A Stochastic Representation for Backward Incompressible Navier-Stokes Equations
aa r X i v : . [ m a t h . P R ] N ov A STOCHASTIC REPRESENTATION FOR BACKWARDINCOMPRESSIBLE NAVIER-STOKES EQUATIONS
XICHENG ZHANG
School of Mathematics and StatisticsThe University of New South Wales, Sydney, 2052, Australia,Department of Mathematics, Huazhong University of Science and TechnologyWuhan, Hubei 430074, P.R.ChinaEmail: [email protected]
Abstract.
By reversing the time variable we derive a stochastic representation for back-ward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths,which is similar to Constantin and Iyer’s forward formulations in [6]. Using this represen-tation, a self-contained proof of local existence of solutions in Sobolev spaces are providedfor incompressible Navier-Stokes equations in the whole space. In two dimensions or largeviscosity, an alternative proof to the global existence is also given. Moreover, a large de-viation estimate for stochastic particle trajectories is presented when the viscosity tendsto zero. Introduction
The classical Navier-Stokes equations describe the evolution of velocity fields of anincompressible fluid, and takes the following form with the external force zero: (cid:26) ∂ t u + ( u · ∇ ) u − ν ∆ u + ∇ p = 0 , t > , ∇ · u = 0 , u (0) = u , (1)where column vector field u = ( u , u , u ) t denotes the velocity field, p is the pressure and ν is the kinematic viscosity. When the viscosity ν vanishes, the above equation becomesthe classical Euler equation: (cid:26) ∂ t u + ( u · ∇ ) u + ∇ p = 0 , t > , ∇ · u = 0 , u (0) = u , (2)which describes the motion of an ideal incompressible fluid. The mathematical theoryabout Navier-Stokes equations and Euler equations has been extensively studied and theexistence of regularity solutions is still a big open problem in modern PDEs.Recently, Constantin and Iyer [6] presented an elegant stochastic representation for in-compressible Navier-Stokes equations based on stochastic particle paths, which is realizedby an implicit stochastic differential equation: the drift term is computed as the expectedvalue of an expression involving the stochastic flow defined by itself. More precisely, let( u, X ) solve the following stochastic system: X t ( x ) = x + Z t u s ( X s ( x ))d s + √ νB t , t > ,A t = X − t ,u t = E P [( ∇ t A t )( u ◦ A t )] , (3) Keywords:
Backward Navier-Stokes Equation, Stochastic Representation, Global Existence, LargeDeviation. here B t is a 3-dimensional Brownian motion, P is the Leray-Hodge projection ontodivergence free vector fields, and ∇ t A t denotes the transpose of Jacobi matrix ∇ A t . Then u satisfies equation (1) with initial data u . One of the proofs given in [6] is basedon a stochastic partial differential equation satisfied by the inverse flow A . By using thisrepresentation, a self-contained proof of the existence of local smooth solutions is providedin [12, 13].Let ( u, p ) solve (1). Notice that if we make the time change:˜ u ( t, x ) := − u ( − t, x ) , ˜ p ( t, x ) = p ( − t, x ) for t , then ˜ u satisfies the following equation (called backward Navier-Stokes equation here): (cid:26) ∂ t ˜ u + (˜ u · ∇ )˜ u + ν ∆˜ u + ∇ p = 0 , t , ∇ · ˜ u = 0 , ˜ u (0) = u . (4)The purpose of the present paper is to give a slightly different representation for ˜ u byusing backward particle paths. More precisely, let (˜ u, X ) solve the following stochasticsystem X t,s ( x ) = x + Z st ˜ u r ( X t,r ( x ))d r + √ ν ( W s − W t ) , t s , ˜ u t = E P [( ∇ t X t, )( u ◦ X t, )] , t . (5)Then ˜ u satisfies backward incompressible Navier-Stokes equation (4) with final value u .Intuitively, by reversing the time, the starting point is changed as the end point. Hence,representation (5) is essentially the same as (3). But, equation (5) is easier to be dealtwith mathematically since the unpleasant term A = X − which usually incurs extramathematical calculations does not appear in (5). Such a representation will be provedin Section 2. We emphasize that representations (3) and (5) are useful in numericalcomputations (cf. [19, 15]). Direct calculations shows that the second equation in (5) isequivalent to ˜ ω t = ∇ × ˜ u t = E [( ∇ − X t, )( ω ◦ X t, )] , (6)˜ u t = − ∆ − ∇ × ˜ ω t , (7)where ˜ ω t is the vorticity, ∇ − X t, is the inverse of Jacobi matrix ∇ X t, and (7) is exactlythe Biot-Savart law.We also mention other stochastic formulations for incompressible Navier-Stokes equa-tions. In [9], a representation formula for the vorticity of three dimensional Navier-Stokes equations was given by using stochastic Largrangian paths, however, there is noa self-contained proof of the existence given there. In [18], Le Jan and Sznitman used abackward-in-time branching process in Fourier space to express the velocity field of a three-dimensional viscous fluid as the average of a stochastic process, which then leads to a newexistence theorem. In [3], basing on Girsanov’s transformation and Bismut-Elworty-Li’sformula, Busnello introduced a purely probabilistic treatment to the existence of a uniqueglobal solution for two dimensional Navier-Stokes equations, where the stretching termdisappears, and the non-linear equation obeyed by the vorticity has the form of Fokker-Planck equation. Later on, Busnello, Flandoli and Romito in [4] carefully analyzed animplicit probabilistic representation for the vorticity of three dimensional Navier-Stokesequations, and a local existence was given. In that paper, much attentions were also paidon a probabilistic representation formula for a general system of linear parabolic equa-tions. Moreover, in [3, 4], an interesting probabilistic representation for the Biot-Savartlaw was also given and analyzed so that they can recover the velocity from the vorticity y probabilistic approach. Recently, Cipriano and Cruzeiro in [5] described a stochasticvariational