Articulo 6.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (77–96). June 2010 Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes S. Albeverio Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany SFB 611,HCM, Bonn, BiBoS, Bielefeld - Bonn, CERFIM, Locarno and USI (Switzerland) email: albeverio@uni-bonn.de and Ya. Belopolskaya St.Petersburg State University for Architecture and Civil Engineering, 2-ja Krasnoarmejskaja 4, 190005, St.Petersburg, Russia email: yana@yb1569.spb.edu ABSTRACT We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a stochastic problem solution. Under suitable conditions we solve the stochastic problem and prove that simultaneously we obtain a weak (generalized) solution to the Cauchy problem for the Navier-Stokes system. RESUMEN Nosotros reducimos la construcción de una solución débil de un problema de Cauchy para el sistema de Navier-Stokes para la construcción de la resolución de un problema estocástico. Bajo condiciones convenientes resolvimos el problema estocástico y probamos que simultáneamente obtenemos una solución débil (generalizada) para el problema de Cauchy del sistema de Navier-Stokes. 78 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) Key words and phrases: Stochastic flows, diffusion processes, nonlinear parabolic equations, Cauchy problem. AMS Subj. Class.: 60H10, 60J60 , 35G05, 35K45 1 Introduction The main purpose of this article is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for the Navier-Stokes (N-S) system ∂u ∂t + (u,∇)u = ν∆u − ∇p, u(0,x) = u0(x), x ∈ R3 (1.1) div u = 0. (1.2) Here u(t,x) ∈ R3,x ∈ R3, t ∈ [0,∞) is the velocity of the fluid at the position x at time t and ν > 0 is the viscosity coefficient. p(t,x) is a scalar field called the pressure which appears in the equation to enforce the incompressibility condition (1.2). There exists a number of papers [1] – [4] and others where the system (1.1), (1.2) was treated from the probabilistic point of view on the base of stochastic models. In particular in our previous paper [1] the system (1.1), (1.2) was reduced to a probabilistic problem presented in the form of the following system of equations dξ(τ) = −u(t − τ,ξ(τ))dτ + σdw(τ), (1.3) u(t,x) = E0,x[u0(ξ(t)) + ∫ t 0 ∇p(t − τ,ξ(τ))dτ] (1.4) p(t,x) = 2E[ ∫ ∞ 0 γ(t,x + B(t))dt] = 2E[ ∫ ∞ 0 tr[∇u]2(t,x + B(t))dt]. (1.5) Here σ = √ 2ν, w(t) and B(t) are independent standard Wiener processes valued in R3, Tr[∇u]2 = ∑3 i,k=1 ∇iuk∇kui. It was shown in [1] that if the initial value u0 is a C3- function the functions u(t,x),p(t,x) given by (1.4), (1.5) are C2+α solutions of (1.1), (1.2) for 0 < α < 1. In the present paper we consider an alternative probabilistic system which allows to construct a weak (distributional) solution to (1.1), (1.2). The approach developed here is based on the theory of stochastic flows due to Kunita [5], [6] and the results due to Belopolskaya and Dalecky [7], [8]. The article is composed as follows. In section 2 we give some preliminary information concerning different analytical approaches to the notion of a solution of the Navier-Stokes system. Here we recall some common ways to exclude the pressure and to obtain a closed equation for the velocity, introduce necessary functional spaces and state various notions of solutions to (1.1), (1.2). In section 3 we state our approach and prove main results. In the last section we compare our approach and results with the Euler-Lagrange approach to incompressible fluids developed by Constantin and Iyer [9],[10]. 2 Preliminaries Within a classical approach to the N-S system one excludes the pressure from (1.1),(1.2) and inves- tigates the resulting nonlinear pseudo-differential equation. To this end first one can derive formally CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 79 from (1.1),(1.2) the relation −∆p(t,x) = γ(t,x), (2.1) where γ(t,x) = 3 ∑ k,j=1 ∇kuj∇juk = Tr[∇u]2 = ∇ · ∇ · u ⊗ u. (2.2) Then given an R3-valued vector field u(t) over R3 the operator P is defined by Pu(t) = u(t) − ∇∆−1∇ · u(t). (2.3) Here and below u · v denotes the inner product in R3 of the vectors u and v. The map P called the Leray projection is a projection of the space L2(R3) ≡ L2(R3)3 of square integrable vector fields to the space of divergence free vector fields. Since the formal solution of the Poisson equation (2.1) is given by p = ∆−1γ = ∆−1∇ · ∇ · u ⊗ u (2.4) one can present ∇p in the form ∇p = ∇∆−1∇ · ∇ · u ⊗ u keeping in mind that divu = 0. Substituting this expression for ∇p into (1.1) one obtains the following Cauchy problem ∂u ∂t = ν∆u − P∇ · (u ⊗ u), u(0) = u0. (2.5) When (2.5) is solved then the pressure is reconstructed from the Poisson equation (2.1). The Leray projection P is used to solve the N-S system both in numerous analytical papers (see, e.g., [11] for references) and in papers where the N-S system is studied