Probab. Theory Relat. Fields (2010) 148:305–332 DOI 10.1007/s00440-009-0234-6
A stochastic representation for backward incompressible Navier-Stokes equations Xicheng Zhang
Received: 2 November 2008 / Revised: 18 May 2009 / Published online: 10 June 2009 © Springer-Verlag 2009
Abstract By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero. Keywords Backward Navier-Stokes equation · Stochastic representation · Global existence · Large deviation Mathematics Subject Classification (2000)
60H30 · 35Q30 · 76D05
1 Introduction The classical Navier-Stokes equations describe the evolution of velocity fields of an incompressible fluid, and takes the following form with the external force zero:
X. Zhang (B) Department of Mathematics, Huazhong University of Science and Technology, 430074 Wuhan, Hubei, People’s Republic of China e-mail:
[email protected] X. Zhang School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia
123
306
X. Zhang
∂t u + (u · ∇)u − νu + ∇ p = 0, t ≥ 0, ∇ · u = 0, u(0) = u 0 ,
(1)
where column vector field u = (u 1 , u 2 , u 3 )t denotes the velocity field, p is the pressure and ν is the kinematic viscosity. When the viscosity ν vanishes, the above equation becomes the classical Euler equation:
∂t u + (u · ∇)u + ∇ p = 0, t ≥ 0, ∇ · u = 0, u(0) = u 0 ,
(2)
which describes the motion of an ideal incompressible fluid. The mathematical theory about Navier-Stokes equations and Euler equations has been extensively studied and the existence of regularity solutions is still a big open problem in modern PDEs. Recently, Constantin and Iyer [6] presented an elegant stochastic representation for incompressible Navier-Stokes equations based on stochastic particle paths, which is realized by an implicit stochastic differential equation: the drift term is computed as the expected value of an expression involving the stochastic flow defined by itself. More precisely, let (u, X ) solve the following stochastic system: ⎧ t ⎪ √ ⎪ ⎪ ⎪ X t (x) = x + u s (X s (x))ds + 2ν Bt , t ≥ 0, ⎪ ⎪ ⎨ 0
(3)
⎪ ⎪ ⎪ At = X t−1 , ⎪ ⎪ ⎪ ⎩ u t = EP[(∇ t At )(u 0 ◦ At )],
where {Bt , t ≥ 0} is a 3-dimensional Brownian motion, P is the Leray-Hodge projection onto divergence free vector fields, and ∇ t At denotes the transpose of Jacobi matrix ∇ At . Then u satisfies Eq. (1) with initial data u 0 . One of the proofs given in [6] is based on a stochastic partial differential equation satisfied by the inverse flow A. By using this representation, a self-contained proof of the existence of local smooth solutions is provided in [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 ≤ 0,
then u˜ satisfies the following equation (called backward Navier-Stokes equation here):
∂t u˜ + (u˜ · ∇)u˜ + νu˜ + ∇ p˜ = 0, t ≤ 0, ∇ · u˜ = 0, u(0) ˜ = u0.
(4)
The purpose of the present paper is to give a slightly different representation for u˜ by using backward particle paths. More precisely, let (u, ˜ X˜ ) solve the following stochastic system
123
A stochastic representation for backward incompressible Navier-Stokes equations
⎧ s ⎪ √ ⎪ ⎪ ⎨ X˜ t,s (x) = x + u˜ r ( X˜ t,r (x))dr + 2ν( B˜ s − B˜ t ), t ≤ s ≤ 0, t ⎪ ⎪ ⎪ ⎩ u˜ = EP(∇ t X˜ )(u ◦ X˜ ) , t ≤ 0, t t,0 0 t,0
307
(5)
where B˜ t := B−t . Then u˜ satisfies backward incompressible Navier-Stokes equation (4) with final value u 0 . Intuitively, by reversing the time, the starting point is changed as the end point. Hence, representation (5) is essentially the same as (3). But, Eq. (5) is easier to be dealt with mathematically since the unpleasant term A = X −1 which usually incurs extra mathematical calculations does not appear in (5). Direct calculations show that the second equation in (5) is equivalent to ω˜ t = ∇ × u˜ t = E (∇ −1 X˜ t,0 )(ω0 ◦ X˜ t,0 ) , −1
u˜ t = −
∇ × ω˜ t ,
(6) (7)
where ω˜ t is the vorticity, ∇ −1 X˜ t,0 is the inverse of Jacobian matrix ∇ X˜ t,0 and (7) is exactly the Biot-Savart law. It should be noticed that representations (3) and (5) are useful in numerical computations (cf. [15,19]). We also mention other stochastic formulations for incompressible Navier-Stokes equations. In [9], a representation formula for the vorticity of three dimensional Navier-Stokes equations was given by using stochastic Largrangian paths, however, there is no a self-contained proof of the existence given there. In [18], Le Jan and Sznitman used a backward-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 new existence theorem. In [3], basing on Girsanov’s transformation and Bismut-Elworty-Li’s formula, Busnello introduced a purely probabilistic treatment to the existence of a unique global solution for two dimensional NavierStokes equations, where the stretching term disappears, and the non-linear equation obeyed by the vorticity has the form of Fokker-Planck equation. Later on, Busnello et al. [4] carefully analyzed an implicit probabilistic representation for the vorticity of three dimensional Navier-Stokes equations, and a local existence was given. In that paper, much attentions were also paid on a probabilistic representation formula for a general system of linear parabolic equations. Moreover, in [3,4], an interesting probabilistic representation for the Biot-Savart law was also given and analyzed so that they can recover the velocity from the vorticity by probabilistic approach. Recently, Cipriano and Cruzeiro in [5] described a stochastic variational principle for two dimensional incompressible Navier-Stokes equations by using the Brownian motions on the group of homeomorphisms on the torus. More recently, Cruzeiro and Shamarova [7] established a connection between Eq. (4) and a system of infinite dimensional forward-backward stochastic differential equations on the group of volume-preserving diffeomorphisms of a flat torus. This paper is organized as follows: In Sect. 2, we prove representation (5). In Sect. 3, we shall give a self-contained proof of local existence in Sobolev spaces. The proof is based on successive approximation or fixed point method as in [12]. Therein, Iyer considered the spatially periodic case and worked in Hölder continuous function
123
308
X. Zhang
spaces. When Sobolev spaces are considered, we have to overcome the difficulty due to the non-closedness of Sobolev spaces under pointwise multiplications and compositions. Thus, it seems to be hard to exhibit the same proof in Sect. 3 for representation (3) due to the presence of A = X −1 . A key point 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 Sect. 4, we shall give an alternative proof to the global existence when the spatial dimension is two or the viscosity is large enough in any dimensions. Such results are well known. For two dimensional Navier-Stokes and Euler equations in the whole space, the global existence of smooth solutions is referred to [19]. The global existence of any dimensional NSEs for small initial values is referred to [16]. Recently, Iyer [14] presented an alternative proof to the global existence for small Reynolds number. His proof is based on the decay of heat flows and stochastic representation (3). Following [14], we will give a different proof to the global existence of small Reynolds number based on Bismut’s formula and representation (5). Let (u ν , X ν ) denote the solution of Eq. (5). In Sect. 5, as ν goes to zero, an asymptotic probability estimate of X ν in diffeomorphism group is presented by the well known large deviation estimate for stochastic diffeomorphism flows. 2 Stochastic representation of backward incompressible Navier-Stokes equations We begin with some notational conventions. Fix d ≥ 2 and put N0 := {0} ∪ N, R− := (−∞, 0], I := (d × d)-unit matrix and
l 2 := σ = (σk )k∈N ∈ RN : σ l22 := σk2 < +∞ . For a differentiable transformation X of Rd , the Jacobi matrix of X is given by ⎛
∂1 X 1 , ∂2 X 1 , · · · ⎜ ∂1 X 2 , ∂2 X 2 , · · · ∇ X := ⎜ ⎝ ··· , ··· , ··· ∂1 X d , ∂2 X d , · · ·
, , , ,
⎞ ∂d X 1 ∂d X 2 ⎟ ⎟, ··· ⎠ ∂d X d
where ∂i = ∂∂xi . We use ∇ t X to denote the transpose of ∇ X . For k ∈ N0 , let Cbk (Rd ; Rd ) denote the space of k-order continuous differentiable vector fields on Rd with the norm: u C k := b
sup |D α u(x)| < +∞,
d |α|≤k x∈R
where D α denotes the derivative with respect to the multi index α.