principle for two dimensional incompressible Navier-Stokes equations by usingthe Brownian motions on the group of homeomorphisms on the torus. More recently,Cruzeiro and Shamarova [7] established a connection between equation (4) and a systemof infinite dimensional forward-backward stochastic differential equations on the group ofvolume-preserving diffeomorphisms of a flat torus.This paper is organized as follows: In Section 3, we shall give a self-contained proofof local existence in Sobolev spaces. The proof is based on successive approximationor fixed point method as in [12]. Therein, Iyer considered the spatially periodic caseand worked in H¨older continuous function spaces. When Sobolev spaces are considered,we have to overcome the difficulty due to the non-closedness of Sobolev spaces underpointwise multiplications and compositions. Thus, it seems to be hard to exhibit thesame proof in Section 3 for representation (3) due to the presence of A = X − . A keypoint in the proof lies in that the flow map x X t,s ( x ) preserves the Lebesgue measure,i.e., det( ∇ X t,s ) = 1.In Section 4, we shall give an alternative proof to the global existence when the spatialdimension is two or the viscosity is large enough in any dimensions. Such results are wellknown. For two dimensional Navier-Stokes and Euler equations in the whole space, theglobal existence of smooth solutions are referred to [19]. The global existence of regularitysolutions for large viscosity is referred to [16]. Recently, Iyer [14] presented an alternativeproof to the global existence for small Reynolds. His proof is based on the decay of heatflows and stochastic representation (3). Following [14], we will give a different proof basedon Bismut’s formula and representation (5).Let ( u ν , X ν ) denote the solution of equation (5). In Section 5, as ν goes to zero, anasymptotic probability estimate of X ν in diffeomorphism group is presented by the wellknown large deviation estimate for stochastic diffeomorphism flows.2. Stochastic Representation of Backward Incompressible Navier-StokesEquations
We begin with some notational conventions. Fix d > N := { } ∪ N , R − := ( −∞ , , I := ( d × d )-unit matrixand l := n σ = ( σ k ) k ∈ N ∈ R N : k σ k l := X σ k < + ∞ o . For a differentiable transformation X of R d , the Jacobi matrix of X is given by ∇ X := ∂ X , ∂ X , · · · , ∂ d X ∂ X , ∂ X , · · · , ∂ d X · · · , · · · , · · · , · · · ∂ X d , ∂ X d , · · · , ∂ d X d , where ∂ i = ∂∂x i . We use ∇ t X to denote the transpose of ∇ X . For k ∈ N , let C kb ( R d ; R d )denote the space of k -order continuous differentiable vector fields on R d with the norm: k u k C kb := X | α | k sup x ∈ R d | D α u ( x ) | < + ∞ , where D α denotes the derivative with respect to the multi index α . et u ∈ C ( R − ; C b ( R d ; R d )) and t σ t = σ ( t ) ∈ L loc ( R − ; R d × l ) satisfy ∞ X k =1 σ · k ( t ) σ · k ( t ) = I . (8)Let { X t,s ( x ) , t s } solve the following SDE X t,s ( x ) = x + Z st u r ( X t,r ( x ))d r + √ ν Z st σ r d B r , (9)where B t := { B kt , t , k ∈ N } is a sequence of independent standard Brownian motionson some probability space (Ω , F , P ). Thanks to (8), the diffusion operator associated toequation (9) is given by L t g ≡ ν ∆ g + ( u t · ∇ ) g. We first prove the following result.
Theorem 2.1.
Let φ : R d → R d be a C -vector field satisfying | D α φ ( x ) | C (1 + | x | β ) , | α | , β > , and f ∈ C ( R − ; C b ( R d ; R d )) . Define Λ φ ( t, x ) := ( ∇ t X t, ( x )) φ ( X t, ( x )) , Λ f ( t, x ) := Z t ( ∇ t X t,r ( x )) f r ( X t,r ( x ))d r and w t ( x ) := E Λ φ ( t, x ) − E Λ f ( t, x ) . Then w ∈ C , (( −∞ , × R d ) satisfies the following backward Kolmogorov’s equation: ∂ t w t + L t w t + ( ∇ t u t ) w t = f t , lim t ↑ w t ( x ) = φ ( x ) . (10) Proof.
Let g be a twice continuously differentiable function satisfying | D α g ( x ) | C (1 + | x | β ) , | α | , β > . For h >
0, by Itˆo’s formula we have E g ( X t − h,t ( x )) = g ( x ) + E (cid:20)Z tt − h ( L r g )( X t − h,r ( x ))d r (cid:21) . From this, it is easy to see that1 h h E g ( X t − h,t ( x )) − g ( x ) i = 1 h E (cid:20)Z tt − h ( L r g )( X t − h,r ( x ))d r (cid:21) → ( L t g )( x ) as h → . (11)Noticing that X t − h, ( x ) = X t, ◦ X t − h,t ( x ) , we have ∇ X t − h, ( x ) = ( ∇ X t, ) ◦ X t − h,t ( x ) · ∇ X t − h,t ( x ) . Thus, by the independence of X t − h,t ( x ) with X t, ( x ), we have E Λ φ ( t − h, x ) = E [( ∇ t X t − h,t ( x ))Λ φ ( t, X t − h,t ( x ))]= E h ( ∇ t X t − h,t ( x ))(( E Λ φ ( t )) ◦ X t − h,t ( x )) i and E Λ f ( t − h, x ) = E (cid:2) ( ∇ t X t − h,t ( x ))Λ f ( t, X t − h,t ( x )) (cid:3) E (cid:20)Z tt − h ( ∇ t X t − h,r ( x )) f ( r, X t − h,r ( x ))d r (cid:21) = E (cid:2) ( ∇ t X t − h,t ( x ))(( E Λ f ( t )) ◦ X t − h,t ( x )) (cid:3) + E (cid:20)Z tt − h ( ∇ t X t − h,r ( x )) f r ( X t − h,r ( x ))d r (cid:21) . Hence, we may write w t − h ( x ) = E h ( ∇ t X t − h,t ( x )) w t ( X t − h,t ( x )) i − E (cid:20)Z tt − h ( ∇ t X t − h,r ( x )) f r ( X t − h,r ( x ))d r (cid:21) and 1 h ( w t ( x ) − w t − h ( x )) = − h E h ( ∇ t X t − h,t ( x ) − I ) w t ( X t − h,t ( x )) i − h h E ( w t ( X t − h,t ( x ))) − w t ( x ) i + 1 h E (cid:20)Z tt − h ( ∇ t X t − h,r ( x )) f r ( X t − h,r ( x ))d r (cid:21) =: I h ( t, x ) + I h ( t, x ) + I h ( t, x ) . Observing that ∇ X t − h,t ( x ) − I = Z tt − h ( ∇ u s ) ◦ X t − h,s ( x ) · ∇ X t − h,s ( x )d s. we deduce lim h ↓ I h ( t, x ) = − [( ∇ t u t ) w t ]( x ) . By (11) we have lim h ↓ I h ( t, x ) = − ( L t w t )( x ) . Moreover, a simple limit procedure also giveslim h ↓ I h ( t, x ) = f t ( x ) . Combining the above calculations, we conclude thatlim h ↓ h ( w t ( x ) − w t − h ( x )) = − ( L t w t )( x ) − [( ∇ t u t ) w t ]( x ) + f t ( x ) . Equation (10) now follows (see [11, p.124] for more details). (cid:3)
Remark 2.2.
