x ) , ∇ u k ( t, x ) we needone more auxiliary estimate. Actually, we have proved that u k ( t ) ∈ Lip with the Lipschitz constant independent of k . It remains to provethat ∇ u k ( t ) have the same property. Lemma 5.8. Assume that C 5.1 and C 5.2 hold. Then thefunction ∇ u g ( t ) satisfies the estimate k∇ u g ( t, x ) − ∇ u g ( t, y ) k ≤ N g ( t ) k x − y k α if t ∈ [0 , T ] for any x, y ∈ G where G is a compact in R and the positive function N g ( t ) depending on parameters in conditions C 5.1 and C 5.2 isbounded over the interval [0 , T ] defined in theorem 5.7. Proof. Applying the integration by parts Bismut – Elworthy – Liformula to (5.33) we deduce the following expression for the gradientof the function u ( t, x ) ∇ u g ( t, x ) = E [ ∇ u ( ψ gt, ( x )) η x,g ( t ) − (5.46) t σ ( t − τ ) ∇ p g ( τ, ψ gt,τ ( x )) Z tτ η x,g ( θ ) d ˆ w ( θ ) dτ ] . It results from (5.46) that k∇ u g ( t, x ) − ∇ u g ( t, y ) k ≤ κ + κ + κ + κ , where κ = E [ k∇ u ( ψ gt, ( x )) − ∇ u ( ψ gt, ( y )) kk η x,g ( t ) k ] ,κ = E [ k∇ u ( ψ gt, ( y )) kk η x,g (0) − η y,g (0) k ] ,κ = Z t σ ( t − τ ) E [ k∇ p g ( τ, ψ gt,τ ( x )) −∇ p g ( τ, ψ gt,τ ( y )) kk Z tτ η x,g ( θ ) d ˆ w ( θ ) k ] dτ,κ = Z t σ ( t − τ ) E (cid:20) k∇ p g ( τ, ψ gt,τ ( y )) k Z tτ [ η x,g ( θ ) − η y,g ( θ )] d ˆ w ( θ ) k (cid:21) dτ. One can easily check using the estimates stated in lemmas 5.3 –5.5 that under conditions C 5.1 , C 5.2 κ ≤ L E k ψ gt, ( x ) − ψ t, ( y ) k e R t K g ( θ ) dθ ≤ k x − y k L e R tτ L g ( θ ) dθ and κ ≤ K E k η x,g ( t ) − η y,g ( t ) k ≤ k x − y k Z tτ K g ( θ ) e R tθ K g ( θ ) dθ dθ. To derive the estimates for κ and κ we recall (see lemma 5.1)that the solution of the Poisson equation − ∆ p g = γ g satisfies theestimates k∇ i ∇ j p g k ∞ ≤ C ( k γ g k q + [ γ g ] α,G ), k∇ i ∇ j p g k r ≤ k γ g k r and k∇ p g k ∞ ≤ C qm ( k γ g k q + k γ g k m ). Hence we obtain the inequalities κ ≤ Z t σ √ t − τ ( E k ψ gt,τ ( x ) − ψ t,τ ( y ) k ) ( k γ g ( τ ) k q + [ γ g ( τ )] α,G ) e R tτ K g ( θ ) dθ dτ ≤ Z t σ √ t − τ ( k x − y k L e R tτ L g ( θ ) dθ ( k γ g ( τ ) k q +[ γ g ( τ )] α,G ) e R tτ K g ( θ ) dθ dτ and κ ≤ Z t σ √ t − τ C qm ( k γ g ( τ ) k q + k γ g ( τ ) k m )( E k η x,g ( τ ) − η y,g ( τ ) k ) dτ ≤k x − y k Z t C qm ( k γ g ( τ ) k q + k γ g ( τ ) k m ) σ √ t − τ Z tτ K g ( θ ) e R tθ K g ( θ ) dθ dθdτ. Denote by Θ( t ) = sup x,y ∈ G k∇ u g ( t,x ) −∇ u g ( t,y ) kk x − y k α and note that[ γ g ( τ )] α,G = sup x,y ∈ G [ K g ( τ ) k∇ u g ( τ, x ) − ∇ u g ( τ, y ) kk x − y k α + u ( τ ) k∇ g ( τ, x ) − ∇ g ( τ, y ) kk x − y k α ] = K g ( τ )Θ( τ ) + K u ( τ )[ g ( τ )] α,G . Then combining the above estimates for κ i , i = 1 , , , , and applyingthe Gronwall lemma we derive the estimateΘ( t ) ≤ N g ( t ) e R t K g ( τ ) dτ = N g ( t ) , where N g ( t ) is a positive bounded function defined on the interval[0 , T ] and depending on parameters in conditions C 5.1 and C 5.2 .The estimates of