123
A stochastic representation for backward incompressible Navier-Stokes equations
309
2 (R ; Rd × l 2 ) satisfy Let u ∈ C(R− ; Cb3 (Rd ; Rd )) and t → σt ∈ L loc − ∞
σt·k σt·k = I.
(8)
k=1
Let {X t,s (x), t ≤ s ≤ 0} solve the following SDE s
√ u r (X t,r (x))dr + 2ν σr dBr , s
X t,s (x) = x + t
(9)
t
where Bt := {Btk , t ≤ 0, k ∈ N} is a sequence of independent standard Brownian motions on some probability space (, F, P). Thanks to (8), the diffusion operator associated to Eq. (9) is given by L t g ≡ νg + (u t · ∇)g. We first prove the following result. Theorem 2.1 Let φ : Rd → Rd be a C 2 -vector field satisfying |D α φ(x)| ≤ C 1 + |x|β , |α| ≤ 2, β > 0, and f ∈ C(R− ; Cb2 (Rd ; Rd )). Define φ (t, x) := ∇ t X t,0 (x) φ X t,0 (x) , 0 f (t, x) :=
t ∇ X t,r (x) fr X t,r (x) dr
(10) (11)
t
and wt (x) := E φ (t, x) − E f (t, x). Then w ∈ C 1,2 ((−∞, 0) × Rd ) satisfies the following backward Kolmogorov’s equation: ∂t wt + L t wt + ∇ t u t wt = f t , lim wt (x) = φ(x). t↑0
(12)
Proof Let g be a twice continuously differentiable function satisfying |D α g(x)| ≤ C(1 + |x|β ), |α| ≤ 2, β > 0.
123
310
X. Zhang
For h > 0, by Itô’s formula, we have ⎡ Eg(X t−h,t (x)) = g(x) + E ⎣
t
⎤ (L r g)(X t−h,r (x))dr ⎦ .
t−h
From this, it is easy to see that ⎡ t ⎤ 1 1 ⎣ Eg(X t−h,t (x)) − g(x) = E (L r g)(X t−h,r (x))dr ⎦ h h t−h
→ (L t g)(x) as h → 0. Noticing that X t−h,0 (x) = X t,0 ◦ X t−h,t (x), we have ∇ X t−h,0 (x) = (∇ X t,0 ) ◦ X t−h,t (x) · ∇ X t−h,t (x). Thus, by the independence of X t−h,t (x) with X t,0 (x), we have E φ (t − h, x) = E ∇ t X t−h,t (x) φ t, X t−h,t (x) = E ∇ t X t−h,t (x) (E φ (t)) ◦ X t−h,t (x) and E f (t − h, x) = E ∇ t X t−h,t (x) f (t, X t−h,t (x)) ⎤ ⎡ t t ∇ X t−h,r (x) fr X t−h,r (x) dr ⎦ +E⎣ t−h
= E ∇ t X t−h,t (x) (E f (t)) ◦ X t−h,t (x) ⎤ ⎡ t t ∇ X t−h,r (x) fr X t−h,r (x) dr ⎦ . +E⎣ t−h
Hence, we may write wt−h (x) = E ∇ t X t−h,t (x) wt X t−h,t (x) ⎤ ⎡ t t ∇ X t−h,r (x) fr X t−h,r (x) dr ⎦ −E ⎣ t−h
123
(13)
A stochastic representation for backward incompressible Navier-Stokes equations
311
and 1 1 (wt (x) − wt−h (x)) = − E ∇ t X t−h,t (x) − I wt X t−h,t (x) h h 1 − E wt (X t−h,t (x)) − wt (x) h ⎡ t ⎤ 1 ∇ t X t−h,r (x) fr X t−h,r (x) dr ⎦ + E⎣ h t−h h I1 (t, x) + I2h (t, x) + I3h (t, x).
=: Observing that
t ∇ X t−h,t (x) − I =
(∇u r ) ◦ X t−h,r (x) · ∇ X t−h,r (x)dr, t−h
we deduce lim I1h (t, x) = − (∇ t u t )wt (x). h↓0
By (13), we have lim I2h (t, x) = −(L t wt )(x). h↓0
Moreover, a simple limit procedure also gives lim I3h (t, x) = f t (x). h↓0
Combining the above calculations, we conclude that 1 lim (wt (x) − wt−h (x)) = −(L t wt )(x) − [(∇ t u t )wt ](x) + f t (x). h↓0 h Equation (12) now follows (see [11, p. 124] for more details).
Remark 2.2 A more general Feynman-Kac formula for a deterministic system of parabolic equations was given in [4]. However, the proof is simpler in our case. In representation (3), if we define wt := E[(∇ t At )(u 0 ◦ At )], then wt also satisfies (12) 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].