A more general Feynman-Kac formula for a deterministic system of para-bolic equations was given in [4] . However, the proof is simpler in our case. In representa-tion (3), if we define w t := E [( ∇ t A t )( u ◦ A t )] , then w t also satisfies (10) with f = 0 (see [6, p.343, (4.5)] ). Basing on this theorem, we can give a stochastic representation for backward Navier-Stokes equation (4) as in [6]. heorem 2.3. Let ν > and u ∈ C b ( R d ; R d ) a deterministic divergence-free vector field,and f ∈ C ( R − ; C b ( R d ; R d )) . Suppose that σ satisfies (8), and ( u, X ) solves the stochasticsystem: X t,s ( x ) = x + Z st u r ( X t,r ( x ))d r + √ ν Z st σ r d B r , t s , (12) u t = P E Λ uu ( t ) − P E Λ uf ( t ) , t , (13) where P is the Leray-Hodge projection onto divergence free vector fields, and Λ uu and Λ uf are given by Λ uu ( t, x ) := ( ∇ t X t, ( x )) u ( X t, ( x )) , Λ uf ( t, x ) := Z t ( ∇ t X t,r ( x )) f r ( X t,r ( x ))d r. Then u satisfies the backward incompressible Navier-Stokes equation: (cid:26) ∂ t u + ( u · ∇ ) u + ν ∆ u + ∇ p = f, t , ∇ · u = 0 , u (0 , x ) = u ( x ) . (14) Conversely, if u solves backward Navier-Stokes equation (14), then u is given by (13).Proof. First of all, let w t ( x ) := E Λ uu ( t, x ) − E Λ uf ( t, x ) . (15)By Theorem 2.1, w ( t, x ) = w t ( x ) solves the following backward Kolmogorov’s equation: ∂ t w + ( u · ∇ ) w + ( ∇ t u ) w + ν ∆ w = f, w (0 , x ) = u ( x ) . (16)In view of u = P w , we may write w = u + ∇ q. Substituting it into equation (16), one finds that ∂ t u + ( u · ∇ ) u + ν ∆ u + ∇ p = f, u (0 , x ) = u ( x ) , where p = ∂ t q + ( u · ∇ ) q + ν ∆ q + 12 | u | . Conversely, let ( u, p ) solve (14). As above, if we define w by (15), then w satisfiesequation (16). We need to show that u = P w , or equivalently, for some scalar valuedfunction q v := w − u = ∇ q. By (16) and (14), v solves the following equation: ∂ t v + ( u · ∇ ) v + ( ∇ t u ) v + ν ∆ v = ∇ p − ∇| u | , v (0 , x ) = 0 . (17)Let q ( t, x ) := E (cid:18)Z t h | u ( r, X t,r ( x )) | − p ( r, X t,r ( x )) i d r (cid:19) . Then q solves the following equation (cf. [11]): ∂ t q + ( u · ∇ ) q + ν ∆ q = p − | u | , q (0 , x ) = 0 . Taking gradients for both sides of the above equation yields ∂ t ∇ q + ( u · ∇ ) ∇ q + ( ∇ t u )( ∇ q ) + ν ∆ ∇ q = ∇ p − ∇| u | . y the uniqueness of solutions to linear equation (17), v = ∇ q . (cid:3) Remark 2.4.
In the above proof, we have assumed that the solutions are regular enoughso that all the calculations are valid. The existence of regular solutions will be proven inthe next section. A Proof of Local Existence in Sobolev Spaces
With a little abuse of notations, in this and next sections we use p to denote theintegrability index since the pressure will not appear below. For k ∈ N and p > W k,p ( R d ; R d ) be the usual R d -valued Sobolev space on R d , i.e, the completion of C ∞ ( R d ; R d ) with respect to the norm: k u k k,p := k u k p + k X j =1 k∇ j u k p , where k · k p is the usual L p -norm, and ∇ j is the j -order gradient operator. Note that W ,p ( R d ; R d ) = L p ( R d ; R d ) and the following Sobolev’s embedding holds: for p > d (cf.[10]) W ,p ( R d ; R d ) ֒ → L ∞ ( R d ; R d ) , i.e. k · k ∞ c k · k ,p , (18)where c = c ( p, d ). Below, we shall use c to denote a constant which may change in differentoccasions, and whose dependence on parameters can be traced carefully from the context.Let W k,ploc ( R d ; R d ) be the local Sobolev space on R d . We introduce the following Banachspace of transformations of R d : X k +2 ,p := n X ∈ W k +2 ,ploc ( R d ; R d ) : | X (0) | + k∇ X k ∞ + k∇ X k k,p < + ∞ o . Definition 3.1.