theorem 5.7 and lemma 5.8 allow to prove theuniform convergence on compacts of the successive approximations(5.10)-(5.14) for the solutions of the system (5.7) – (5.9) in C ([0 , T ] ,C ,α ( K )) ∩ C ([0 , T ] , L m ( G ) ∩ L q ( G )) for 1 < q < < < m < ∞ and arbitrary compact G in R .To this end we differentiate the system (5.10)-(5.14) and add tothis system the following relations dη k,xt,θ = −∇ u k ( θ, ψ kt,θ ) η x,kt,θ dθ, η x,kt,t = I, (5.47)where I is the identity matrix acting in R and ∇ u k +1 ( t, x ) = E [ ∇ u ( ψ k +1 t, ( x )) η x,kt, − Z t σ ( t − τ ) ∇ p k +1 ( τ, ψ kt,τ ( x )) Z tτ η x,kt,θ d ˆ w ( θ ) dτ ] , (5.48) − ∇ p k +1 ( t, x ) = Z ∞ τ E [ γ k +1 ( t, x + B ( τ )) B ( τ )] dτ, (5.49)where γ k +1 = ∇ u k +1 ∇ u k . Now we can prove the following assertion. Theorem 5.9. Assume that C 5.2 holds. Then if k → ∞ thefunctions u k ( t ) , ∇ u k ( t, x ) determined by (5.8) and (5.48) uniformlyconverge on compacts to a limiting function u ( t ) ∈ C ([0 , T ] , C ,α ) , <α ≤ for all t ∈ [0 , T ] , where [0 , T ] is the interval such that thesolution of (5.45), (5.46) is bounded on [0 , T ] . In addition on thisinterval the limiting function satisfies the estimates sup x k∇ u ( t, x ) k ≤ α ( t ) , k∇ u ( t ) k q,loc ≤ β ( t ) for < q < where α ( t ) and β ( t ) solve(5.45), (5.46). Proof. By theorem 5.7 we know that the mappingΦ( t, x, g ) = E (cid:20) u ( ψ gt, ( x )) − Z t ∇ p g ( τ, ψ gt,τ ( x )) dτ (cid:21) acts in the space C ,α ∩ L q,loc ∩ L m,loc (for a fixed t ∈ [0 .T ]) with1 < q < < < m < ∞ . Consider the successive approximations (5.10) –(5.14) and (5.47)– (5.49), denote by S k +1 ( t, x ) = k u k +1 ( t, x ) − u k ( t, x ) k ,n k +1 ( t, x ) = k∇ u k +1 ( t, x ) − ∇ u k ( t, x ) k and let l k ( t ) = k S k ( t ) k ∞ , m kr ( t ) = k S k ( t ) k r , k ( t ) = k n k ( t ) k ∞ , ζ kr ( t ) = k n k ( t ) k r . Then we obtain n k +1 ( t, x ) ≤ L ( E [ k ψ kt, ( x ) − ψ k − t, ( x ) kk η x,kt, k ]+ E [ k ψ kt, ( x ) kk η x,kt, − η x,k − t, k ]) + Z t σ ( t − τ ) E [ k∇ p k +1 ( τ, ψ kt,τ ( x )) −∇ p k ( τ, ψ k − t,τ ( x )) kk Z tτ η x,kt,θ d ˆ w ( θ ) k ] dτ + Z t σ ( t − τ ) E (cid:20) k∇ p k ( τ, ψ kt,τ ( x )) k Z tτ [ η x,kt,θ − η x,k − t,θ ] d ˆ w ( θ ) k (cid:21) dτ. (5.50)Recall that by lemmas 5.2, 5.3 we know thatsup x E k ψ kt, ( x ) − ψ k − t, ( x ) k ≤ Z t [ k u k ( τ ) − u k − ( τ ) k ∞ ] dτ e R t α ( τ ) dτ , sup x E k η x,kt, − η x,k − t, k ≤ Z t k∇ u k ( τ ) − ∇ u k − ( τ ) k ∞ dτ e R t α ( τ ) dτ + sup x Z t E k∇ u k − ( τ, ψ kt,τ ( x )) − ∇ u k − ( τ, ψ k − t,τ ( x )) k dτ e R t α ( τ ) dτ and applying the estimates from theorem 5.7 we get ρ k +1 ( t ) ≤ e R t α ( τ ) dτ [ L Z t sup x E k u k ( τ, ψ kt,τ ( x )) − u k − ( τ, ψ k − t,τ ( x )) k dτ + Z t ρ k ( τ ) dτ + sup x Z t E k∇ u k − ( τ, ψ kt,τ ( x )) − ∇ u k − ( τ, ψ k − t,τ ( x )) k dτ ]+ Z t σ √ t − τ C [ k∇ u k ( τ ) ∇ u k − ( τ ) k q + k∇ u k ( τ ) ∇ u k − ( τ ) k m ]( E k η k ( τ ) − η k − ( τ ) k ∞ ) dτ + Z t e R tτ α ( θ ) dθ σ √ t − τ sup x E k∇ p k +1 ( τ, ψ kt,τ ( x )) − ∇ p k ( τ, ψ k − t,τ ( x )) k ) dτ. To derive the estimate for the last term we recall ( see lemma 5.1)that for 1 < q < the inequality k∇ p k ( t, x ) − ∇ p k ( t, y ) k ≤ k∇ p k ( t ) k ∞ k x − y k ≤ C [ k γ k ( t ) k q,loc + [ γ k ( t )] α,G ] k x − y k holds and as a result we obtain E k∇ p k ( τ, ψ kt,τ ( x )) − ∇ p k ( τ, ψ k − t,τ ( x )) k ≤ C [ β ( τ ) + Θ( τ )] E k ψ kt,τ ( x ) − ψ k − t,τ ( x ) k . In addition k∇ p k +1 ( t ) − ∇ p k ( t ) k ∞ ≤ C qm [ k γ k +1 ( t ) − γ k ( t ) k q,loc + k γ k +1 ( t ) − γ k ( t ) k m,loc ] ≤ C qm α ( t )[ k∇ u k +1 ( t ) − ∇ u k ( t ) k q,loc + ∇ u k ( t ) − ∇ u k − ( t ) k q,loc + k∇ u k +1 ( t ) − ∇ u k ( t ) k m,loc + k∇ u k ( t ) − ∇ u k − ( t ) k m,loc ] . It results from (5.50) that n k +1 ( t, x ) ≤ C ( t )[ Z t E k∇ u k ( τ, ψ kt,τ ( x )) − ∇ u k − ( τ, ψ k − t,τ ( x )) k dτ + Z t n k ( τ, x ) dτ ] + Z t σ √ t − τ C [ k∇ u k ( τ ) ∇ u k − ( τ ) k q + k∇ u k ( τ ) ∇ u k − ( τ ) k m ] r ( E k η x,k ( τ ) − η x,k − ( τ ) k ) dτ + Z t σ √ t − τ e R tτ α ( θ ) dθ ( E k∇ p k +1 ( τ, ψ kt,τ ( x )) −∇ p k ( τ, ψ k − t,τ ( x )) k ) dτ. Note that by the H¨older inequality we can prove that for any pos-itive f ( τ ) ∈ L r and m + r = 1 Z G [ Z t f ( τ, x ) dτ ] r dx ≤ Z G t rm Z t f r ( τ, x ) dτ dx = t rm Z t Z G f r ( τ, x ) dxdτ and for m + r = 1 and m < Z G [ Z t σ √ t − τ f ( τ, x ) dτ ] r dx ≤ t r (2 − m m Z t Z G f r ( τ, x ) dxdτ. (5.51)Then from (5.50) and (5.51) we have for r > ζ k +1 r ( t ) ≤ C ( t )[ Z t Z G [ E k u k ( τ, ψ kt,τ ( x )) − u k − ( τ, ψ k − t,τ ( x )) k r dxdτ ]+ Z t ζ kr ( τ ) dτ + Z t Z G k∇ u k − ( τ, ψ kt,τ ( x )) − ∇ u k − ( τ, ψ k − t,τ ( x )) k r dxdτ ]+ Z t σ √ t − τ C [[ k∇ u k ( τ ) ∇ u k − ( τ ) k q + k∇ u k ( τ ) ∇ u k − ( τ ) k m ] r Z G ( E k η x,k ( τ ) − η x,k − ( τ ) k ) r dx ] dτ + Z t σ √ t − τ e R tτ α ( θ ) dθ Z G ( E k∇ p k +1 ( τ, ψ kt,τ ( x )) − ∇ p k ( τ, ψ k − t,τ ( x )) k ) r dxdτ. In addition for m kr ( t ) = k u k ( t ) − u k − ( t ) k r,loc using the aprioriestimates proved in lemmas 5.2 – 5.8 and theorem 5.9 we obtain m k +1 r ( t ) ≤ C ( t )[( Z t Z G E k u k ( τ, ψ kt,τ ( x )) − u k − ( τ, ψ k − t,τ ( x )) k r dxdτ ) r +( 1 σ t m − Z t Z G E k∇ u k +1 ( τ, ψ k +1 t,τ ( x )) ∇ u k ( τ, ψ kt,τ ( x )) −∇ u k ( τ, ψ kt,τ ( x )) ∇ u k − ( τ, ψ k − t,τ ( x )) k r dxdτ ) r ] ≤ ( t )[( Z t m kr ( τ ) dτ ) r + ( Z t Z G α ( τ ) E k ψ kt,τ ( x ) − ψ k − t,τ ( x ) k r dxdτ ) r +1 σ t m − ( Z t [ ρ k +1 ( τ ) + ρ k ( τ )] ζ kr ( τ ) dτ ) r ] . Since u k and ∇ u k are uniformly bounded on [0 , T ] and k∇ u ( t, · ) − ∇ u ( · ) k r,loc ≤ const < ∞ , both for r = m and r = q we obtain that there exists a bounded on[0 , T ] positive function C ( t ) such that the function κ n ( t ) = ρ n ( t ) + ζ nm ( t ) + m nr satisfies the estimate κ n ( t ) ≤ [ C ( t )] n n !and hence lim n →∞ κ n ( t ) = 0 , since C ( t ) is bounded on [0 , T ]. Fi-nally we obtain that for each t ∈ [0 , T ) the family u n ( t, · ) uniformlyconverges to a limiting function u ( t, · ) ∈ C ,α ∩ L m,loc . In addition, wecan check that the limiting function ∇ u ( t, x ) is Lipschitz continuousin x . In fact, by lemma 2.8 and theorem 2.9 for each t ∈ [0 , T ], wehave for any x, y ∈ G k∇ u n ( t, x ) − ∇ u n ( t, y ) k ≤ N ( t ) k x − y k , where N ( t ) and T were defined above in lemmas 5.8 and theorem 5.7respectively and the estimate is uniform in n . This allows to statethat the limiting function is Lipschitz continuous as well.To prove the uniqueness of the solution of (2.8)-(2.10) constructedabove we assume first that there exist two solutions u ( t, x ), u ( t, x ) to(5.7)-(5.9) possessing the same initial data u (0 , x ) = u (0 , x ) = u ( x ).Computations similar to those used to prove the convergence ofthe family ( u n ( t ) , ∇ u n ( t )) allow to check that[ ∇ u ( t ) − ∇ u ( t )] α,G = 0 and k∇ u ( t ) − ∇ u ( t ) k m,loc = 0 . Finally, we know that the Cauchy problemfor a stochastic equationwith Lipschitz coefficients has a unique solution. This implies theuniqueness of the solution to (5.7)-(5.9).Summarizing the above results we see that the following statementis valid. Theorem 5.10. Assume that C 5.2 holds. Then there existsa unique solution ψ t,x ( s ) , u ( t, x ) , p ( t, x ) of the system (5.7)-(5.9), forall t from the interval the [0 , T ] , with T given by theorem 5.7 and x ∈ G for any compact G ⊂ R . In addition ψ t,x ( s ) is a Markovprocess in R and u ∈ C ([0 , T ] , C ,α ( G )) ∩ C ([0 , T ] , L q,loc ∩ L m,loc ) for < q < < < m < ∞ . To fulfill our program we have to check that the conditions of the-orem 2.8 are sufficient to verify that the functions u ( t, x ) , p ( t, x ) givenby (5.8), (5.9) define a weak solution of the Navier -Stokes system.To this end we have to apply the results of the Kunita theory ofstochastic flows. amely we check that given a distribution valued process λ ( t ) ofthe form λ ( t ) = u − Z t ∇ p u ( τ ) ◦ φ u ,τ dτ (5.52)the function λ ( t ) ◦ ψ ut, = u ◦ ψ ut, − Z t ∇ p u ( τ ) ◦ ψ ut,τ dτ gives rise to a solution of (5.1).To this end we apply the generalized Ito formula [19], [20] to derive λ ( t ) ◦ ψ ut, = u + Z t σ u ( θ ) ◦ ψ uθ, ] dθ + (5.53) Z t ∇ [ u ( θ ) ◦ ψ uθ, ] σdw ( θ ) − Z t ∇ [ u ( θ ) ◦ ψ uθ, ] u ( θ ) dθ − Z t ∇ p u ( θ ) dθ. Note that for Lu = − ( u, ∇ ) u + σ ∆ u we have E (cid:20)Z R Z t ( L ( u ( τ ) ◦ ψ uτ,s ( x )) dτ, h ( x )) dx (cid:21) = E (cid:20)Z t h u ( τ ) ◦ ψ uτ, , L ∗ h i dτ (cid:21) = Z t L h E [ u ( τ ◦ ψ uτ, )] , h i dτ. Hence u ( t ) = E [ λ ( t ) ◦ ψ ut, ] = u + Z t LE [ u ( τ ) ◦ ψ uτ, ] dτ − Z t ∇ p u ( τ ) dτ. Differentiating each term with respect to t we can check that thefunction u ( t ) = E [ λ ( t ) ◦ ψ ut, ] (5.54)solves the Cauchy problem (5.1),(5.2).To summarize the obtained results we can state the following as-sertion. Theorem 5.11. Assume