123
312
X. Zhang
Theorem 2.3 Let ν ≥ 0 and u 0 ∈ Cb2 (Rd ; Rd ) be a deterministic divergence-free vector field, and f ∈ C(R− ; Cb2 (Rd ; Rd )). Suppose that σ satisfies (8) and (u, X ) solves the stochastic system: s X t,s (x) = x +
u r (X t,r (x))dr +
√
s σr dBr , t ≤ s ≤ 0,
2ν
t
(14)
t
u t = PE u 0 (t) − PE f (t), t ≤ 0,
(15)
where P is the Leray-Hodge projection onto divergence free vector fields, and u 0 and f are given by (10) and (11). Then u satisfies the backward incompressible Navier-Stokes equation:
∂t u + (u · ∇)u + νu + ∇ p = f, t ≤ 0, ∇ · u = 0, u(0, x) = u 0 (x).
(16)
Conversely, if u solves backward Navier-Stokes equation (16), then u is given by (15). Proof First of all, let wt (x) := E u 0 (t, x) − E f (t, x).
(17)
By Theorem 2.1, w(t, x) = wt (x) solves the following backward Kolmogorov’s equation: ∂t w + (u · ∇)w + (∇ t u)w + νw = f, w(0, x) = u 0 (x).
(18)
In view of u = Pw, we may write w = u + ∇q. Substituting it into Eq. (18), one finds that ∂t u + (u · ∇)u + νu + ∇ p = f, u(0, x) = u 0 (x), where 1 p = ∂t q + (u · ∇)q + νq + |u|2 . 2 Conversely, let (u, p) solve (16). As above, if we define w by (17), then w satisfies Eq. (18). We need to show that u = Pw, or equivalently, for some scalar valued function q v := w − u = ∇q.
123
A stochastic representation for backward incompressible Navier-Stokes equations
313
By (18) and (16), v solves the following equation: 1 ∂t v + (u · ∇)v + ∇ t u v + νv = ∇ p − ∇|u|2 , v(0, x) = 0. 2 Let
⎛ q(t, x) := E ⎝
0
(19)
⎞ 1 |u(r, X t,r (x))|2 − p(r, X t,r (x)) dr ⎠ . 2
t
Then q solves the following equation (cf. [11, p. 148, Theorem 5.3]): 1 ∂t q + (u · ∇)q + νq = p − |u|2 , q(0, x) = 0. 2 Taking gradients for both sides of the above equation yields 1 ∂t ∇q + (u · ∇)∇q + ∇ t u (∇q) + ν∇q = ∇ p − ∇|u|2 . 2 By the uniqueness of solutions to linear Eq. (19), v = ∇q.
Remark 2.4 In the above proof, we have assumed that the solutions are regular enough so that all the calculations are valid. The existence of regular solutions will be proven in the next section. 3 A proof of local existence in Sobolev spaces With a little abuse of notations, in this and next sections we use p to denote the integrability index since the pressure will not appear below. For k ∈ N0 and p ≥ 1, let Wk, p (Rd ; Rd ) be the usual Rd -valued Sobolev space on Rd , i.e, the completion of C0∞ (Rd ; Rd ) with respect to the norm: u k, p := u p +
k
∇ j u p ,
j=1
where · p is the usual L p -norm, and ∇ j is the j-order gradient operator. Note that W0, p (Rd ; Rd ) = L p (Rd ; Rd ) and the following Sobolev’s embedding holds: for p > d (cf. [10]) W1, p Rd ; Rd → L ∞ Rd ; Rd , i.e., · ∞ ≤ c · 1, p ,
(20)
where c = c( p, d). Below, we shall use c to denote an unrelated constant which may k, p change in different occasions. Let Wloc Rd ; Rd be the local Sobolev space on Rd .
123
314
X. Zhang
We introduce the following Banach space of transformations of Rd : k+2, p Xk+2, p := X ∈ Wloc (Rd ; Rd ) : X Xk+2, p < +∞ , where X Xk+2, p := |X (0)| + ∇ X ∞ + ∇ 2 X k, p . k+2, p
Let Xp.vol. ⊂ Xk+2, p be the class of all transformations preserving the volume, which is closed under · Xk+2, p , and so a Polish space. Definition 3.1 The Weber operator W : L p (Rd ; Rd ) × X2, p → L p (Rd ; Rd ) is defined by W(v, ) := P[(∇ t )v], where P is the Leray-Hodge projection onto divergence free vector fields. Remark 3.2 P = I − ∇(−)−1 div is a singular integral operator (SIO) which is bounded in L p -space for p ∈ (1, ∞) (cf. [22]). We now prepare several lemmas for later use. Lemma 3.3 (i) For any k ∈ N0 and p > d, there exists a constant c = c(k, p, d) > 0 such that for all v ∈ Wk+2, p (Rd ; Rd ) and ∈ Xk+2, p , ∇W(v, ) k+1, p ≤ c ∇ ∞ + ∇ 2 k, p · ∇v k+1, p .
(21)
(ii) For p > d, there exists a constant c = c( p, d) > 0 such that for all v1 , v2 ∈ W2, p (Rd ; Rd ) and 1 , 2 ∈ X2, p with 1 − 2 ∈ L p (Rd ; Rd ), W(v1 , 1 )−W(v2 , 2 ) p ≤ c ∇v1 1, p 1 −2 p + ∇2 ∞ v1 −v2 p . (22) Proof (i) Noting that P((∇ t )v) + P((∇ t v)) = P(∇( · v)) = 0, we have ∂i W(v, ) = P (∇ t ∂i )v + (∇ t )∂i v = P −(∇ t v)∂i + (∇ t )∂i v and ∂ j ∂i W(v, ) = P −(∇ t v)∂ j ∂i − (∇ t ∂ j v)∂i + (∇ t )∂ j ∂i v + (∇ t ∂ j )∂i v .
123
(23)
A stochastic representation for backward incompressible Navier-Stokes equations
315
Hence, by (20) we have ∂i W(v, ) p ≤ ≤ ≤
c (∇ t v)∂i p + (∇ t )∂i v p c ∇v p · ∂i ∞ + ∇ ∞ ∂i v p c ∇v p · ∇ ∞
and ∂ j ∂i W(v, ) p ≤ c ∇ 2 p · ∇v ∞ + ∇ ∞ · ∇ 2 v p ≤ c ∇ 2 p · ∇v 1, p + ∇ ∞ · ∇ 2 v p ≤ c ∇ ∞ + ∇ 2 p · ∇v 1, p , which produce ∇W(v, ) 1, p ≤ c ∇ ∞ + ∇ 2 p · ∇v 1, p . The higher derivatives can be estimated similarly. (ii) By (23), we have W(v1 , 1 ) − W(v2 , 2 ) = P (∇ t (1 − 2 ))v1 + P (∇ t 2 )(v1 − v2 ) = −P (∇ t v1 )(1 − 2 ) + P (∇ t 2 )(v1 − v2 ) . So, W(v1 , 1 ) − W(v2 , 2 ) p ≤ c (∇ t v1 )(1 − 2 ) p + c (∇ t 2 )(v1 − v2 ) p ≤ c ∇v1 ∞ 1 − 2 p + c ∇2 ∞ v1 − v2 p ,
which yields (22) by (20).