The Weber operator W : L p ( R d ; R d ) × X ,p → L p ( R d ; R d ) is defined by W ( v, ℓ ) := P [( ∇ t ℓ ) v ] , where P is the Leray-Hodge projection onto divergence free vector fields. Remark 3.2. P = I − ∇ ( − ∆) − div is a singular integral operator(SIO) which is boundedin L p -space for p ∈ (1 , ∞ ) (cf. [22] ). We now prepare several lemmas for later use.
Lemma 3.3. (i) For any k ∈ N and p > d , there exists a constant c = c ( k, p, d ) > such that for all v ∈ W k +2 ,p ( R d ; R d ) and ℓ ∈ X k +2 ,p , k∇ W ( v, ℓ ) k k +1 ,p c ( k∇ ℓ k ∞ + k∇ ℓ k k,p ) · k∇ v k k +1 ,p (19) (ii) For p > d , there exists a constant c = c ( p, d ) > such that for all v , v ∈ W ,p ( R d ; R d ) and ℓ , ℓ ∈ X ,p with ℓ − ℓ ∈ L p ( R d ; R d ) k W ( v , ℓ ) − W ( v , ℓ ) k p c (cid:16) k v k ,p k ℓ − ℓ k p + k∇ ℓ k ∞ k v − v k p (cid:17) . (20) Proof. (i) Noting that P (( ∇ t ℓ ) v ) + P (( ∇ t v ) ℓ ) = P ( ∇ ( ℓ · v )) = 0 , (21)we have ∂ i W ( v, ℓ ) = P [( ∇ t ∂ i ℓ ) v + ( ∇ t ℓ ) ∂ i v ] = P [ − ( ∇ t v ) ∂ i ℓ + ( ∇ t ℓ ) ∂ i v ]and ∂ j ∂ i W ( v, ℓ ) = P [ − ( ∇ t v ) ∂ j ∂ i ℓ − ( ∇ t ∂ j v ) ∂ i ℓ + ( ∇ t ℓ ) ∂ j ∂ i v + ( ∇ t ∂ j ℓ ) ∂ i v ] . ence, by (18) we have k ∂ i W ( v, ℓ ) k p c ( k ( ∇ t v ) ∂ i ℓ k p + k ( ∇ t ℓ ) ∂ i v k p ) c ( k∇ v k p · k ∂ i ℓ k ∞ + k∇ ℓ k ∞ k ∂ i v k p ) c k∇ v k p · k∇ ℓ k ∞ and k ∂ j ∂ i W ( v, ℓ ) k p c ( k∇ ℓ k p · k∇ v k ∞ + k∇ ℓ k ∞ · k∇ v k p ) c ( k∇ ℓ k p · k∇ v k ,p + k∇ ℓ k ∞ · k∇ v k p ) c ( k∇ ℓ k ∞ + k∇ ℓ k p ) · k∇ v k ,p , which produces k∇ W ( v, ℓ ) k ,p c ( k∇ ℓ k ∞ + k∇ ℓ k p ) · k∇ v k ,p . The higher derivatives can be estimated similarly.(ii) By (21), we have W ( v , ℓ ) − W ( v , ℓ ) = P (( ∇ t ( ℓ − ℓ )) v ) + P (( ∇ t ℓ )( v − v ))= − P (( ∇ t v )( ℓ − ℓ )) + P (( ∇ t ℓ )( v − v )) . So, k W ( v , ℓ ) − W ( v , ℓ ) k p c k ( ∇ t v )( ℓ − ℓ ) k p + c k ( ∇ t ℓ )( v − v ) k p c k∇ v k ∞ k ℓ − ℓ k p + c k∇ ℓ k ∞ k v − v k p , which yields (20) by (18). (cid:3) Lemma 3.4. (i) For k ∈ N and p > d , there exist constants c = c ( k, p, d ) > and α k ∈ N such that for all u ∈ W k +2 ,p ( R d ; R d ) and all X ∈ X k +2 ,p preserving the volume, k∇ ( u ◦ X ) k k +1 ,p c k∇ u k k +1 ,p (1 + k∇ X k k +2 ∞ + k∇ X k α k k,p ) . (22) (ii) For p > d , there exists a constant c = c ( p, d ) > such that for all u ∈ W ,p ( R d ; R d ) and X, ˜ X ∈ X ,p with X − ˜ X ∈ L p ( R d ; R d ) , k u ◦ X − u ◦ ˜ X k p c k∇ u k ,p · k X − ˜ X k p . (23) Proof. (i) Since X preserves the volume, we have k u ◦ X k p = k u k p . (24)Observe that for m > ∇ k ( u ◦ X ) = ( ∇ m u ) ◦ X · ( ∇ X ) m + · · · + ( ∇ u ) ◦ X · ∇ m X. (25)(22) follows by (18) and (24).(ii) It follows from u ◦ X − u ◦ ˜ X = Z ( ∇ u ) ◦ ( sX + (1 − s ) ˜ X )) · ( X − ˜ X )d s (26)and (18). (cid:3) Lemma 3.5.