from the probabilistic point of view [2],[3], [10]. In this paper we avoid the direct application of the Leray projection and construct the solution of (1.1), (1.2) via stochastic processes associated with (1.1) and (2.1). To give a rigorous definition of a solution for the N-S system we have to specify the required functional spaces. Let D = D(R3) = C∞c denote the space of all infinitely differentiable real valued functions on R3 with compact support equipped with the Schwartz topology and let D′ be its topological dual. Let 〈φ,ψ〉 = ∫ R3 φ(x)ψ(x)dx denote the natural coupling between φ ∈ D and ψ ∈ D′. If it will not lead to misunderstandings we will use the same notation for vector fields u and v as well, that is 〈h,u〉 = ∫ R3 3 ∑ k=1 hk(x)uk(x)dx. Let D((0,T )×R3) = (D′((0,T )×R3))3 denote the space of R3-valued vector fields h with components hk ∈ D and D′ denote the space dual to D(R3). The Leray weak solution of the N-S system on [0,T ] × R3 is a vector field u(t,x) in (D′((0,T ) × R3))3 such that u is locally square integrable on (0,T ) × R3, satisfies div u = 0 and there is a distribution p ∈ D′((0,T ) × R3) such that ∂u ∂t = ν∆u − ∇ · (u ⊗ u) − ∇p, lim t→0 u(t) = u0 (2.6) 80 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) holds in the sense of distributions. The Kato mild solution is a solution u to the following integral equation u(t) = et∆u0 − ∫ t 0 e(t−s)∆P∇ · (u ⊗ u)(s)ds. (2.7) Note that instead of looking for u(t,x) and p(t,x) one may look for their Fourier images û(λ) = (2π)− 3 2 ∫ R3 e−iλ·xu(x)dx. The Leray and Kato approaches stated in original terms and in terms of the Fourier transformation of the Navier-Stokes system were developed in a number of papers (see, e.g., references in the book by Lemarie-Rieusset [11]). Below we will need as well the following functional spaces: the space C(Rd,Rn) of bounded continuous functions mapping Rd to Rn, the space C(R3,R1) = C(R3) of bounded continuous real functions f with the norm ‖f‖∞ = supx∈R3|f(x)|; the space C(R3) of bounded continuous vector functions with the norm ‖u‖∞ = supx∈R3‖u(x)‖, where ‖ · ‖ is the norm in R3; the space C0(R 3) of continuous vector functions with compact supports; the Banach space Lq(R3) of integrable functions f with norm ‖f‖q = ( ∫ R3 ‖f(x)‖qdx) 1 q ; the space Ck(R3) of k-times differentiable functions with the norm ‖g‖Ck = ∑ |β|≤k ‖Dβg‖∞; the space Ck,α(R3) (for a natural number k) of vector fields whose k-th derivatives are Hölder continuous with exponent α, 0 < α ≤ 1 with norm ‖g‖Ck,α = ‖g‖Ck + [g]k+α and [g]k+α = ∑ |β|=k sup x,y∈R3 |Dβg(x) − Dβg(y)| |x − y|α . Let Z denote the set of all integers, and suppose that k ∈ Z is positive and 1 < q < ∞. Denote by Wk,q = Wk,q(R3) the set of all real functions h defined on R3 such that h and all its distributional derivatives ∇αh of order |α| = ∑ αj ≤ k belong to Lq(R3). It is a Banach space with norm ‖h‖k,p = ( ∑ |α|≤k ∫ R3 |Dαh(x)|qdx) 1 q . (2.8) Denote by W k,q 0 the subspace of functions from W k,q = Wk,q(R3) with compact supports. Finally we will need some spaces of locally integrable functions. Let G ⊂ R3 be a bounded domain, p be a positive integer and f : G → R1 be a Lebesgue measurable function. The set of functions {f : ∫ K |f(x)|pdx < ∞ for all compact subsets K ⊂ G} is denoted by Lp loc and called a space of p- locally integrable functions. Although L p loc (G) are not normed spaces they are readily topologized. Namely a sequence {un} converges to u in Lploc(G) if {un} → u in Lp(K) for each open K having compact closure in G and ‖u‖p,loc = ( ∫ K ‖u(x)‖pdx) 1 p < ∞. In a natural way one can define the spaces Wk,q and L p loc (G) of vector fields with components in Wk,p and in L p loc (G). CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 81 3 A probabilistic approach to the Navier-Stokes system Let us come back to the Navier-Stokes system written in the form ∂u ∂t + (u,∇)u = σ 2 2 ∆u − ∇p, u(0,x) = u0(x), x ∈ R3 (3.1) −∆p = γ (3.2) with γ defined by (1.3). Our main purpose in this section is to construct a diffusion process that allows us to obtain a a weak solution to (3.1), (3.2) via its probabilistic representation. To be more precise we intend to reduce the system (3.1), (3.2) to a certain system of stochastic equations and to construct its solution. Then we have to verify that in this way we have constructed a weak solution of (3.1), (3.2). As above let w(t),B(t) be standard R3-valued independent Wiener processes defined on a prob- ability space (Ω,F,P). Given a bounded measurable function f(x) and a stochastic process ξ(t) we denote Es,xf(ξ(t)) ≡ Ef(ξs,x(t)) the conditional expectation under the condition ξ(s) = x. Given a function g(t,x) ∈ R3, a smooth (in x) function q(t,x) ∈ R1, t ∈ (0,∞),x ∈ R3 and a constant σ we consider stochastic processes ξg(t) and λ(t) satisfying the stochastic equations