Lemma 3.4 (i) For k ∈ N0 and p > d, there exist constants c = c(k, p, d) > 0 k+2, p and αk ∈ N0 such that for all u ∈ Wk+2, p (Rd ; Rd ) and all X ∈ Xp.vol. , αk 2 ∇(u ◦ X ) k+1, p ≤ c ∇u k+1, p 1 + ∇ X k+2 ∞ + ∇ X k, p .
(24)
(ii) For p > d, there exists a constant c = c( p, d) > 0 such that for all u ∈ W2, p (Rd ; Rd ) and X, X˜ ∈ X2, p with X − X˜ ∈ L p (Rd ; Rd ), u ◦ X − u ◦ X˜ p ≤ c ∇u 1, p · X − X˜ p .
(25)
(iii) For k ∈ N0 and p > d, let u ∈ Wk+2, p (Rd ; Rd ). Then k+2, p
Xp.vol. X → u ◦ X ∈ Wk+2, p (Rd ; Rd ) is continuous.
(26)
123
316
X. Zhang
Proof (i) Since X preserves the volume, we have u ◦ X p = u p .
(27)
Observe that by the chain rule ∇ m (u ◦ X ) =
m
c j,α · (∇ j u)(X ) · ∇ α1 X · · · ∇ α j X
j=1 |α|=m m
= (∇ u) ◦ X · (∇ X )m + · · · + (∇u) ◦ X · ∇ m X,
(28)
where α = (α1 , . . . , α j ), αi ≥ 1, |α| = α1 + · · · + α j and c j,α ∈ N. Thus, (24) follows by (20) and (27). (ii) It follows from u ◦ X − u ◦ X˜ =
1
(∇u) ◦ (s X + (1 − s) X˜ )) · (X − X˜ )ds
(29)
0
and (20). (iii) Using expression (28), we only prove that k+2, p
Xp.vol. X → (∇ k+2 u) ◦ X · (∇ X )k+2 ∈ L p ,
(30)
k+2, p Xp.vol.
(31)
X → (∇u) ◦ X · ∇
k+2
X∈L
p
are continuous. The other terms are analogous. k+2, p Let X n converge to X in Xp.vol. . Then, k+2 (∇ u) ◦ X n · (∇ X n )k+2 − (∇ k+2 u) ◦ X · (∇ X )k+2 p k+2 k+2 k+2 ≤ (∇ u) ◦ X n p · (∇ X n ) − (∇ X ) ∞ k+2 k+2 k+2 + (∇ u) ◦ X n − (∇ u) ◦ X p · ∇ X ∞ . By (27), the first term clearly converges to zero. Choose G m ∈ C0∞ (Rd ) such that lim sup (G m − ∇ k+2 u) ◦ X n = lim G m − ∇ k+2 u = 0.
m→∞ n
p
m→∞
p
Noting that for any x ∈ Rd 1 X n (x) − X (x) = X n (0) − X (0) +
(∇ X n − ∇ X )(sx) · xds, 0
123
(32)
A stochastic representation for backward incompressible Navier-Stokes equations
317
we have lim X n (x) = X (x).
n→∞
Thus, for each m, by the dominated convergence theorem, we have for any ϕ ∈ C0∞ (Rd ) (G m ◦ X n (x) − G m ◦ X (x))ϕ(x)dx = 0,
lim
n→∞
Rd
which implies that G m ◦ X n weakly converges to G m ◦ X in L p as n → ∞. By virtue of G m ◦ X n p = G m p , we then get lim G m ◦ X n − G m ◦ X p = 0.
(33)
lim (∇ k+2 u) ◦ X n − (∇ k+2 u) ◦ X = 0.
(34)
n→∞
Thus, by (32) and (33),
n→∞
p
The continuity of (30) now follows. For the continuity of (31), we have (∇u) ◦ X n · ∇ k+2 X n − (∇u) ◦ X · ∇ k+2 X p k+2 ≤ (∇u) ◦ X ∞ · ∇ X n − ∇ k+2 X p k+2 + (∇u) ◦ X n − (∇u) ◦ X ∞ · ∇ Xn p k+2 ≤ (∇u) ◦ X 1, p · ∇ X n − ∇ k+2 X p
+ c · (∇u) ◦ X n − (∇u) ◦ X 1, p . It is clear that the first term converges to zero as n → ∞. The second term converging to zero can be proved as (34). Lemma 3.5 For k ∈ N0 , U > 0 and T := U1 , there exist constants c1 = c1 ( p, d) > 0 and c2 = c2 (k, p, d) > 0 such that for any divergence free vector field u ∈ C([−T, 0]; Wk+2, p ) satisfying supt∈[−T,0] ∇u t k+1, p ≤ U , the solution X t,s to (14) k+2, p
belongs to Xp.vol. a.s., and for all −T ≤ t ≤ s ≤ 0 ∇ X t,s ∞ ≤ c1 , ∇ 2 X t,s
k, p
≤ c2 .
(35)
123
318
X. Zhang
Proof Noting that s ∇ X t,s = I +
(∇u r ) ◦ X t,r · ∇ X t,r dr,
(36)
t
we have s ∇ X t,s ∞ ≤ 1 +
∇ X t,r ∞ · ∇u r ∞ dr. t
By Gronwall’s inequality, we obtain by (20) ⎡ sup
−T ≤t≤s≤0
∇ X t,s ∞ ≤ exp ⎣
0
⎤ ∇u r ∞ dr ⎦ ≤ ecU T = ec =: c1 .
(37)
−T
On the other hand, from (36), an elementary calculation shows that ⎡
⎤
s
det(∇ X t,s ) = exp ⎣ (∇ · u r ) ◦ X t,r dr ⎦ = 1. t
So, x → X t,s (x) preserves the volume. Now, ∇ 2 X t,s =
s
! 2 ∇ 2 u r ◦ X t,r · ∇ X t,r + (∇u r ) ◦ X t,r · ∇ 2 X t,r dr.
t
Hence, by (37) and (20) s 2 ∇ X t,s ≤ c p
" # 2 2 2 ∇ u r ∇ X t,r ∞ + ∇u r ∞ ∇ X t,r dr p
p
t
s ≤ cU T + cU ∇ 2 X t,r dr. p
t
By Gronwall’s inequality again we get 2 ∇ X t,s ≤ cU T ecU T = cec =: c2 . p
Higher derivatives can be estimated similarly step by step.
123
A stochastic representation for backward incompressible Navier-Stokes equations
319
Lemma 3.6 Keep the same settings as in Lemma 3.5, the mapping [−T, 0] t → k+2, p X t,0 ∈ Xp.vol. is continuous. Proof First of all, it is easy to prove that for fixed x ∈ Rd and s ∈ [−T, 0] [−T, s] t → X t,s (x) ∈ Rd is continuous.