For k ∈ N , U > and T := U , there exist constants c = c ( p, d ) > and c = c ( k, p, d ) > such that for any divergence free vector field u ∈ C ([ − T, W k +2 ,p ) satisfying sup t ∈ [ − T, k∇ u t k k +1 ,p U , the solution X t,s to (12) belongs to X k +2 ,p a.s., andfor all t ∈ [ − T, k∇ X t, k ∞ c , k∇ X t, k k,p c . (27) roof. Noting that ∇ X t,s = I + Z st ( ∇ u r ) ◦ X t,r · ∇ X t,r d r, (28)we have k∇ X t,s k ∞ Z st k∇ X t,r k ∞ · k∇ u r k ∞ d r. By Gronwall’s inequality, we obtain by (18)sup − T t s k∇ X t,s k ∞ exp (cid:20)Z − T k∇ u r k ∞ d r (cid:21) e cUT = e c =: c . (29)On the other hand, from (28), an elementary calculation shows thatdet( ∇ X t,s ) = exp (cid:20)Z st ( ∇ · u r ) ◦ X t,r d r (cid:21) = 1 . So, x X t,s ( x ) preserves the volume. Now, ∇ X t,s = Z st [( ∇ u r ) ◦ X t,r · ( ∇ X t,r ) + ( ∇ u r ) ◦ X t,r · ∇ X t,r ]d r. Hence, by (29) and (18) k∇ X t,s k p c Z st (cid:16) k∇ u r k p k∇ X t,r k ∞ + k∇ u r k ∞ k∇ X t,r k p (cid:17) d r cU T + cU Z st k∇ X t,r k p d r. By Gronwall’s inequality again we get k∇ X t,s k p cU T e cUT = ce c =: c . Higher derivatives can be estimated similarly step by step. (cid:3)
Lemma 3.6.
For p > d and
T > , let u, ˜ u ∈ C ([ − T, W ,p ( R d ; R d )) , and X, ˜ X solve SDE (9) with drifts u and ˜ u respectively. Then for some c = c ( p, d ) > and any t ∈ [ − T, , k X t, − ˜ X t, k p exp h c sup t ∈ [ − T, k∇ u t k ,p i · Z t k u r − ˜ u r k p d r. (30) Proof.
We have X t,s ( x ) − ˜ X t,s ( x ) = Z st ( u r ( X t,r ( x )) − ˜ u r ( ˜ X t,r ( x )))d r = Z st ( u r ( X t,r ( x )) − u r ( ˜ X t,r ( x )))d r + Z st ( u r ( ˜ X t,r ( x )) − ˜ u r ( ˜ X t,r ( x )))d r. For
R >
0, let B R := { x ∈ R d : | x | R } . By virtue of x ˜ X t,r ( x ) preserving the volumeand formula (26), we have k X t,s − ˜ X t,s k L p ( B R ) Z st k u r ◦ X t,r − u r ◦ ˜ X t,r k L p ( B R ) d r + Z st k u r − ˜ u r k p d r sup r ∈ [ − T, k∇ u r k ∞ Z st k X t,r − ˜ X t,r k L p ( B R ) d r Z t k u r − ˜ u r k p d r, (31)By Gronwall’s inequality and (18), we get k X t,s − ˜ X t,s k L p ( B R ) exp h c sup t ∈ [ − T, k∇ u t k ,p i · Z t k u r − ˜ u r k p d r. Letting R go to infinity gives (30). (cid:3) We are now in a position to prove the following local existence result.
Theorem 3.7.
For ν > , k ∈ N and p > d , there exists a constant c = c ( k, p, d ) > independent of ν such that for any u ∈ W k +2 ,p ( R d ; R d ) divergence free and T :=( c k∇ u k k +1 ,p ) − , there is a unique pair ( u, X ) with u ∈ C ([ − T, W k +2 ,p ) satisfying X t,s ( x ) = x + Z st u r ( X t,r ( x ))d r + √ ν Z st σ r d B r , t s ,u t = P E [( ∇ t X t, )( u ◦ X t, )] , t . (32) Moreover, for any t ∈ [ − T, k∇ u t k k +1 ,p c k∇ u k k +1 ,p . (33) Proof.
Set u r ( x ) := u ( x ). Consider the following Picard’s iteration sequence X nt,s ( x ) = x + Z st u nr ( X nt,r ( x ))d r + √ ν Z st σ ( r )d B r , t s ,u n +1 t = P E [( ∇ t X nt, )( u ◦ X nt, )] , t . (34)Noting that P E [( ∇ t X nt, )( u ◦ X nt, )] = E W ( u ◦ X nt, , X nt, ) , we have by (19) and (22) k∇ u n +1 t k k +1 ,p E k∇ W ( u ◦ X nt, , X nt, ) k k +1 ,p c E h ( k∇ X nt, k ∞ + k∇ X nt, k k,p ) · k∇ ( u ◦ X nt, ) k k +1 ,p i c E h (1 + k∇ X nt, k k +3 ∞ + k∇ X nt, k β k k,p ) · k∇ u k k +1 ,p i , where β k ∈ N only depends on k and c = c ( k, p, d ) > c := c (1 + c k +31 + c β k ) > , where c and c are from Lemma 3.5. Choosing U := c k∇ u k k +1 ,p and T := 1 /U inLemma 3.5, we have by induction and Lemma 3.5sup t ∈ [ − T, k∇ u nt k k +1 ,p U, ∀ n ∈ N . (35)On the other hand, we also have by (27) k u n +1 t k p c E k ( ∇ t X nt, )( u ◦ X nt, ) k p c E h k∇ t X nt, k ∞ k u ◦ X nt, k p i c k u k p , which together with (35) gives the following uniform estimate:sup n ∈ N sup t ∈ [ − T, k u nt k k +2 ,p < + ∞ . (36) ow by (20) and (23) (30), we have k u n +1 t − u m +1 t k p c E h k u ◦ X nt, k ,p · k X nt, − X mt, k p + k∇ X mt, k ∞ · k u ◦ X nt, − u ◦ X mt, k p i c Z t k u nr − u mr k p d r, where c = c ( p, d, U ) is independent of n, m . From this we derive thatlim sup n,m →∞ sup t ∈ [ − T, k u nt − u mt k p = 0 . By (36) and interpolation inequality, we further havelim sup n,m →∞ sup t ∈ [ − T, k u nt − u mt k k +1 ,p = 0 . Therefore, there is a u ∈ C ([ − T, W k +1 ,p ( R d ; R d )) such thatlim sup n →∞ sup t ∈ [ − T, k u nt − u t k k +1 ,p = 0 . Taking limits for (34), one finds that u is a solution of (32). Estimate (33) follows from(35). (cid:3) Remark 3.8.