dξgy(t) = g(t,ξ g y(t))dt − σdw(t), ξgy(0) = y ∈ R3, λ(t) = u0 − ∫ t 0 ∇q(τ,φg0,τ )dτ, (3.3) where φ g 0,t denotes the stochastic map in R 3 generated by the process ξg(t), φ g 0,t(y) = ξ g y(t). The map φ g 0,t : R 3 → R3 is called a stochastic flow. The processes ξg(t) and λ(t) are auxiliary ones. The main role in our considerations is played by the stochastic flow ψt,0 which is an inverse flow to φ0,t, ψt,0(φ0,t(y)) = y. To construct the flow ψt,0 we need the process ŵ(θ) = w(t − θ) − w(t) which is proved to be the standard Wiener process. Here we use the results of the Kunita theory of stochastic flows [5],[6] and extend them to the case of stochastic processes associated with nonlinear PDEs. Actually we consider the closed system dψt,θ(x) = −u(θ,ψt,θ(x))dθ + σdŵ(θ), ψt,t(x) = x, (3.4) u(t,x) = E[u0(ψt,0(x)) − ∫ t 0 ∇p(τ,ψt,τ (x))dτ], (3.5) −2∇p(t,x) = E[ ∫ ∞ 0 1 τ γ(t,x + B(τ))B(τ)dτ], (3.6) where γ is given by (1.3) and look for a solution u(t,x),p(t,x),ψt,θ(x) of this system under some assumptions on the initial data u0 to be specified below. To construct the solution of (3.4)– (3.6) we consider its differential prolongation. Namely, we consider the following formal relation dη x(θ) = −∇u(θ,ψt,θ(x))ηx(θ)dθ, ηx(t) = I, (3.7) 82 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) where I is the identity matrix acting in R3, and one of two formal relations for ∇u(t,x) ∇u(t,x) = E [ ∇u0(ψt,0(x))ηx(t) − ∫ t 0 ∇2p(τ,ψt,τ (x))ηx(τ)dτ ] . (3.8) or ∇u(t,x) = E [ ∇u0(ψt,0(x))ηx(t) − ∫ t 0 ∇p(τ,ψt,τ (x)) σ(t − τ) ∫ t τ ηx(θ)dŵ(θ)dτ ] . (3.9) Note that to derive the second term in the right hand side of (3.9) we need a specific integration by parts formula called the Bismut-Elworthy-Li formula [12]. Since the system (3.3)– (3.8) is a closed system with respect to (ψt,0(x),η x(t),u(t,x),p(t,x),∇u(t,x)), we aim to prove the existence and uniqueness theorem for its solution. At the end we check that the functions (u(t,x),p(t,x)) given by (3.4)– (3.5) satisfy (3.1), (3.2). To construct the solution of (3.4)– (3.8) we consider a system of successive approximations and prove their convergence. Set u 1(t,x) = u0(x), ψ 0 t,0(x) = x, p 1(t,x) = 0 (3.10) and consider stochastic processes ψkt,θ(x), vector fields u k(t,x) and scalar functions pk(t,x) given by the following relations dψkt,θ = −uk(θ,ψkt,θ)dθ + σdŵ(θ), ψkt,t = x, (3.11) uk+1(t,x) = E[u0(ψ k t,0(x)) − ∫ t 0 ∇pk+1(τ,ψkt,τ (x))dτ], (3.12) −2pk+1(t,x) = ∫ ∞ 0 E[γk(t,x + B(τ))]dτ, (3.13) where γk(t,x) = Tr[∇uk]2(t,x)]. (3.14) Finally, we consider η x,k t,θ , ∇uk+1(t,x) and ∇pk+1(t,x) defined respectively by dη x,k t,θ = −∇uk(θ,ψkt,θ)η x,k t,θ dθ, η x,k t,t = I, (3.15) and ∇uk+1(t,x) = E[∇u0(ψk+1t,0 (x))η x,k t,0 − ∫ t 0 ∇2pk+1(τ,ψkt,τ(x))η x,k t,τ dτ], (3.16) −2∇pk+1(t,x) = ∫ ∞ 0 1 τ E[γk(t,x + B(τ))B(τ)]dτ. (3.17) Note that for k = 1 we have dψ 1 t,θ = −u0(θ,ψ1t,θ)dθ + σdŵ(θ), ψ1t,t = x, CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 83 that is we can solve the stochastic equation (3.4) independently on (3.5)-(3.6). Then given the process ψ1t,0(x) and keeping in mind the properties of the function p 1 that satisfies the Poisson equation −∆p1(t,x) = γ0(t,x), (3.18) we compute u1(t,x) from (3.12). Next we compute ∇u1(t,x), ∇p1(t,x) from (3.16), (3.17) and proceed to k = 2. To prove the convergence of the successive approximations obtained in this way we need to derive some apriori estimates. Let g(t,x) ∈ R3 be a given bounded Lipschitz continuous function on [0,∞)×R3. Set g(t,ψ(t,x)) = g(t) ◦ ψ(t)(x) for any functions ψ(t,x) ∈ R3). Consider the stochastic equation dψ g t,θ = −g(θ) ◦ ψg t,θ dθ + σdŵ(θ), ψ g t,t(x) = x (3.19) and define the vector fields ug(t,x) and ∇pg(t,x) by ug(t,x) = E[u0(ψ g t,0(x)) − ∫ t 0 ∇pg(τ,ψgt,τ(x))dτ], (3.20) −2pg(t,x) = ∫ ∞ 0 E[γg(t,x + B(τ))]dτ, (3.21) where γg(t,x) = Tr[∇g]2(t,x). (3.22) We derive formally from (3.21) by the integration by parts formula (Bismut – Elworthy – Li formula [12]) that −2∇pg(t,x) = ∫ ∞ 0 E[ 1 τ γg(t,x + B(τ))B(τ)]dτ. (3.23) Below we will describe the conditions on γ which justify (3.23). Condition C 3.1 Let g(t,x) ∈ R3 be a divergent free vector field depending on time and defined on [0,T ] × R3 for a certain constant T > 0. We assume that g(t) belongs to C1,α(R3), 0 < α ≤ 1 for a fixed t ∈ [0,T ] and satisfies the following estimates: 1. ‖g(t)‖q,loc ≤ Ng(t) for some q to be specified below, ‖g(t)‖∞ ≤ Kg(t) and ‖g(t,x) − g(t,y)‖ ≤ Lg(t)‖x − y‖, ‖∇g(t,x) − ∇g(t,y)‖ ≤ L1g(t)‖x − y‖. 