that C 5.2 holds. Then the functions u ( t, x ) , p ( t, x ) given by (5.8),(5.9) are defined on the interval [0 , T ] with T determined by theorem 5.8 and satisfy (5.1)-(5.2) in a weaksense on this interval. Remark 5.12. We have proved that under condition C 5.2 thesystem (5.7)-(5.9) gives rise to a weak solution of (5.1)-(5.2) . Notethat if the initial data are smoother, say u ∈ C α , α ∈ [0 , 1] similarconsiderations can be applied to verify that the pair u ( t, x ) , p ( t, x )given by (5.8)-(5.9) stands for a classical C -smooth solution of (5.1)-(5.2). In fact in this case applying the generalized Ito formula for theverification assertion we may treat the action of the operator L in theclassical sense rather then in the weak sense. Lagrangian and stochastic approachto the Euler and the N-S system The probabilistic approach developed in the previous section is ina sense an analogue of the Lagrangian approach to the Euler andthe Navier-Stokes systems. A rather close model was constructed inpapers by Constantin and Iyer [11],[12]. To make it easier to comparewe rewrite the results from these papers in terms similar to those usedin the previous section. We consider first the Euler system ∂u∂t + ( u, ∇ ) u = −∇ p, u (0 , x ) = u ( x ) , x ∈ R (6.1) div u = 0 . (6.2)and recall that the corresponding Lagrangian path starting at y isgoverned by the Newton equation ∂ ˜ φ ,t ( y ) ∂t = F ˜ φ ( t, y ) . (6.3)The incompressibility condition for the map φ yieldsdet( ∇ ˜ φ ,t ( y )) = 1 . (6.4)The force F in (3.3) has the form F ˜ φ ( t, y ) = −∇ p ( t, ˜ φ ,t ( y )) = − [( ∇ ˜ φ ,t ( y )) ∗ ] − ∇ [ p ( t, ˜ φ ,t ( y ))] . (6.5)One can deduce from (6.3) that ∂∂t [ ∂ ˜ φ k ,t ( y ) ∂t ∂ ˜ φ k ,t ( y ) ∂y i ] = − ∂q ( t, ˜ φ ,t ( y )) ∂y i , (6.6)where q ( t, y ) = p ( t, y ) − k ∂ ˜ φ ,t ( y ) ∂t k . (6.7)We recall that in (6.6) and below summation over the repeated indicesis assumed. Integrating (6.6) in time we get ∂ ˜ φ k ,t ( y ) ∂t ∂ ˜ φ k ,t ( y ) ∂y i = u ( y ) − ∂n ( t, ˜ φ ,t ( y )) ∂y i , (6.8)where n ( t, y ) = Z t q ( τ, y ) dτ (6.9)and u ( y ) = ∂ ˜ φ ,t ( y ) ∂t | t =0 (6.10)is the initial velocity.Consider the inverse diffeomorphism ˜ ψ t, = [ ˜ φ ,t ] − , come back to(6.7), multiply it by [ ∇ ˜ ψ t, ] and put y = ˜ ψ t, ( x ). As a result we obtainby the chain rule the relation u i ( t, x ) = ( u j ( ˜ ψ t, ( x )) ∇ x i ˜ ψ jt, ( x ) − Z t ∇ x i q ( τ, ˜ ψ t,τ ( x )) dτ. (6.11) he equation (6.11) shows that the general Euler velocity may bewritten in the form that generalizes the Clebsch variable representa-tion u = [ ∇ ˜ ψ t, ] ∗ C − ∇ n, where C = u ( ψ t, ( x )) is an active vector and n is defined by theincompressibility condition divu = 0.Note that a vector A is called active if ddt A = ∂A∂t + ( u, ∇ ) A = 0 . It is easy to check by the chain rule that ddt ˜ ψ t,θ ( x ) = ∂ ˜ ψ t,θ ( x ) ∂t + ( u, ∇ ) ˜ ψ t,θ ( x ) = 0 , (6.12)that is ψ t, ( x ) is an active vector.Hence the Euler