The constant c in (33) is usually strictly greater than . If c equals ,then we can invoke the standard bootstrap method to obtain the global existence. This willbe studied in the next section when the periodic boundary is considered and the viscosityis large enough. Since the existence time interval in Theorem 3.7 is independent of the viscosity ν , wealso obtain the local existence of solutions to Euler equation (2). Moreover, as ν →
0, thesolution of Navier-Stokes equation converges to the solution of Euler equation as givenbelow.
Proposition 3.9.
Keep the same assumptions as in Theorem 3.7. For ν > and u ∈ W k +2 ,p ∩ L , let ( u ν , X ν ) be the solution of (32) corresponding to viscosity ν and initialvalue u . Then for any j = 0 , · · · , k + 1 , there exists c = c ( k, j, p, d, k u k k +2 ,p , k u k ) > such that for all ν > and t ∈ [ − T, k u νt − u t k C jb c ( ν | t | ) ( k +2 − jd − p ) / ( + k +2 d − p ) . Proof.
Note that u ∈ W k +2 ,p ∩ L guarantees u νt ∈ W k +2 ,p ∩ L . By ∂ t ( u νt − u t ) + νu νt + P [( u νt · ∇ ) u νt − ( u t · ∇ ) u t ] = 0 , we have − ∂ t k u νt − u t k = ν h ∆ u νt , u νt − u t i + h ( u νt · ∇ ) u νt − ( u t · ∇ ) u t , u νt − u t i = ν h ∆ u νt , u νt − u t i + h (( u νt − u t ) · ∇ ) u νt , u νt − u t i ν k ∆ u νt k · k u νt − u t k + k∇ u νt k ∞ k u νt − u t k , i.e., − ∂ t k u νt − u t k ν k ∆ u νt k + k∇ u νt k ∞ k u νt − u t k . By Gronwall’s inequality and (33) we obtain k u νt − u t k ν Z t k ∆ u νs k d s · exp (cid:20)Z t k∇ u νs k ∞ d s (cid:21) cν | t | . he desired estimate now follows by the Sobolev embedding(cf. [10]): for u ∈ W k +2 ,p ∩ L k∇ j u k ∞ c k,j,p,d k u k αk +2 ,p k u k − α , where α = ( jd + ) / ( + k +2 d − p ). (cid:3) Remark 3.10.
We cannot prove a convergence rate O ( √ νt ) as in [12] starting from (32)because x ( X νt, ( x ) − X t, ( x )) does not belong to any L p -spaces. Existence of Global Solutions
Global Existence in Two Dimensions.
First of all, we recall the following Beale-Kato-Majda’s estimate about SIOs, which can be proved as in [19, p.117, Proposition 3.8],we omit the details.
Lemma 4.1.
For p > d , let u ∈ W ,p ( R d ; R d ) be a divergence free vector field and ω := curl u . Then, for some c = c ( p, d ) k∇ u k ∞ c (1 + log + k ω k ,p )(1 + k ω k ∞ ) , (37) where log + x := max { log x, } for x > . In two dimensional case, taking the curl for the second equation in (32), one finds that ω t := curl u t := ∂ u t − ∂ u t = E [ ω ◦ X t, ] . (38)From this, we clearly have k ω t k p k ω k p , p ∞ . (39)Basing (37) and representation (38), we may prove the following global existence for 2DNavier-Stokes and Euler equations. Theorem 4.2.
In two dimensions, for ν > , k ∈ N , p > and u ∈ W k +2 ,p ( R ; R ) divergence free, there exists a unique global solution ( u, X ) to equation (32).Proof. We only need to prove the following a priori estimate: for all t ∈ R − k u t k k +2 ,p c ( k u k k +2 ,p , k, p, t ) < + ∞ , where c ( k u k k +2 ,p , k, p, t ) continuously depends on its parameters.Following the proof of Lemma 3.5, we have k∇ X t, k ∞ exp (cid:20)Z t k∇ u r k ∞ d r (cid:21) . (40)Noting that ∇ ω t = E ( ∇ ω ◦ X t, · ∇ X t, ) , we have k∇ ω t k p k∇ ω k p · E k∇ X t, k ∞ and by (39) and (40) k ω t k ,p k ω k ,p · (cid:18) (cid:20)Z t k∇ u r k ∞ d r (cid:21)(cid:19) . (41)Hence, by (37) (39) and (41) k∇ u t k ∞ c (1 + log + k ω t k ,p )(1 + k ω t k ∞ ) c + c Z t k∇ u r k ∞ d r, here c = c ( k ω k ,p , p ). By Gronwall’s inequality we obtain k∇ u t k ∞ ce c | t | . Substituting this into (40) and (41) gives k∇ X t, k ∞ e c | t | e c | t | , and by Calderon-Zygmund’s inequality about SIOs (cf. [22]) k∇ u t k ,p k ω t k ,p k ω k ,p · (cid:16) e c | t | e c | t | (cid:17) . Moreover, k u t k p c E ( k∇ X t, k ∞ · k u ◦ X t, k p ) c k u k p · e c | t | e c | t | . Thus, k u t k ,p c ( k u k ,p , p, t ) < + ∞ . Starting from (38) and as in Lemma 3.5, higher derivatives can be estimated similarly. (cid:3)
Global Existence for Large Viscosity.
In this section, we study the existenceof global solutions for large viscosity and work on the d -dimensional torus T d = R d / Z d .Let W k,p ( T d , R d ) be the R d -valued Sobolev spaces on T d with vanishing mean. Instead of(12), we consider X t,s ( x ) = x + Z st u r ( X t,r ( x ))d r + √ ν ( B s − B t ) , (42)where B is the standard Wiener process on Ω := C ( R − ; R d ), i.e., for ω ∈ Ω, B · ( ω ) = ω ( · ).We recall the following Bismut’s formula (cf. [2, 8]). For the reader’s convenience, ashort proof is provided here. Theorem 4.3.