2. ‖∇g(t)‖∞ ≤ K1g(t), ‖∇g(t)‖r,loc ≤ N1g (t). Here Kg(t),Lg(t), Ng(t) and K 1 g(t),L 1 g(t),N 1 g (t) are positive continuous functions defined on an interval [0,T ] with T > 0, r = m and r = q for 1 < q < 3 2 < 3 < m < ∞. Set ψg(τ) = ψ g t,τ(x) and consider the stochastic equation ψg(τ) = x − ∫ t τ g(τ1,ψ g(τ1))dτ1 + ∫ t τ σdŵ(τ1), (3.24) 84 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) with 0 ≤ τ ≤ t < T . When we are interested in the particular dependence of the process ψg(τ) on the parameters t,x we write ψg(τ) = ψ g t,x(τ) or ψ g(τ) = ψ g t,τ(x). Lemma 3.1 Assume that C 3.1 holds. Then there exists a unique solution ψg(τ) of (3.24) and the following estimates E‖ψg(τ)‖2 ≤ 3[‖x‖2 + σ2(t − τ) + (t − τ) ∫ t τ [K2g(τ1)]dτ1], (3.25) E‖ψgt,x(τ) − ψ g t,y(τ)‖ ≤ ‖x − y‖e R t τ Lg(θ)dθ, (3.26) E‖ψg(τ) − ψg1 (τ)‖ ≤ ∫ t τ ‖g(τ1) − g1(τ1)‖∞dτ1e R t τ Lg(θ)dθ (3.27) hold. Proof. The proof of the estimates of this lemma is standard and based on estimates of classical and stochastic integrals. We only show the proof of (3.26). In view of C 3.1 we have E‖ψgt,x(τ) − ψ g t,y(τ)‖ ≤ ‖x − y‖ + ∫ t τ Lg(τ1)‖ψgt,x(τ1) − ψ g t,y(τ1)‖dτ1 where 0 ≤ τ ≤ t ≤ T with some constant T to be chosen later. Finally, by Gronwall’s lemma, we get E‖ψgt,x(τ) − ψ g t,y(τ)‖ ≤ ‖x − y‖e R t τ Lg(θ)dθ. 2 Along with (3.19)-(3.22) we need the equations for the mean square derivative ηg(t) = ∇ψgt,0(x) of the diffusion process ψ g t,0(x) that satisfies (3.19), and the gradient v(t,x) = ∇ug(t,x) of the function ug(t,x) given by (3.20). Lemma 3.2 Assume that C 3.1 holds. Then the process ηx,g(τ) = ∇ψgt,τ(x) satisfies the stochastic equation dηx,g(τ) = −∇g(τ,ψx,gt,τ (x))ηx,g(τ)dτ, ηx,g(t) = I. (3.28) The process ηx,g(τ) possesses the following properties. The determinant det ηg(τ) is equal to 1, i. e. det ηg(τ) = Jt,τ = 1 and E‖ηx,g(τ)‖ ≤ e R t τ K 1 g (θ)dθ (3.29) E‖ηx,g(τ) − ηy,g(τ)‖ ≤ C‖x − y‖ (3.30) with some positive constant C depending on t,τ and g. In addition the following integration by part formula is valid ∫ R3 f(ψ g t,x(τ))dx = ∫ R3 f(x)dx, f ∈ L1(R3). (3.31) Proof. Under C 3.1 the first statement immediately follows from general results of the stochastic differential equation theory. By direct computation one can check that Jt,τ satisfies the linear equation dJt,τ = −div g(ψgt,τ)Jt,τdτ, Jt,t = I CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 85 and since div g = 0 we get the second statement. Besides ψ g t,τ is a C 1 stochastic diffeomorphism (see [5]) and hence the integration by part formula (3.31) holds. Finally (3.29) is deduced from the inequality E‖ηx,g(τ)‖ ≤ 1 + ∫ t τ K1g(θ)E‖ηx,g(θ)‖dθ by the Gronwall lemma. One can easily check that for the solution ηx,g(t) of (3.28) we have E‖ηx,g(τ) − ηy,g(τ)‖ ≤ ∫ t τ E‖∇g(θ,ψg t,θ (x)) − ∇g(θ,ψg t,θ (y))‖dθe R t τ K 1 g (θ)dθ ≤ E ∫ t τ L1g(θ)E‖ψ g t,θ (x)) − ψg t,θ (y))‖dθ and by (3.26) we derive (3.30). 2 Let us state conditions on the initial data u0 of the N-S system. We say that C 3.2 holds when i) for some 0 < α ≤ 1 the initial vector field u0 ∈ C1+α0 (R3) satisfies the estimates ‖u0‖∞ ≤ K0, ‖∇u0‖∞ ≤ K10, ‖u0‖r,loc ≤ M0, ‖∇u0‖r,loc ≤ M10 with some positive constants K0,K 1 0,M0,M 1 0 and r. ii) u0 and ∇u0 are Lipschitz-continuous with positive Lipschitz constants L0 and L10 respectively. Keeping in mind conditions C 3.1 and C 3.2 we derive estimates for ug(t) defined by (3.20) on a certain time interval [0,T ] and its gradient ∇ug(t,x). Lemma 3.3 Assume that g(t,x) satisfies C 3.1 and u0 satisfies C 3.2 with r = q and r = m for 1 < q < 3 2 < 3 < m < ∞. Then the vector field ug(t,x) given by (3.20) satisfies the estimate ‖ug(t)‖∞ ≤ K0 + ∫ t 0 CqmK 1 g(τ)[‖∇g(τ)‖q,loc + ‖∇g(τ)‖m,loc]dτ. (3.32) Under the conditions of this lemma the proof of (3.32) can be easily obtained by a direct compu- tation from (3.20) using the estimates of the Newton potential given in lemma 3.4 below. Lemma 3.4 Let G ⊂ R3 be a bounded domain and γg ∈ Lq(G) ∩ Lm(G) for some 1 ≤ q < 3 2 < 3 < m < ∞ and −∆p(t,x) = γg(t,x), x ∈ G. Then ‖∇pg‖∞ ≤ Cqm(‖γg‖q,loc + ‖γg‖m,loc) and ‖∇i∇jpg‖∞ ≤ C(‖γg‖q,loc + [γg]α). 3. Let γg ∈ Lr(G) for 1 < r < ∞.Then pg ∈ W 2,r(R3) and the Calderon- Zygmund inequality holds ‖∇2pg‖r,loc ≤ C1‖γg‖r,loc. 86 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) The proof of these estimates for a solution of the Poisson equation can be found in the book by Gilbarg and Trudinger ([13] Th 9.9). The probabilistic proof of some of these estimates can be found in [4]. 2 Lemma 3.5 Assume that the conditions of lemma 3.3 hold and ug(t,x) is given by (3.20). Then the function ∇ug(t,x) admits a representation of the form ∇ug(t,x) = E[∇u0(ψgt,0(x))η x,g(t) − ∫ t 0 ∇2pg(τ,ψgt,τ (x))ηx,g(τ)dτ] (3.33) and the estimate ‖∇ug(t)‖∞ ≤ e R t 0 K 1 g (θ)dθ K 1 0 + ∫ t 0 e R t τ K 1 g (θ)dθ K 1 g(τ)[‖∇g(τ)‖q,loc + ‖∇g(τ)‖m,loc]dτ (3.34) holds for 1 < q < 3 2 < 3 < m < ∞, 0 ≤ t ≤ T . Proof. The formal differentiation of (3.20) in x justified by C 3.1, C 3.2 and the results of lemma 3.4 yields (3.33). To verify the estimate (3.34) we use the above estimates for the process ηx,g(t) and the estimates of the Newton potential derivative from lemma 3.4. Hence we obtain ‖∇ug(t)‖∞ ≤ K10 ∫ t 0 K1g(θ)dθ+ (3.35) ∫ t 0 Cqme R t τ K 1 g (θ)dθ[‖Tr[∇g(τ)]2‖m,loc + ‖Tr[∇g(τ)]2‖q,loc]dτ] that immediately leads to (3.34). 