(38)
Let −T ≤ t ≤ t ≤ s ≤ 0. Then s ∇X
t ,s
− ∇ X t,s =
(∇u r ) ◦ X t ,r − (∇u r ) ◦ X t,r · ∇ X t ,r dr
t
s t + (∇u r ) ◦ X t,r · ∇ X t ,r −∇ X t,r dr + (∇u r ) ◦ X t,r · ∇ X t,r dr. t
t
Thus, by (35), we have ∇ X t ,s − ∇ X t,s ≤ ∞
s
(∇u r ) ◦ X t ,r − (∇u r ) ◦ X t,r · ∇ X t ,r dr ∞ ∞
t
s +
∇u r ∞ · ∇ X t ,r − ∇ X t,r ∞ dr
t
t + t 0
≤c
∇u r ∞ · ∇ X t,r ∞ dr
(∇u r ) ◦ X t ,r − (∇u r ) ◦ X t,r dr + c|t − t| ∞
t
s +c
∇ X t ,r − ∇ X t,r dr, ∞
t
which gives by Gronwall’s inequality and (20) that ∇ X t ,s − ∇ X t,s ≤ cT ∞
0
(∇u r ) ◦ X t ,r − (∇u r ) ◦ X t,r dr + cT |t − t|. 1, p
t
Using limit (38) and as in proving (34), we may prove that lim ∇ X t ,s − ∇ X t,s ∞ = 0.
t→t
(39)
123
320
X. Zhang
For the second order derivative, we have s ! 2 ∇ 2 u r ◦ X t ,r − (∇ 2 u r ) ◦ X t,r · ∇ X t ,r dr =
∇ X t ,s − ∇ X t,s 2
2
t
s ! + ∇ 2 u r ◦ X t,r · ∇ X t ,r − ∇ X t,r · ∇ X t ,r dr t
s +
(∇u r ) ◦ X t,r − (∇u r ) ◦ X t,r · ∇ 2 X t ,r dr
t
s +
! (∇u r ) ◦ X t,r · ∇ 2 X t ,r − ∇ 2 X t,r dr
t
s + ∇ 2 u r ◦ X t,r · ∇ X t,r · ∇ X t ,r − ∇ X t,r dr t
t ! ∇ 2 u r ◦ X t,r · (∇ X t,r )2 + (∇u r ) ◦ X t,r · ∇ 2 X t,r dr + t
=: I1 + I2 + I3 + I4 + I5 + I6 . For I1 , as (34) we have 0 I1 p ≤ c
(∇ 2 u r ) ◦ X t ,r − (∇ 2 u r ) ◦ X t,r p dr → 0 as t → t .
t
For I2 , by limit (39) we have s I2 p + I5 p ≤ c
∇ 2 u r p · ∇ X t ,r − ∇ X t,r ∞ dr → 0 as t → t .
t
For I3 , as (34) we have 0 I3 p ≤ c
(∇u r ) ◦ X t ,r − (∇u r ) ◦ X t,r ∞ · ∇ 2 X t ,r p dr t
0 ≤c t
123
(∇u r ) ◦ X t ,r − (∇u r ) ◦ X t,r 1, p dr → 0 as t → t .
A stochastic representation for backward incompressible Navier-Stokes equations
321
For I4 , we have s I4 p ≤
∇u r ∞ · ∇ 2 X t ,r − ∇ 2 X t,r dr. p
t
For I6 , we have t 2 2 2 I6 p ≤ ∇ u r · ∇ X t,r ∞ + ∇u r ∞ · ∇ X t,r dr p
p
t
≤ c|t − t|. Combining the above calculations and by Gronwall’s inequality again, we get lim ∇ 2 X t ,s − ∇ 2 X t,s = 0. p
t→t
Higher derivatives can be calculated similarly.
Lemma 3.7 For p > d and T > 0, let u, u˜ ∈ C([−T, 0]; W2, p (Rd ; Rd )) be divergence free, and X, X˜ solve SDE (9) with drifts u and u˜ respectively. Then for some c = c( p, d) > 0 and any t ∈ [−T, 0], $ X t,0 − X˜ t,0 ≤ exp cT
% 0 sup ∇u t 1, p · u r − u˜ r p dr.
p
t∈[−T,0]
(40)
t
Proof We have s X t,s (x) − X˜ t,s (x) = u r (X t,r (x)) − u˜ r X˜ t,r (x) dr t
s u r (X t,r (x)) − u r X˜ t,r (x) dr = t
s u r X˜ t,r (x) − u˜ r X˜ t,r (x) dr. + t
For R > 0, let B R := {x ∈ Rd : |x| ≤ R}. By virtue of formula (29) and x → X˜ t,r (x) preserving the volume, we have
123
322
X. Zhang
X t,s − X˜ t,s
L p (B R )
s ≤ u r ◦ X t,r − u r ◦ X˜ t,r t
≤
sup
r ∈[−T,0]
s L p (B R )
u r − u˜ r p dr
dr + t
s ∇u r ∞ X t,r − X˜ t,r t
L p (B R )
dr
0 u r − u˜ r p dr,
+
(41)
t
By Gronwall’s inequality and (20), we get X t,s − X˜ t,s
$ L p (B R )
≤ exp cT
% 0 sup ∇u t 1, p · u r − u˜ r p dr. t∈[−T,0]
t
Letting R go to infinity gives (40). We are now in a position to prove the following local existence result.
Theorem 3.8 For ν ≥ 0, k ∈ N0 and p > d, there exists a constant c0 = c0 (k, p, d) > 0 independent of ν such that for any u 0 ∈ Wk+2, p (Rd ; Rd ) divergence free and T := (c0 ∇u 0 k+1, p )−1 , there is a unique pair of (u, X ) with u ∈ C([−T, 0]; Wk+2, p ) satisfying ⎧ s s ⎪ √ ⎪ ⎨ X t,s (x) = x + u r (X t,r (x))dr + 2ν σr dBr , t ≤ s ≤ 0, t t ⎪ ⎪ ⎩ t u t = PE[(∇ X t,0 )(u 0 ◦ X t,0 )], t ∈ [−T, 0].
(42)
Moreover, for any t ∈ [−T, 0] ∇u t k+1, p ≤ c0 ∇u 0 k+1, p .