For any t < and f ∈ C b ( T d ; R ) , it holds that ( ∇ E f ( X t, ))( x ) = 1 t √ ν E (cid:20) f ( X t, ( x )) Z t (cid:16) s ( ∇ u s ) ◦ X t,s ( x ) − I (cid:17) d B s (cid:21) . (43) In particular, for any p > d and some c = c ( p, d ) k∇ E f ( X t, ) k p c p ν | t | k f k p " | t | · sup s ∈ [ t, k∇ u s k ,p + 1 . (44) Proof.
Fix t < y ∈ R d below and define h ( s ) := 1 t √ ν (cid:20) ( t − s ) y + Z st [( ∇ u r ) ◦ X t,r ( x )] · ( ry )d r (cid:21) , s ∈ [ t, . Consider the Malliavin derivative of X t,s with respect to the sample path along the direc-tion h , i.e., D h X t,s ( x, ω ) = lim ε → X t,s ( x, εh + ω ) − X t,s ( x, ω ) ε , ω ∈ Ω . From (42) one sees that D h X t,s ( x ) = Z st [( ∇ u r ) ◦ X t,r ( x )] · D h X t,r ( x )d r + √ νh ( s )= ( t − s ) yt + Z st [( ∇ u r ) ◦ X t,r ( x )] · h D h X t,r ( x ) + ryt i d r. n the other hand, we have ∇ X t,s ( x ) · y = y + Z st [( ∇ u r ) ◦ X t,r ( x )] · ∇ X t,r ( x ) · y d r. By the uniqueness of solutions, we get ∇ X t,s ( x ) · y = D h X t,s ( x ) + syt . In particular, ∇ X t, ( x ) · y = D h X t, ( x ) . Now ∇ E f ( X t, ) · y = E (cid:2) [( ∇ f ) ◦ X t, ] · ∇ X t, · y (cid:3) = E (cid:2) ( ∇ f ) ◦ X t, · D h X t, (cid:3) = E (cid:2) D h ( f ◦ X t, ) (cid:3) = E (cid:20) ( f ◦ X t, ) Z t ˙ h ( s )d B s (cid:21) , where the last step is due to the integration by parts formula in the Malliavin calculus(cf. [20]). Formula (43) now follows.For estimation (44), by H¨older’s inequality and Itˆo’s isometry, the square of the righthand side of (43) is controlled by12 νt E | f ( X t, ( x )) | E (cid:20)Z t | s ( ∇ u s ) ◦ X t,s ( x ) − I | d s (cid:21) cνt E | f ( X t, ( x )) | " | t | sup s ∈ [ t, k∇ u s k ∞ + | t | . Hence, by (18) k∇ E f ( X t, ) k p c p ν | t | k f k p " | t | · sup s ∈ [ t, k∇ u s k ∞ + 1 c p ν | t | k f k p " | t | · sup s ∈ [ t, k∇ u s k ,p + 1 . The proof is complete. (cid:3)
We now prove the following global existence result (see also [16, 14]).
Theorem 4.4.
Let k ∈ N and p > d , u ∈ W k,p ( T d ; R d ) be divergence free and meanzero. Let ( u, X ) be the local solution of (32) in Theorem 3.7. Then, there exist T = T ( k, p, d, k∇ u k k +1 ,p ) < and δ = δ ( k, p ) > such that if ν > δ k∇ u k k +1 ,p , then k∇ u T k k +1 ,p k∇ u k k +1 ,p , and there is a global solution to equation (32).Proof. Let ( u, X ) be the local solution of (32) on [ − T,
0] in Theorem 3.7, where T =( c k∇ u k k +1 ,p ) − . Recalling the estimations in Lemma 3.5 and Theorem 3.7, we have forall t ∈ [ − T, k∇ X t, k ∞ c , k∇ X t, k k,p c (45) nd k∇ u t k k +1 ,p c k∇ u k k +1 ,p . (46)Write u t = P E [( ∇ t X t, − I )( u ◦ X t, )] + P E ( u ◦ X t, ) . (47)We separately deal with the first term and the second term.For the first term in (47), using (45) (46) and as in Lemma 3.5, one may prove that forsome c = c ( k, p, d ) and all t ∈ [ − T, k∇ t X t, − I k k +2 ,p c k∇ u k k +1 ,p · | t | . Using this estimate as well as (22) (45) and (46), one finds that k∇ P E [( ∇ t X t, − I )( u ◦ X t, )] k k +1 ,p c k∇ u k k +1 ,p · | t | . (48)For the second term in (47), by (44), (46) and Poincare’s inequality, we have k∇ P E ( u ◦ X t, ) k p c p ν | t | k u k p c p ν | t | k∇ u k p . (49)Note that ∇ E ( u ◦ X t, ) = ∇ E (( ∇ u ) ◦ X t, ) + ∇ E [(( ∇ u ) ◦ X t, )( ∇ X t, − I )] . As above, we have k∇ E (( ∇ u ) ◦ X t, ) k p c p ν | t | k∇ u k p and k∇ E [(( ∇ u ) ◦ X t, )( ∇ X t, − I )] k k,p c k∇ u k k +1 ,p · | t | . So, k∇ P E ( u ◦ X t, ) k p c k∇ E ( u ◦ X t, ) k p c p ν | t | k∇ u k p + c k∇ u k k +1 ,p · | t | . (50)Continuing the above calculations we get k∇ k +2 P E ( u ◦ X t, ) k p c p ν | t | k∇ k +1 u k p + c k∇ u k k +1 ,p · | t | . (51)Combining (49) (50) and (51), we find k∇ P E ( u ◦ X t, ) k k +1 ,p c p ν | t | k∇ u k k,p + c k∇ u k k +1 ,p · | t | . (52)Summarizing (47) (48) and (52) yields k∇ u t k k +1 ,p " c p ν | t | + c k∇ u k k +1 ,p · | t | k∇ u k k +1 ,p , where c = c ( k, p, d ) and c = c ( k, p, d ) > c . Now, taking T = − c k∇ u k k +1 ,p and δ = 8 c c , we have for ν > δ k∇ u k k +1 ,p k∇ u T k k +1 ,p k∇ u k k +1 ,p . The proof is thus finished. (cid:3) . A Large Deviation Estimate for Stochastic Particle Paths
Let G k denote the k -order diffeomorphism group on R d , which is endowed with thelocally uniform convergence topology together with its inverse for all derivatives up to k .Then G k is a Polish space. Let G k be the subspace of G k in which each transformationpreserves the Lebesgue measure, equivalently, G k := { X ∈ G k : det( ∇ X ) = 1 } . Then G k is a closed subspace of G k , therefore, a Polish space.It is clear that t X νt ( · ) ∈ G k is continuous by the theory of stochastic flow (cf. [17]). We now state a large deviationprinciple of Freidlin-Wentzell’s type, which follows from the results in [1, 21] by usingProposition 3.9. Theorem 5.1.