2 Now we have to derive the estimate for the function ‖∇u(t)‖r,loc. Lemma 3.6 Assume that the conditions of lemma 3.3 hold. Then for 1 < r < ∞ the gradient of the function ug(t,x) given by (3.20) satisfies the estimate ‖∇ug(t)‖r,loc ≤ e2 R t 0 K 1 g (θ)dθ [ ‖∇u0‖r,loc + C ∫ t 0 ‖∇g(τ)‖r,locdτ ] , (3.36) where 0 ≤ t ≤ T and C depends on r and T . Proof. Let us derive the Lp- estimate for ∇ug(t,x) given by (3.33). To derive the estimate for ‖∇ug(t)‖rr,loc = ∫ K ‖∇ug(t,x)‖rdx (where K is an arbitrary compact in G) we apply first the triangle inequality to obtain ‖∇ug(t)‖r,loc ≤ α1 + α2, where α1 = ( ∫ K E[‖∇u0(ψgt,0(x))ηx,g(t)‖r]dx ) 1 r , α2 = ( ∫ K ∫ t 0 E‖∇2pg(τ,ψgt,τ(x)))ηx,g(τ)‖rdτdx ) 1 r . To estimate α1 we apply the Hölder inequality and take into account the inequality (3.29) for the process ηx,g(τ). Besides we recall that ψt,τ(x) preserves the volume. As a result we have α1 ≤ ( ∫ K (E[‖∇u0(ψgt,0(x))‖2]E[‖ηx,g(t)‖2]) r 2 dx ) ) 1 r ≤ ‖∇u0‖r,loce R t 0 K 1 g (θ)dθ. CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 87 To derive the estimate for α2 we deduce from the Calderon-Zygmund inequality (see lemma 3.4) and the estimate of ηx,g(t) that αr2 ≤ Cr ∫ t 0 e R τ 0 K 1 g (θ)dθK1g(τ) ∫ K ‖∇g(τ,x)‖rdxdτ. Combining the above estimates for α1 and α2 we obtain the required estimate ‖∇ug(t)‖r,loc ≤ e R t 0 K 1 g (θ)dθ [‖∇u0‖r,loc+ Cr ∫ t 0 e R τ 0 K 1 g (θ)dθ K 1 g(τ)‖∇g(τ)‖r,locdτ ] . Finally we get ‖∇ug(t)‖r,loc ≤ e2 R t 0 K 1 g (θ)dθ [ ‖∇u0‖r,loc + C ∫ t 0 ‖∇g(τ)‖r,locdτ ] , where C depends on r and T. 2 Theorem 3.7 Assume that conditions C 3.1 and C 3.2 hold. Then there exists an interval ∆1 = [0,T1] and functions α(t), β(t), κ(t) bounded for t ∈ ∆1, such that, if for all t ∈ ∆1, ‖g(t)‖∞ ≤ κ(t) and ‖∇g(t)‖∞ ≤ α(t), ‖∇g(t)‖r,loc ≤ βr(t) then the function ‖∇ug(t,x)‖ (where ug(t,x) is given by (3.20)) satisfies the estimates ‖ug(t)‖∞ ≤ κ(t), ‖∇ug(t)‖2∞ ≤ α(t), ‖∇ug(t)‖2r,loc ≤ βr(t) (3.37) for r = q and r = m and 1 < m < 3 2 < 3 < q < ∞. Proof. Analyzing the above estimates (3.35), (3.36) for the functions ug(t,x) and ∇ug(t,x) we note that to prove the required estimates it is enough to construct the solutions of the following integral equations α(s) = e R t s α(θ)dθK10 + Cqm ∫ t s e R τ s α(θ)dθα(τ)[nq(τ) + nm(τ)]dτ, (3.38) nr(s) = e R t s α(θ)dθ‖∇u0‖r + Cr ∫ t s e R τ s α(θ)dθ nr(τ)α(τ)dτ (3.39) for r = q and r = m and Cqm = max(Cq,Cm) and β(s) = e R t s α(θ)dθ β0 + Cqm ∫ t s e R τ s α(θ)dθ α(τ)β(τ)dτ, (3.40) where β(τ) = nq(τ) + nm(τ), and ‖∇u0‖q,loc + ‖∇u0‖m,loc = nq(0) + nm(0) = β0. To construct the solution of the above system of integral equations (3.40)-(3.42) we consider the system of ODEs dα ds = −α2(s) − Cqmα(s)β(s), α(t) = K10, (3.41) dβ ds = −α(s)β(s) − Cqmα(s)β(s), β(t) = β0. (3.42) 88 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) By classical results of the ODE theory we know that there exists an interval [0,T1] depending on K 1 0,N 1 0 and C,Cqm such that the system (3.41), (3.42) has a bounded solution defined on this interval. To prove the convergence for k → ∞ of functions uk(t,x),∇uk(t,x) we need one more auxiliary estimate. Actually, we have proved that uk(t) is Lipschitz-continuous with the Lipschitz constant independent of k. It remains to prove that ∇uk(t) has the same property. Lemma 3.8 Assume that C 3.1 and C 3.2 hold. Then the function ∇ug(t,x) defined in lemma 3.5 admits a representation of the form ∇ug(t,x) = E[∇u0(ψgt,0(x))ηx,g(t)− ∫ t 0 1 σ(t − τ) ∇pg(τ,ψgt,τ (x)) ∫ t τ ηx,g(θ)dŵ(θ)dτ] (3.43) and satisfies the estimate ‖∇ug(t,x) − ∇ug(t,y)‖ ≤ Ng1 (t)‖x − y‖ if t ∈ [0,T1] for any x,y ∈ G where G is a compact in R3 and the positive function Ng1 (t) depending on the parameters in conditions C 3.1 and C 3.2 is bounded over the interval [0,T1] defined in theorem 3.7. Proof. To derive (3.43) we compute directly the gradient of the first term in (3.20) and apply the Bismut-Elworthy -Li formula [12] to compute the gradient of the second term in this relation. Next we use the representation (3.43) to deduce the Lipschitz estimate for the gradient of the function u(t,x) . As a result we have ‖∇ug(t,x) − ∇ug(t,y)‖ ≤ κ1 + κ2 + κ3 + κ4, where κ1(t) = E[‖∇u0(ψgt,0(x)) − ∇u0(ψ g t,0(y))‖‖η x,g(t)‖], κ2(t) = E[‖∇u0(ψgt,0(y))‖‖ηx,g(0) − ηy,g(0)‖], κ3(t) = ∫ t 0 E [ ‖∇pg(τ,ψgt,τ (x)) − ∇pg(τ,ψ g t,τ (y))‖ σ(t − τ) ‖ ∫ t τ ηx,g(θ)dŵ(θ)‖ ] dτ, κ4(t) = ∫ t 0 1 σ(t − τ) E [ ‖∇pg(τ,ψgt,τ(y))‖ ∫ t τ [ηx,g(θ) − ηy,g(θ)]dŵ(θ)‖ ] dτ. One can easily check using the estimates stated in lemmas 3.3 – 3.5 that under the conditions C 3.1, C 3.2 κ1(t) ≤ L10E‖ψ g t,0(x) − ψ g t,0(y)‖e R t 0 K 1 g (θ)dθ ≤ ‖x − y‖L10e R t τ [Lg(θ)+K 1 g (θ)]dθ = Θ1‖x − y‖ and κ2(t) ≤ K10E‖ηx,g(t) − ηy,g(t)‖ ≤ ‖x − y‖ ∫ t τ K1g(θ)e R t θ K 1 g (θ1)dθ1dθ = Θ2‖x − y‖. CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 