(43)
Proof Set u r1 (x) := u 0 (x). Consider the following Picard’s iteration sequence ⎧ ⎪ ⎪ ⎪ ⎨
s n X t,s (x)
n u rn (X t,r (x))dr
=x+
+
√
s σr dBr , t ≤ s ≤ 0,
2ν
⎪ ⎪ ⎪ ⎩ u n+1 = PE ∇ X n u ◦ X n , t ≤ 0. 0 t t,0 t,0 t t
t
Noting that n n n n u 0 ◦ X t,0 = EW u 0 ◦ X t,0 , , X t,0 PE ∇ t X t,0
123
(44)
A stochastic representation for backward incompressible Navier-Stokes equations
323
we have by (21) and (24) n+1 ∇u t
k+1, p
n n ≤ E ∇W(u 0 ◦ X t,0 , X t,0 ) k+1, p " # 2 n n ∇ u 0 ◦ X n · ≤ cE ∇ X t,0 + X ∇ t,0 t,0 k+1, p ∞ k, p β n k+3 n k ≤ c3 E 1 + ∇ X t,0 + ∇ 2 X t,0 · ∇u 0 k+1, p , ∞ k, p
where βk ∈ N only depends on k and c3 = c3 (k, p, d) ≥ 1. Set β
c0 := c3 (1 + c1k+3 + c2 k ) ≥ 1, where c1 and c2 are from Lemma 3.5. Choosing U := c0 ∇u 0 k+1, p and T := 1/U in Lemma 3.5, we have by induction and Lemma 3.5 sup ∇u nt k+1, p ≤ U, ∀n ∈ N.
(45)
t∈[−T,0]
On the other hand, we also have by (35) n+1 n n u 0 ◦ X t,0 u t ≤ cE ∇ t X t,0 p p t n ! n ≤ cE ∇ X t,0 ∞ u 0 ◦ X t,0 p ≤ c u 0 p , which together with (45) gives the following uniform estimate: sup
sup u nt k+2, p < +∞.
n∈N t∈[−T,0]
(46)
The continuity of [−T, 0] t → u nt ∈ Wk+2, p follows from Lemma 3.6 and (21) (26). Now by (22) and (25) (40), we have n n+1 n m ) 1, p · X t,0 − X t,0 u t − u tm+1 ≤ cE ∇(u 0 ◦ X t,0 p p ! m u 0 ◦ X n − u 0 ◦ X m + ∇ X t,0 · t,0 t,0 ∞ p 0 ≤c
n u − u m dr, r r p
t
123
324
X. Zhang
where c = c( p, d, U ) is independent of n, m. From this we derive that lim sup sup u nt − u m t p = 0. n,m→∞ t∈[−T,0]
By (46) and interpolation inequality, we further have lim sup sup u nt − u m t k+1, p = 0. n,m→∞ t∈[−T,0]
So, there is a u ∈ C([−T, 0]; Wk+1, p (Rd ; Rd )) such that lim sup sup u nt − u t k+1, p = 0. n→∞ t∈[−T,0]
Taking limits for (44), one finds that u is a solution of (42). Estimate (43) follows from (45). Moreover, u ∈ C([−T, 0], Wk+2, p ) follows from Lemma 3.6 and (21) (26). Remark 3.9 The constant c0 in (43) is usually strictly greater than 1. If c0 equals 1, then we can invoke the standard bootstrap method to obtain the global existence. This will be studied in the next section when the periodic boundary is considered and the viscosity is large enough. Since the existence time interval in Theorem 3.8 is independent of the viscosity ν, we also obtain the local existence of solutions to Euler equation (2). Moreover, as ν → 0, the solution of Navier-Stokes equation converges to the solution of Euler equation as given below. Proposition 3.10 Keep the same assumptions as in Theorem 3.8. For ν ≥ 0 and u 0 ∈ Wk+2, p ∩ L 2 , let (u ν , X ν ) be the solution of (42) corresponding to viscosity ν and initial value u 0 . Then for any j = 0, . . . , k + 1, there exists c = c(k, j, p, d, u 0 k+2, p , u 0 2 ) > 0 such that for all ν ≥ 0 and t ∈ [−T, 0] ν u t − u 0t
j
Cb
≤ c (ν|t|)
k+2− j − 1p d
&
1 k+2 1 2+ d − p
.
Proof Note that u 0 ∈ Wk+2, p ∩ L 2 guarantees u νt ∈ Wk+2, p ∩ L 2 . By ! ∂t u νt − u 0t + νu νt + P u νt · ∇ u νt − u 0t · ∇ u 0t = 0, we have ( ( ' ' 2 −∂t u νt − u 0t 2 = ν u νt , u νt − u 0t + (u νt · ∇)u νt − (u 0t · ∇)u 0t , u νt − u 0t 2 (2 ' ( ' ν ν 0 ν 0 ν ν 0 = ν u t , u t − u t + (u t − u t ) · ∇ u t , u t − u t 2
≤
123
ν u νt 2
· u νt
− u 0t 2
2
+ ∇u νt ∞ u νt
− u 0t 22 ,
A stochastic representation for backward incompressible Navier-Stokes equations
325
i.e., −∂t u νt − u 0t 2 ≤ ν u νt 2 + ∇u νt ∞ u νt − u 0t . 2
By Gronwall’s inequality and (43) we obtain 0
ν u − u 0 ≤ ν t t 2
⎡ 0 ⎤ ν ν u ds · exp ⎣ ∇u ds ⎦ ≤ cν|t|. s 2 s ∞
t
t
The desired estimate now follows by Gagliardo-Nirenberge’s inequality (cf. [10]): for u ∈ Wk+2, p ∩ L 2 j ∇ u where α = ( dj + 21 )/( 21 +
∞
k+2 d
≤ ck, j, p,d u αk+2, p u 1−α 2 ,
− 1p ).
√ Remark 3.11 We cannot prove a convergence rate O( ν|t|) as in [12] starting from ν (x) − X 0 (x)) does not belong to any L p -spaces. (42) because x → (X t,0 t,0 4 Existence of global solutions 4.1 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 ∈ W2, p (Rd ; Rd ) be a divergence free vector field and ω := curlu. Then, for some c = c( p, d) > 0 ∇u ∞ ≤ c 1 + log+ ω 1, p (1 + ω ∞ ),
(47)
where log+ x := max{log x, 0} for x > 0. In two dimensional case, taking the curl for the second equation in (42), one finds that ωt := curlu t := ∂1 u 2t − ∂2 u 1t = E[ω0 ◦ X t,0 ].
(48)
From this, we clearly have ωt p ≤ ω0 p , 1 ≤ p ≤ ∞.
(49)
Basing (47) and representation (48), we may prove the following global existence for 2D Navier-Stokes and Euler equations.
123
326
X. Zhang
Theorem 4.2 In two dimensions, for ν ≥ 0, k ∈ N0 , p > 2 and u 0 ∈ Wk+2, p (R2 ; R2 ) divergence free, there exists a unique global solution (u, X ) to equation (42). Proof We only need to prove the following a priori estimate: for all t ∈ R− u t k+2, p ≤ c( u 0 k+2, p , k, p, t) < +∞, where c( u 0 k+2, p , k, p, t) continuously depends on its parameters. Following the proof of Lemma 3.5, we have ∇ X t,0 ∞
⎡ 0 ⎤ ≤ exp ⎣ ∇u r ∞ dr ⎦ .