Keep all the things as in Proposition 3.9. For any Borel set E ⊂ C ([ − T, G k ) , we have − inf Y ∈ E o I ( Y ) lim inf ν → ν log P ( X ν ∈ E ) lim sup ν → ν log P ( X ν ∈ E ) − inf Y ∈ ¯ E I ( Y ) , where E o and ¯ E denotes the interior and the closure respectively in C ([ − T, G k ) , and I ( Y ) is the rate function defined by I ( Y ) := 12 inf { h ∈ L ( − T, l ): S ( h )= Y } Z − T k h s k l d s, Y ∈ C ([ − T, G k ) , where S ( h ) = Y solves the following ODE: Y s ( x ) = x + Z s − T u r ( Y r ( x ))d r + Z s − T h σ r , h r i l d r, s ∈ [ − T, . Remark 5.2.
In two dimensions, the T in the above theorem can be arbitrarily large byTheorem 4.2. Acknowledgements:
The author would like to thank Professor Benjamin Goldys for providing him an excel-lent environment to work in the University of New South Wales. His work is supportedby ARC Discovery grant DP0663153 of Australia.
References [1] G. Ben Arous and F. Castell: Flow decomposition and large deviations.
J. Funct. Anal.
140 (1996),no. 1, 23–67.[2] J. M. Bismut,
Large Deviations and the Malliavin Calculus , Birkh¨auser, Basel, 1984.[3] B. Busnello: A probabilistic approach to the two-dimensional Navier-Stokes equations.
Ann. Probab.
27 (1999), no. 4, 1750–1780.[4] B. Busnello, F. Flandoli and M. Romito: A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations.
Proc. Edinb. Math. Soc. (2) 48 (2005), no. 2, 295–336.[5] F. Cipriano and A.B. Cruzeiro: Navier-Stokes equation and Diffusions on the Group of Homeomor-phisms of the Torus.
Commun. Math. Phys. , 275, 255-269(2007).[6] P. Constantin and G. Iyer: A Stochsatic Lagrangian Representation of the Three-DimensionalIncompressible Navier-Stokes Equations.
Comm. Pure and Appl. Math. , Vol. LXI, 330-345, 2008.[7] A.B. Cruzeiro and E. Shamarova: Navier-Stokes equations and forward-backward SDEs on thegroup of diffeomorphisms of a torus. http://arxiv.org/abs/0807.0421.
8] A.B. Cruzeiro and X. Zhang: Bismut type formulae for diffusion semigroups on Riemannian mani-folds,
Potential Analysis , Volume 25, Number 2, Pages: 121 - 130(2006).[9] R. Esposito, R. Marra, M. Pulvirenti, C. Sciarretta: A stochastic Lagrangian picture for the three-dimensional Navier-Stokes equation.
Comm. Partial Differential Equations . 13 (1988), no. 12, 1601–1610.[10] A. Friedman:
Partial Differential Equations . Holt, Rinehart and Winston, INC., NewYork, 1969.[11] A. Friedman:
Stochastic Differential Equations and Applications . Volume 1, Academic Press, NewYork, 1975.[12] G. Iyer: A stochastic perturbation of inviscid flows.
Comm. Math. Phys. , 266(2006), N0.3 631-645.[13] G. Iyer: A stochastic Lagrangian formulation of the Navier-Stokes and related transport equations.Doctoral dissertation, Univ. of Chicago, 2006.[14] G. Iyer: A Stochastic Lagrangian Proof of Global Existence of Navier-Stokes Equations for Flowswith Small Reynolds Number. http://arxiv.org/abs/math/0702506.[15] G. Iyer and J. Mattingly: A stochastic-Lagrangian particle system for the Navier-Stokes equations.
Nolinearity , 21(2008)2537-2553.[16] H. Koch and D. Tataru: Well-posedness for the Navier-Stokes equations.
Adv. Math. , 157(2001),No. 1, 22-35.[17] H. Kunita:
Stochastic flows and stochastic differential equations . Cambridge Studies in Adva. Math.,24, Cambridge Univ. Press, New York, 1990.[18] Y. Le Jan and A.S. Sznitman: Stochastic cascades and 3-dimensional Navier-Stokes equations.
Proab. Theory and Rela. Fields , 109(1997), No.3, 343-366.[19] A.J. Majda and A.L. Bertozzi:
Vorticity and Incompressible Flow . Cambridge Texts in AppliedMathematics, Cambridge University Press, 2002.[20] P. Malliavin:
Stochastic Analysis , Grundlehren der Mathematischen Wissenschaften 313. Springer-Verlag, Berlin, 1997.[21] J. Ren and X. Zhang: Freidlin-Wentzell’s large deviations for homeomorphism flows of non-LipschitzSDEs,
Bull. Sci. Math. 2 Serie , Vol 129/8 pp 643-655(2005).[22] E.M. Stein:
Singular Integrals and Differentiability Properties of Functions.