89 To derive the estimates for κ3 and κ4 we apply the inequalities ‖∇i∇jpg‖∞ ≤ C(‖γg‖q,loc + [γg]1,G), ‖∇i∇jpg‖r,loc ≤ ‖γg‖r,loc from lemma 3.4. This yields κ3(t) ≤ ∫ t 0 E [ ‖∇2pg(τ)‖∞ ‖ψgt,0(x) − ψ g t,0(y)‖ σ √ t − τ ‖ηx,g(τ)‖ ] dτ ≤ ∫ t 0 (E‖ψgt,τ (x) − ψt,τ (y)‖2) 1 2 σ √ t − τ (‖γg(τ)‖q,loc + [γg(τ)]1,G)e R t τ K 1 g (θ)dθ dτ ≤ Θ1‖x − y‖ ∫ t 0 ‖γg(τ)‖q,loc + [γg(τ)]1,G σ √ t − τ e R t τ K 1 g (θ)dθdτ ≤ Θ3Θ1‖x − y‖σ−1 ( sup0≤τ≤t[β(τ)] √ t + ∫ t 0 s(τ)√ t − τ dτ ) and κ4(t) ≤ ∫ t 0 Cqm(‖γg(τ)‖q,loc + ‖γg(τ)‖m,loc) σ √ t − τ (E‖ηx,g(τ) − ηy,g(τ)‖2) 1 2 dτ ≤ 2sup0≤τ≤tβ(τ)‖x − y‖ ∫ t 0 Θ2Cqm σ √ t − τ dτ = 2Θ4‖x − y‖ Θ2Cqm σ √ t. Here Θ3 = e R t 0 K 1 g (τ)dτ, s(τ) = [γg(τ)]1,G and β(τ) is defined in theorem 3.7. Finally, combining the above estimates for κi, i = 1, 2, 3, 4, we obtain s(t) ≤ Θ5 + Θ6 ∫ t 0 s(τ)√ t − τ dτ and applying the Gronwall lemma we derive the estimate s(t) ≤ Θ5eΘ6 √ t where Θi, i = 5, 6 depend on the parameters in conditions C 3.1 and C 3.2, σ and T1 for 0 ≤ t ≤ T1, where T1 is defined in theorem 3.7. 2 The estimates of theorem 3.7 and lemma 3.8 allow to prove the uniform convergence on compacts of the successive approximations (3.10)-(3.14) for the solutions of the system (3.4) – (3.6) in C([0,T1], C1,α(K)) ∩ C([0,T1],Lm(K) ∩ Lq(K)) for 1 < q < 32 < 3 < m < ∞ and arbitrary compact K in G. In particular, they justify the possibility to differentiate the system (3.10)-(3.14) in x for each k and to consider the following equations dη k,x t,θ = −∇uk(θ,ψkt,θ)η x,k t,θ dθ, η x,k t,t = I, (3.44) where I is the identity matrix acting in R3 and ∇uk+1(t,x) = E[∇u0(ψk+1t,0 (x))η x,k t,0 − ∫ t 0 1 σ(t − τ) ∇pk+1(τ,ψkt,τ (x)) ∫ t τ η x,k t,θ dŵ(θ)dτ], (3.45) −2∇pk+1(t,x) = ∫ ∞ 0 1 τ E[γk(t,x + B(τ))B(τ)]dτ, (3.46) where γk = Tr[∇uk]2. 90 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) Now we can prove the following assertion. Theorem 3.9 Assume that C 3.2 holds. Then if k → ∞ the functions uk(t),∇uk(t,x) deter- mined by (3.11) and (3.45) uniformly converge on compacts to limiting functions u(t), ∇u(t) satisfying (3.4) and (3.8)and u(t) ∈ C([0,T1], C1,α), ∇u(t) ∈ C([0,T1], C0,α), 0 < α ≤ 1 for all t ∈ [0,T1]. Here [0,T1] is the interval where the solution (α(t),β(t)) of the system (3.41), (3.42) is bounded. In addition the estimates ‖∇u(t)‖∞ ≤ α(t) , ‖∇u(t)‖q,loc ≤ β(t) hold for 1 < q < 32, t ∈ [0,T1]. Proof. By theorem 3.7 we know that the mapping Φ(t,x,g) = E [ u0(ψ g t,0(x)) − ∫ t 0 ∇pg(τ,ψgt,τ(x))dτ ] acts in the space C1,α ∩ Lq,loc ∩ Lm,loc (for a fixed t ∈ [0.T1]) with 1 < q < 32 < 3 < m < ∞. Consider the successive approximations (3.10) –(3.14) and (3.44) – (3.46), set Sk+1(t,x) = ‖uk+1(t,x) − uk(t,x)‖, n k+1(t,x) = ‖∇uk+1(t,x) − ∇uk(t,x)‖ and l k(t) = ‖Sk(t)‖∞, mkr(t) = ‖Sk(t)‖r,loc, ρk(t) = ‖nk(t)‖∞, ζkr (t) = ‖nk(t)‖r,loc. Then we obtain n k+1(t,x) ≤ L10(E[‖ψkt,0(x) − ψk−1t,0 (x)‖‖η x,k t,0 ‖]+ E[‖ψkt,0(x)‖‖η x,k t,0 − η x,k−1 t,0 ‖]) + ∫ t 0 1 σ(t − τ) E[‖∇pk+1(τ,ψkt,τ (x))− ∇pk(τ,ψk−1t,τ (x))‖‖ ∫ t τ η x,k t,θ dŵ(θ)‖]dτ+ ∫ t 0 1 σ(t − τ) E [ ‖∇pk(τ,ψkt,τ (x))‖ ∫ t τ [η x,k t,θ − ηx,k−1 t,θ ]dŵ(θ)‖ ] dτ. (3.47) Recall that by lemmas 3.2, 3.3 we know that sup x E‖ψkt,0(x) − ψk−1t,0 (x)‖ ≤ ∫ t 0 [ ‖uk(τ) − uk−1(τ)‖∞]dτe R t 0 α(τ)dτ, sup x E‖ηx,kt,0 − η x,k−1 t,0 ‖ ≤ ∫ t 0 ‖∇uk(τ) − ∇uk−1(τ)‖∞dτe R t 0 α(τ)dτ + sup x ∫ t 0 E‖∇uk−1(τ,ψkt,τ (x)) − ∇uk−1(τ,ψk−1t,τ (x))‖dτe R t 0 α(τ)dτ and applying the estimates from theorem 3.7 we get ρ k+1(t) ≤ e R t 0 α(τ)dτ[L10 ∫ t 0 sup x E‖uk(τ,ψkt,τ(x)) − uk−1(τ,ψk−1t,τ (x))‖dτ + ∫ t 0 ρk(τ)dτ + sup x ∫ t 0 E‖∇uk−1(τ,ψkt,τ(x)) − ∇uk−1(τ,ψk−1t,τ (x))‖dτ]+ CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 91 ∫ t 0 C[‖∇uk(τ)∇uk(τ)‖q + ‖∇uk(τ)∇uk(τ)‖m](E‖ηk(τ) − ηk−1(τ)‖2∞) 1 2 σ √ t − τ dτ+ ∫ t 0 1 σ √ t − τ sup x E‖∇pk+1(τ,ψkt,τ(x)) − ∇pk(τ,ψk−1t,τ (x))‖2) 1 2 e R t τ α(θ)dθdτ. To derive the estimate for the last term of the above inequality we recall (see lemma 3.1 and lemma 3.4) that for 1 < q < 3 2 the estimate ‖∇pk(t,x) − ∇pk(t,y)‖ ≤ ‖∇2pk(t)‖∞‖x − y‖ ≤ C[‖γk(t)‖q,loc + [γk(t)]1,G]‖x − y‖ holds and as a result we obtain E‖∇pk(τ,ψkt,τ (x)) − ∇pk(τ,ψk−1t,τ (x))‖ ≤ C[β(τ) + s(τ)]E‖ψkt,τ (x) − ψk−1t,τ (x)‖. In addition ‖∇pk+1(t) − ∇pk(t)‖∞ ≤ Cqm[ ‖γk+1(t) − γk(t)‖q,loc+ ‖γk+1(t) − γk(t)‖m,loc] ≤ Cqmα(t)[ ‖∇uk+1(t) − ∇uk(t)‖q,loc+ ‖∇uk(t) − ∇uk−1(t)‖q,loc+ ‖∇uk+1(t) − ∇uk(t)‖m,loc + ‖∇uk(t) − ∇uk−1(t)‖m,loc]. It follows from (3.47) that nk+1(t,x) ≤ C(t)[ ∫ t 0 E‖∇uk(τ,ψkt,τ(x)) − ∇uk−1(τ,ψk−1t,τ (x))‖dτ+ ∫ t 0 nk(τ,x)dτ] + ∫ t 0 1 σ √ t − τ C1[‖∇uk(τ)∇uk(τ)‖q+ ‖∇uk−1(τ)∇uk−1(τ)‖m]r(E‖ηx,k(τ) − ηx,k−1(τ)‖2) 1 2 dτ + ∫ t 0 1 σ √ t − τ e R t τ α(θ)dθ(E‖∇pk+1(τ,ψkt,τ(x)) − ∇pk(τ,ψk−1t,τ (x))‖2) 1 2 dτ. By the Hölder inequality we derive that for any positive f(τ) ∈ Lr,loc and 1m1 + 1 r = 1 and m1 < 2 we have for any compact G ⊂ R3 ∫ K [ ∫ t 0 1 σ √ t − τ f(τ,x)dτ]rdx ≤ 1 σr t r(2−m1) 2m1 ∫ t 0 ∫ K f r(τ,x)dxdτ. (3.48) Then from (3.47) and (3.48) we have for r > 2 ζk+1r (t) ≤ C2(t)[ ∫ t 0 ∫ K [E‖uk(τ,ψkt,τ(x)) − uk−1(τ,ψk−1t,τ (x))‖rdxdτ]+ ∫ t 0 ζkr (τ)dτ + ∫ t 0 ∫ K ‖∇uk−1(τ,ψkt,τ (x)) − ∇uk−1(τ,ψk−1t,τ (x))‖rdxdτ] 92 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) + ∫ t 0 1 σ √ t − τ C[[‖∇uk(τ)∇uk−1(τ)‖q + ‖∇uk(τ)∇uk−1(τ)‖m]r ∫ K (E‖ηx,k(τ) − ηx,k−1(τ)‖2) r2 dx]dτ + ∫ t 0 1 σ √ t − τ e R t τ α(θ)dθ ∫ K (E‖∇pk+1(τ,ψkt,τ (x)) − ∇pk(τ,ψk−1t,τ (x))‖2) r 2 dxdτ. For the function mkr(t) = ‖uk(t) − uk−1(t)‖r,loc using the apriori estimates proved in lemmas 3.2 – 3.8 and theorem 3.9 we obtain m k+1 r (t) ≤ C(t)[( ∫ t 0 ∫ K E‖uk(τ,ψkt,τ(x)) − uk−1(τ,ψk−1t,τ (x))‖rdxdτ) 1 r +( 1 σ t 2−m1 2m1 ∫ t 0 ∫ K E‖∇uk+1(τ,ψkt,τ (x))∇uk(τ,ψkt,τ (x))− ∇uk−1(τ,ψkt,τ(x))∇uk−1(τ,ψk−1t,τ (x))‖rdxdτ) 1 r ] ≤ C1(t) [ ( ∫ t 0 mkr(τ)dτ ) 1 r + ( ∫ t 0 ∫ K α(τ)E‖ψkt,τ (x) − ψk−1t,τ (x)‖rdxdτ ) 1 r + 1 σ t 1 m1 − 1 2 ( ∫ t 0 [ρk+1(τ) + ρk(τ)]ζkr (τ)dτ ) 1 r ] . Since uk and ∇uk are proved to be uniformly bounded on [0,T1] and ‖∇u1(t, ·) − ∇u0(·)‖r,loc ≤ const < ∞, ‖u1(t, ·) − u0(·)‖r,loc ≤ const < ∞, both for r = m and r = q we obtain that there exists a bounded on [0,T1] positive function C2(t) such that the function κk(t) = ρk(t) + ζkm(t) + m k r satisfies the estimate κk(t) ≤ [C2(t)] k k! and hence limk→∞ κ k(t) = 0. Since all summands defining κk(t) are positive we deduce that all of them converges to 0 as k → ∞. As a results we deduce that for each t ∈ [0,T1) the family uk(t, ·) converges uniformly on compacts and the limiting function u(t, ·) ∈ C1,α ∩ Lm,loc. In addition, we can check that the limiting function ∇u(t,x) is Lipschitz continuous in x. In fact, by lemma 3.8 and theorem 3.9 for each t ∈ [0,T1], we have for any x,y ∈ G ‖∇uk(t,x) − ∇uk(t,y)‖ ≤ s(t)‖x − y‖, where s(t) and T1 were defined above in lemma 3.8 and the estimate is uniform in k. To prove the uniqueness of the solution of (3.4)-(3.6) constructed above we assume first that there exist two solutions u1(t,x), u2(t,x) to (3.4)-(3.6) possessing the same initial data u1(0,x) = u2(0,x) = u0(x). CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 93 Computations similar to those used to prove the convergence of the family (uk(t),∇uk(t)) allow to check that both [∇u1(t) − ∇u2(t)]∞ = 0 and ‖∇u1(t) − ∇u2(t)‖m,loc = 0. Finally, we know that a stochastic equation with Lipschitz coefficients has a unique solution of the Cauchy problem. This implies the uniqueness of the solution to (3.4)-(3.6). 2 Summarizing the above results we see that the following statement is valid. Theorem 3.10 Assume that C 3.2 holds. Then there exists a unique solution ψt,x(s),u(t,x),p(t,x) of the system (3.4)-(3.6) for all t from the interval [0,T1], with T1 given by theorem 3.7 and x ∈ K for any compact K ⊂ G. In addition the process ψt,x(s) is the Markov process in R3 and u ∈ C([0,T1], C 1,α(K)) ∩ C([0,T1], Lq,loc ∩ Lm,loc) for 1 < q < 32 < 3 < m < ∞. Proof. First we note that as soon as we know that u(t,x) is locally Lipschitz continuous by classical SDE theory we know that the silution ψt,0(x) of the equation (3.4) is the Markov process in R3. All other assertions of the theorem are already proved above. To fulfill our program we have to check that the functions u(t,x),p(t,x) that satisfy (3.5) and (3.6) define a weak solution of (1.5),(1.2). Let us come back to the Kunita theory of stochastic flows [5], [6] and recall that given a dis- tribution u0 ∈ D′ and a stochastic flow ψut,0 one can define a stochastic flow u0 ◦ ψut,0 as another distribution satisfying 〈u0 ◦ ψut,0,h〉 = 〈u0,h ◦ φu0,tJ0,t〉. Here φu0,t is the inverse flow to ψut,0. Since any locally integrable function is a distribution, given u0 and the solution ψt,0,u(t),p(t) of (3.4)-(3.6) constructed above we consider a process λ(t) ∈ D′ of the form λ(t) = u0 − ∫ t 0 ∇pu(τ) ◦ φu0,τdτ. Next we consider the process λ(t) ◦ ψut,0 = u0 ◦ ψut,0 − ∫ t 0 ∇pu(τ) ◦ ψut,τdτ and verify that a weak solution u(t) of (3.1) admits the representation u(t) = E[λ(t)◦ψut,0] and satisfies (3.2). By the generalized Ito formula we derive λ(t) ◦ ψut,0 = u0 + ∫ t 0 σ2 2 ∆[u(θ) ◦ ψuθ,0]dθ+ (3.49) ∫ t 0 ∇[u(θ) ◦ ψuθ,0]σdw(θ) − ∫ t 0 ∇[u(θ) ◦ ψuθ,0]u(θ)dθ − ∫ t 0 ∇pu(θ)dθ, where (3.49) is considered in a weak sense. Hence for Lu = −(u,∇)u + σ 2 2 ∆u and the test function h ∈ D we have E [ ∫ R3 ∫ t 0 L(u(τ) ◦ ψuτ,s(x))dτ · h(x)dx ] = (3.50) E [ ∫ t 0 〈u(τ) ◦ ψuτ,0,L∗h〉dτ ] = ∫ t 0 L〈E[u(τ ◦ ψuτ,0)],h〉dτ. 94 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) As a result we deduce from (3.49) and (3.50) u(t) = E[λ(t) ◦ ψut,0] = u0 + ∫ t 0 LE[u(τ) ◦ ψuτ,0]dτ − ∫ t 0 ∇pu(τ)dτ. Differentiating each term with respect to t we check that the function u(t) = E[λ(t) ◦ ψut,0] solves the Cauchy problem (1.1). As soon as the function p(t) was constructed as a solution of the Poisson equation (3.2) we can verify that (1.2) holds as well. To summarize the obtained results we state the following: Theorem 3.11. Assume that C 3.2 holds. Then the functions u(t), p(t) that solve (3.5),(3.6) satisfy (3.1)-(3.2) in a weak sense for t ∈ [0,T1] where T1 is defined in theorem 3.9. Remark 3.13. We have proved that under condition C 3.2 the system (3.4)-(3.6) gives rise to a weak solution of (3.1)-(3.2). Moreover, when the initial data are smoother, say u0 ∈ C2,α, α ∈ [0, 1] similar considerations can be applied to verify that the pair u(t,x),p(t,x) given by (3.5)-(3.6) stands for a classical C2-smooth solution of (3.1), (3.2). 