(50)
t
Noting that ∇ωt = E ∇ω0 ◦ X t,0 · ∇ X t,0 , we have ∇ωt p ≤ ∇ω0 p · E ∇ X t,0 ∞ and by (49) and (50) ⎛
⎡
ωt 1, p ≤ ω0 1, p · ⎝1 + exp ⎣
0
⎤⎞ ∇u r ∞ dr ⎦⎠ .
t
Hence, by (47) (49) and (51) ∇u t ∞ ≤ c 1 + log+ ωt 1, p (1 + ωt ∞ ) 0 ≤ c+c
∇u r ∞ dr, t
where c = c( ω0 1, p , p). By Gronwall’s inequality we obtain ∇u t ∞ ≤ cec|t| . Substituting this into (50) and (51) gives c|t|
∇ X t,0 ∞ ≤ ec|t|e , and by Calderon-Zygmund’s inequality about SIOs (cf. [22]) c|t| . ∇u t 1, p ≤ ωt 1, p ≤ ω0 1, p · 1 + ec|t|e
123
(51)
A stochastic representation for backward incompressible Navier-Stokes equations
327
Moreover, c|t| u t p ≤ cE ∇ X t,0 ∞ · u 0 ◦ X t,0 p ≤ c u 0 p · ec|t|e . Thus, u t 2, p ≤ c( u 0 2, p , p, t) < +∞. Starting from (48) and as in Lemma 3.5, higher derivatives can be estimated similarly. 4.2 Global existence for large viscosity In this section, we study the existence of global solutions for large viscosity and work on the d-dimensional torus Td = Rd /Zd . Let Wk, p (Td , Rd ) be the Rd -valued Sobolev space on Td with vanishing mean. Instead of (14), we consider s X t,s (x) = x +
u r (X t,r (x))dr +
√ 2ν(Bs − Bt ),
(52)
t
where B is the standard Wiener process on := C(R− ; Rd ), i.e., for ω ∈ , B· (ω) = ω(·). We first recall the following Bismut’s formula (cf. [2,8]). For the reader’s convenience, a short proof is provided here. Theorem 4.3 For any t < 0 and f ∈ Cb1 (Td ; R), it holds that ⎡ 1 (∇E f (X t,0 ))(x) = √ E ⎣ f (X t,0 (x)) t 2ν
0
⎤ s(∇ t u s ) ◦ X t,s (x) − I dBs ⎦ .
(53)
t
In particular, for any p > d and some c = c( p, d) % $ c ∇E f (X t,0 ) p ≤ √ f p |t| · sup ∇u s 1, p + 1 . ν|t| s∈[t,0]
(54)
Proof Fix t < 0 and y ∈ Rd below and define ⎤ ⎡ s 1 ⎣ (t − s)y + [(∇u r ) ◦ X t,r (x)] · (r y)dr ⎦ , s ∈ [t, 0]. h(s) := √ t 2ν t
Consider the Malliavin derivative of X t,s with respect to the sample path along the direction h, i.e.,
123
328
X. Zhang
Dh X t,s (x, ω) = lim
ε→0
X t,s (x, εh + ω) − X t,s (x, ω) , ω ∈ . ε
From (52) one sees that s Dh X t,s (x) =
[(∇u r ) ◦ X t,r (x)] · Dh X t,r (x)dr +
√ 2νh(s)
t
=
(t − s)y + t
s [(∇u r ) ◦ X t,r (x)] · Dh X t,r (x) +
ry ! dr. t
t
On the other hand, we have s ∇ X t,s (x) · y = y +
[(∇u r ) ◦ X t,r (x)] · ∇ X t,r (x) · ydr. t
By the uniqueness of solutions, we get ∇ X t,s (x) · y = Dh X t,s (x) +
sy . t
In particular, ∇ X t,0 (x) · y = Dh X t,0 (x). Now ∇E f (X t,0 ) · y = E [(∇ f ) ◦ X t,0 ] · ∇ X t,0 · y = E (∇ f ) ◦ X t,0 · Dh X t,0 = E Dh ( f ◦ X t,0 ) ⎤ ⎡ 0 ˙ = E ⎣( f ◦ X t,0 ) h(s)dB s⎦ , t
where the last step is due to the integration by parts formula in the Malliavin calculus (cf. [20]). Formula (53) now follows. For estimation (54), by Hölder’s inequality and Itô’s isometry, the square of the right hand side of (53) is controlled by ⎡ 0 ⎤ 1 E| f (X t,0 (x))|2 E ⎣ |s(∇ t u s ) ◦ X t,s (x) − I|2 ds ⎦ 2νt 2 t % $ c ≤ 2 E| f (X t,0 (x))|2 |t|3 sup ∇u s 2∞ + |t| . νt s∈[t,0]
123
A stochastic representation for backward incompressible Navier-Stokes equations
329
Hence, by (20) % $ c ∇E f (X t,0 ) p ≤ √ f p |t| · sup ∇u s ∞ + 1 ν|t| s∈[t,0] % $ c ≤ √ f p |t| · sup ∇u s 1, p + 1 . ν|t| s∈[t,0]
The proof is complete. We now prove the following global existence result (see also [14,16]).
Theorem 4.4 Let k ∈ N0 and p > d, u 0 ∈ Wk+2, p (Td ; Rd ) be divergence free and mean zero. Let (u, X ) be the local solution of (42) in Theorem 3.8. Then, there exist T0 = T0 (k, p, d, ∇u 0 k+1, p ) < 0 and δ = δ(k, p, d) > 0 such that if ν ≥ δ ∇u 0 k+1, p , then ∇u T0 k+1, p ≤ ∇u 0 k+1, p , and so there is a global solution to equation (42). Proof Let (u, X ) be the local solution of (42) on [−T, 0] in Theorem 3.8, where T = (c0 ∇u 0 k+1, p )−1 . Recalling the estimations in Lemma 3.5 and Theorem 3.8, we have for all t ∈ [−T, 0] ∇ X t,0 ∞ ≤ c1 , ∇ 2 X t,0 k, p ≤ c2
(55)
∇u t k+1, p ≤ c0 ∇u 0 k+1, p .
(56)
u t = PE[(∇ t X t,0 − I)(u 0 ◦ X t,0 )] + PE(u 0 ◦ X t,0 ).
(57)
and
Write
We separately deal with the first term and the second term. For the first term in (57), using (55) (56) and as in Lemma 3.5, one may prove that for some c = c(k, p, d) and all t ∈ [−T, 0] ∇ t X t,0 − I k+1, p ≤ c ∇u 0 k+1, p · |t|. Using this estimate as well as (24) (55) and (56), one finds that ∇PE[(∇ t X t,0 − I)(u 0 ◦ X t,0 )] k+1, p ≤ c ∇u 0 2k+1, p · |t|.