equations are equivalent to the system consistingof (3.9) and the following relation∆ n ( t, x ) = ∂∂x i { u k ( ˜ ψ t, ( x )) ∂ ˜ ψ kt, ( x ) ∂x i } , (6.13)where n is given by (6.9).Now one can assume the periodic boundary conditions or thezero boundary conditions at infinity. Note that in the periodic case n ( t, x ) , u ( t, x ) and δ ( t, x ) = x − ˜ ψ t, ( x ) (6.14)are periodic functions in each spatial direction. Finally due to divu =0 one can rewrite the equation of state (6.11) in the form u ( t ) = Π { u j ( ˜ ψ t, ) ∇ ˜ ψ jt, } = Π { [ ∇ ˜ ψ t, ] ∗ u ( ˜ ψ t, ) } , (6.15)where Π = I − ∇ ∆ − ∇ is the Leray-Hodge projector (with corre-sponding boundary conditions) on divergence free vector fields. TheEuler pressure is determined up to additive constants by p ( t, x ) = ∂n ( t, x ) ∂t + ( u ( t, x ) , ∇ ) n ( t, x ) + 12 k u ( t, x ) k . Note that (6.11), (6.12) made a closed system and may be used todetermine u ( t ).Let us compare (6.11), (6.12) with the alternative representationfor the state u ( t ) developed in the previous section.To this end we choose φ ,t : y → φ ,t ( y ) to be a volume preservingdiffeomorphism that satisfies the equation dφ ,τ ( y ) = u ( t − τ, φ ,τ ( y )) dτ, φ , ( y ) = y, (6.16)with div u ( t ) = 0.Consider the system dψ t,θ ( x ) = − u ( θ, ψ t,θ ( x )) dθ, ψ t,t ( x ) = x, (6.17) ( t, x ) = u ( ψ t, ( x )) − Z t ∇ p ( τ, ψ t,τ ( x )) dτ, (6.18) − p ( t, x ) = E [ Z ∞ γ ( t, x + B ( τ )) dτ ] , (6.19)where γ is given by (1.3).If the fields u ( t, x ) , p ( t, x ) are regular enough then we may con-struct the representation of the solution to the Euler system in theform (6.18), (6.19).To check this we consider a volume preserving diffeomorphism φ ,t : y → ˜ φ ,t ( y ) that satisfies (6.16).Next we consider the system dψ t,θ ( x ) = − u ( θ, ψ t,θ ( x )) dθ, ψ t,t ( x ) = x, (6.20) u ( t, x ) = u ( ψ t, ( x )) − Z t ∇ p ( τ, ψ t,τ ( x )) dτ, (6.21) − p ( t, x ) = E [ Z ∞ γ ( t, x + B ( τ )) dτ ] , (6.22)where γ is given by (1.4).If u ( t, x ) is regular enough then we may construct the representa-tion of the solution to the Euler system (6.1), (6.2) in the form (6.21),(6.22).To this end we consider a vector field λ ( t ) satisfying the equation dλ ( t ) dt = −∇ p ( t ) ◦ φ ,t λ (0) = u , where φ ,t satisfies the ODE (6.16), and let the process ψ t, be itsinverse. Applying the Kunita approach [19] to the process ψ t, we canverify that ψ t, along with (6.20) satisfies the equation ψ t,τ ( x ) = x + Z tτ ∇ φ gθ,t ( ψ t,θ ) − u ( θ, x ) dθ, (6.23)that allows to prove that u ( t ) given by (6.21) satisfies (6.1).Comparing (6.15) and (6.21) we note that they give different ex-pressions for the velocity field. Actually (6.21) includes the Eulerpressure p ( t, x ) instead of q ( t, x ) used in (6.15). Besides the proba-bilistic representation of the solution p to the Poisson equation − ∆ p = ∇ i u k ∇ k u i is used instead of the Leray projection.Coming back to