4 Lagrangian and stochastic approach to the N-S system The probabilistic approach developed in the previous section is in a sense an analogue of the Lagrangian approach developed for the Euler system which coincides with (1.1), (1.2) when σ = 0. The classical Lagrangian path starting at y is governed by the Newton equation ∂2φ̃0,t(y) ∂t2 = F φ̃ (t,y). (4.1) The force F in (4.1) has the form F φ̃ (t,y) = −∇p(t, φ̃0,t(y)) = −[(∇φ̃0,t(y))∗]−1∇[p(t, φ̃0,t(y))] (4.2) and the incompressibility condition yields det(∇φ̃0,t(y)) = 1. One can deduce from (4.1) that ∂ ∂t [ ∂φ̃k0,t(y) ∂t ∂φ̃k0,t(y) ∂yi ] = − ∂q(t, φ̃0,t(y)) ∂yi , (4.3) where q(t,y) = p(t,y) − 1 2 ‖ ∂φ̃0,t(y) ∂t ‖2 (4.4) summation over repeated indices is assumed. Integrating (4.3) in time we get ∂φ̃k0,t(y) ∂t ∂φ̃k0,t(y) ∂yi = u0(y) − ∂n(t, φ̃0,t(y)) ∂yi , (4.5) where n(t,y) = ∫ t 0 q(τ,y)dτ (4.6) and u0(y) = ∂φ̃0,t(y) ∂t |t=0 is the initial velocity. CUBO 12, 2 (2010) Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes 95 Consider the inverse diffeomorphism ψ̃t,0 = [φ̃0,t] −1, come back to (4.4), multiply it by [∇ψ̃t,0] and put y = ψ̃t,0(x). As a result we obtain by the chain rule the relation u i(t,x) = (u j 0(ψ̃t,0(x))∇xiψ̃ j t,0(x) − ∫ t 0 ∇xiq(τ,ψ̃t,τ (x))dτ. (4.7) Hence the Euler equations are equivalent to the system consisting of (4.7) and the relation ∆n(t,x) = ∂ ∂xi {uk0(ψ̃t,0(x)) ∂ψ̃ k t,0(x) ∂xi }, where n is given by (4.6). Finally due to divu = 0 one can rewrite the equation of state (4.7) in the form u(t) = P{u0(ψ̃t,0)∇ψ̃t,0} = P{[∇ψ̃t,0]∗u0(ψ̃t,0)}, (4.8) where P = I − ∇∆−1∇ is the Leray projector. The Euler pressure is determined up to additive constants by p(t,x) = ∂n(t,x) ∂t + (u(t,x),∇)n(t,x) + 1 2 ‖u(t,x)‖2. When σ 6= 0 one can develop an analogue of the Lagrange approach as follows. Let us choose φ0,t to be generated by the stochastic equation dφ0,θ = u(θ,φ0,θ)dθ + σdw(θ), φ0,0(y) = y, (4.9) next set ψθ,0 = [φ0,θ] −1, (4.10) and finally obtain the closed system by choosing u(t) = EP[(∇ψt,0)(u0 ◦ ψt,0)]. (4.11) The system (4.9) – (4.11) was studied by Constantin and Iyer [9], [10]. In [14] the existence and uniqueness of the solution to (4.9) – (4.11) was proved by the successive approximation technique. The main result due to Constantin and Iyer reads as follows: Theorem 4.1 Let k ≥ 1 and u0 ∈ Ck+1,α be divergence free. Then there exists a time interval [0,T ] with T = T (k,α,L,‖u0‖k+1,α) but independent of viscosity σ and a pair φ0,t(x),u(t,x) such that u ∈ C([0,T ],Ck+1,α) and (u,φ) satisfy (4.9)-(4.11). Further there exists U = U(k,α,L,‖u0‖k+1,α) such that ‖u(t)‖k+1,α ≤ U for t ∈ [0,T ] and u satisfies the N-S system. Comparing the system (3.4) – (3.6) and the system (4.9) – (4.11) we can check that the process ψt,0 given by (4.10) has the same distribution as the solution of (3.4). At the other hand the representations for the velocity u and the pressure p in the above systems are different. In the system (3.4) – (3.6) we avoid using the Leray projection and use instead the probabilistic representation of the Poisson equation. This allows us to construct both strong (classical) and weak (distributional) solutions of the Cauchy problem for the N-S system. At the very end we remark that the approach developed in the previous section does not allow to construct a solution to the Euler system as a limit of the solution to (1.1), (1.2) when σ goes to 0, since the appriori estimates in lemma 3.5 and lemma 3.8 cease to be valid. Acknowledgement. The authors gratefully acknowledge the financial support of DFG Grant 436 RUS 113/823. Received: February 2009. Revised: March 2009. 96 S. Albeverio and Ya. Belopolskaya CUBO 12, 2 (2010) References [1] S. Albeverio and Ya. Belopolskaya, Probabilistic approach to hydrodynamic equations. In the book Probabilistic Methods in Hydrodynamics. World Scientific (2003) 1-21. [2] Y. Le Jan and A. Sznitman, Stochastic cascades and 3-dimensional Navier Stokes equations, Prob. Theory Relat. Fields, 109 (1997) 343-366. [3] M. Ossiander, A probabilistic representation of solution of the incompressible Navier-Stokes equations in R3. Prob. Theory Relat. Fields, 133 2 (2005) 267-298. [4] B. Busnello, F. 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