(58)
For the second term in (57), by (54), (56) and Poincare’s inequality, we have ∇PE(u 0 ◦ X t,0 ) p ≤ c ∇E(u 0 ◦ X t,0 ) p ≤ √
c c u 0 p ≤ √ ∇u 0 p . (59) ν|t| ν|t|
Note that ∇ 2 E(u 0 ◦ X t,0 ) = ∇E((∇u 0 ) ◦ X t,0 ) + ∇E[((∇u 0 ) ◦ X t,0 )(∇ X t,0 − I)].
123
330
X. Zhang
As above, we have c ∇E((∇u 0 ) ◦ X t,0 ) p ≤ √ ∇u 0 p ν|t| and ∇E[((∇u 0 ) ◦ X t,0 )(∇ X t,0 − I)] k, p ≤ c ∇u 0 2k+1, p · |t|. So, ∇ 2 PE(u 0 ◦ X t,0 ) p ≤ c ∇ 2 E(u 0 ◦ X t,0 ) p c ≤ √ ∇u 0 p + c ∇u 0 2k+1, p · |t|. ν|t|
(60)
Continuing the above calculations we get c ∇ k+2 PE(u 0 ◦ X t,0 ) p ≤ √ ∇ k+1 u 0 p + c ∇u 0 2k+1, p · |t|. ν|t|
(61)
Combining (59) (60) and (61), we find c ∇PE(u 0 ◦ X t,0 ) k+1, p ≤ √ ∇u 0 k, p + c ∇u 0 2k+1, p · |t|. ν|t|
(62)
Summarizing (57) (58) and (62) yields c3 ∇u t k+1, p ≤ √ + c4 ∇u 0 k+1, p · |t| ∇u 0 k+1, p , t ∈ [−T, 0], ν|t| where c3 = c3 (k, p, d) and c4 = c4 (k, p, d) > c0 . Now, taking T0 = − 2c4 ∇u10 k+1, p
and δ = 8c32 c4 , we have for ν ≥ δ ∇u 0 k+1, p
∇u T0 k+1, p ≤ ∇u 0 k+1, p .
The proof is thus finished. 5 A large deviation estimate for stochastic particle paths
Let Gk denote the k-order diffeomorphism group on Rd , which is endowed with the locally uniform convergence topology together with its inverse for all derivatives up to k. Then Gk is a Polish space. Let Gk0 be the subspace of Gk in which each transformation preserves the Lebesgue measure, i.e., Gk0 := {X ∈ Gk : det(∇ X ) = 1}. Then Gk0 is a closed subspace of Gk , therefore, a Polish space. It is clear that t → X tν (·) ∈ Gk0
123
A stochastic representation for backward incompressible Navier-Stokes equations
331
is continuous by the theory of stochastic flow (cf. [17]). We now state a large deviation principle of Freidlin-Wentzell’s type, which follows from the results in [1,21] by using Proposition 3.10. Theorem 5.1 Keep all the things as in Proposition 3.10. For any Borel set E ⊂ C([−T, 0]; Gk0 ), we have − inf o I (Y ) ≤ lim inf ν log P(X ν ∈ E) ≤ lim sup ν log P(X ν ∈ E) ≤ − inf I (Y ), ν→0
Y ∈E
ν→0
Y ∈ E¯
where E o and E¯ denotes the interior and the closure respectively in C([−T, 0]; Gk0 ), and I (Y ) is the rate function defined by I (Y ) :=
1 inf 2 {h∈L 2 (−T,0;l 2 ):S(h)=Y }
0
h s l22 ds, Y ∈ C [−T, 0]; Gk0 ,
−T
where S(h) = Y solves the following ODE: s Ys (x) = x + −T
s u r0 (Yr (x))dr
+
σr , h r l 2 dr, s ∈ [−T, 0].
−T
Remark 5.2 In two dimensions, the T in the above theorem can be arbitrarily large by Theorem 4.2. Acknowledgments The author would like to thank Professor Benjamin Goldys for providing him an excellent environment to work in the University of New South Wales. His work is supported by ARC Discovery grant DP0663153 of Australia and NSF of China (No. 10871215).
References 1. Ben Arous, G., Castell, F.: Flow decomposition and large deviations. J. Funct. Anal. 140(1), 23–67 (1996) 2. Bismut, J.M.: Large Deviations and the Malliavin Calculus. Birkhäuser, Basel (1984) 3. Busnello, B.: A probabilistic approach to the two-dimensional Navier-Stokes equations. Ann. Probab. 27(4), 1750–1780 (1999) 4. Busnello, B., Flandoli, F., Romito, M.: 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(2), 295–336 (2005) 5. Cipriano, F., Cruzeiro, A.B.: Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus. Commun. Math. Phys. 275, 255–269 (2007) 6. Constantin, P., Iyer, G.: A stochsatic lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. Comm. Pure Appl. Math. LXI, 330–345 (2008) 7. Cruzeiro, A.B., Shamarova, E.: Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus. http://arxiv.org/abs/0807.0421 8. Cruzeiro, A.B., Zhang, X.: Bismut type formulae for diffusion semigroups on Riemannian manifolds. Potential Anal. 25(2), 121–130 (2006) 9. Esposito, R., Marra, R., Pulvirenti, M., Sciarretta, C.: A stochastic Lagrangian picture for the three-dimensional Navier-Stokes equation. Comm. Partial Differ. Equ. 13(12), 1601–1610 (1988)
123
332
X. Zhang
10. Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, INC., NewYork (1969) 11. Friedman, A.: Stochastic Differential Equations and Applications, vol. 1. Academic Press, New York (1975) 12. Iyer, G.: A stochastic perturbation of inviscid flows. Comm. Math. Phys. 266(3), 631–645 (2006) 13. Iyer, G.: A stochastic Lagrangian formulation of the Navier-Stokes and related transport equations. Doctoral dissertation, University of Chicago (2006) 14. Iyer, G.: A Stochastic Lagrangian proof of global existence of Navier-Stokes equations for flows with small Reynolds Number. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 181–189 (2009) 15. Iyer, G., Mattingly, J.: A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nonlinearity 21, 2537–2553 (2008) 16. Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001) 17. Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge Studies in Adv. Math., 24. Cambridge University Press, New York (1990) 18. Le Jan, Y., Sznitman, A.S.: Stochastic cascades and 3-dimensional Navier-Stokes equations. Proab. Theory Relat. Fields 109(3), 343–366 (1997) 19. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002) 20. Malliavin, P.: Stochastic Analysis, Grundlehren der Mathematischen Wissenschaften 313. Springer, Berlin (1997) 21. Ren, J., Zhang, X.: Freidlin-Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs. Bull. Sci. Math. 2 Ser. 129/8, pp 643–655 (2005) 22. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
123