the Navier-Stokes system ((5.1),(5.2) we recallhere the approach due to Constantin and Iyer [12]. The stochasticcounterpart of the Navier-Stokes equations in the version of Iyer [25]looks like the following.Consider the closed stochastic system dφ ,θ = u ( θ, φ ,θ ) dt + σdw ( θ ) , φ , ( y ) = y, (6.24) ψ θ, = [ φ ,θ ] − , (6.25) ( t ) = E Π[( ∇ ψ t, )( u ◦ ψ t, )] . (6.26)The existence and uniqueness of the solution to this system is provedin [12] by the successive approximation technique. As a result theauthors constructed a strong local in time solution of the Cauchyproblem for the Navier-Stokes system for regular enough initial data.The main result due to Constantin and Iyer reads as follows Theorem 6.1. Let k ≥ and u ∈ C k +1 ,α be divergence free.Then there exists a time interval [0 , T ] with T = T ( k, α, L, k u k k +1 ,α ) but independent of viscosity σ and a pair φ ,t ( x ) , u ( t, x ) such that u ∈ C ([0 , T ] , C k +1 ,α ) and ( u, φ ) satisfy (6.24)-(6.26). Further there exists U = U ( k, α, L, k u k k +1 ,α ) such that k u ( t ) k k +1 ,α ≤ U for t ∈ [0 , T ] and u satisfies the N-S system. As we have mentioned above an approach close to the one of [12]was developed in our previous paper [7]. Both these approaches allowto construct a classical (local in time) solution to the Cauchy problemfor the Navier-Stokes system and prove the uniqueness of the solution.On the other hand the approach developed in section 5 allows toconstruct a weak (local in time) solution to (5.1), (5.2) and prove theuniqueness of this solution in the corresponding functional classes.The stochastic counterpart of the Navier-Stokes system consideredin section 2 has the form dψ t,θ ( x ) = − u ( θ, ψ t,θ ( x ) dθ + σd ˆ w ( θ ) , ψ t,t ( x ) = x, (6.27) u ( t, x ) = E [ u ( ψ t, ( x )) − Z t ∇ p ( τ, ψ t,τ ( x )) dτ ] , (6.28)2 p ( t, x ) = − Z ∞ E [ γ ( t, x + B ( τ ))] dτ. (6.29)Note that we can use the relation − ∇ p ( t, x ) = E [ Z ∞ τ γ ( t, x + B ( τ )) B ( τ ) dτ ] (6.30)to eliminate the pressure from the above system (6.27) – (6.29).We can see that the difference between (6.27) – (6.29) and (6.24)– (6.26) has the same nature as the difference between (6.11), (6.12)and (6.17) – (6.19).Finally we note that the approach developed in section 5 allows usto construct both strong (classical) and weak (distributional) solutionsof the Cauchy problem for the N-S system. Acknowledgement. The authors gratefully acknowledge the fi-nancial support of DFG Grant 436 RUS 113/823. References [1] McKean H. A class of Markov processes associated with nolinearparabolic equations . Proc. Nat. Acad. Sci. USA 2] Freidlin M. Quasilinear parabolic equations and measures in func-tional spaces. Funct. Anal. and Appl. , N 3 (1967) 237-240.[3] Freidlin M. Functional integration and partial differential equa-tions. Princeton Univ. Press 1985.[4] Belopolskaya Ya., Dalecky Yu. 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