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arXiv:0803.1222v3 [math.PR] 16 Aug 2008

A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR THE

NAVIER-STOKES EQUATIONS.

GAUTAM IYER AND JONATHAN MATTINGLY

Abstract. This paper is based on a formulation of the Navier-Stokes equa-

tions developed by P. Constantin and the first author (arxiv:math.PR/0511067,

to appear), where the velocity field of a viscous incompressible fluid is written

as the expected value of a stochastic process. In this paper, we take N copies

of the above process (each based on independent Wiener processes), and re-

place the expected value with

Ntimes the sum over these N copies. (We

remark that our formulation requires one to keep track of N stochastic flows

of diffeomorphisms, and not just the motion of N particles.)

We prove that in two dimensions, this system of interacting diffeomor-

phisms has (time) global solutions with initial data in the space C1,αwhich

consists of differentiable functions whose first derivative is α H¨ older continuous

(see Section 3 for the precise definition). Further, we show that as N → ∞

the system converges to the solution of Navier-Stokes equations on any finite

interval [0,T]. However for fixed N, we prove that this system retains roughly

O(1

N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞

do not commute. For general flows, we only provide a lower bound to this ef-

fect. In the special case of shear flows, we compute the behaviour as t → ∞

explicitly.

1

1. Introduction

The Navier-Stokes equations

∂tu + (u · ∇)u − ν△u + ∇p = 0

∇ · u = 0

(1.1)

(1.2)

describe the evolution of a velocity field of an incompressible fluid with kinematic

viscosity ν > 0. These equations have been used to model numerous physical

problems, for example air flow around an airplane wing, ocean currents and meteo-

rological phenomena to name a few [4,15,18]. The mathematical theory (existence

and regularity [5,14]) of these equations have been extensively studied and is still

one of the outstanding open problems in modern PDE’s [7,8].

The questions addressed in this paper are motivated by a formalism of (1.1)–

(1.2) developed in [9] (equations (2.1)–(2.2) below). This formalism essentially

superimposes Brownian motion onto particle trajectories, and then averages with

respect to the Wiener measure. In this paper, we take N independent copies of

the Wiener process and replace the expected value in the above formalism with

2000 Mathematics Subject Classification. Primary 60K40, 76D05.

Key words and phrases. stochastic Lagrangian, incompressible Navier-Stokes, Monte-Carlo.

GI was partly supported by NSF grant (DMS-0707920), and thanks the mathematics de-

partment at Duke for its hospitality. JCM was partial supported by an NSF PECASE awawrd

(DMS-0449910) and a Sloan foundation fellowship.

1

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2 GAUTAM IYER AND JONATHAN MATTINGLY

1

Ntimes the sum over these N independent copies (see equations (2.5)–(2.6) for

the exact details). In the original formulation the random trajectory of a particle

induced by a single Brownian motion interacts with its own law. This is essentially

a self-consistent, mean-field interaction. In this paper, we replace this with N copies

or replica whose average is used to approximate the interaction with the processes

own law. This technique has been extensively used in numerical computation (e.g.

[2,17,19]). We remark that in our formulation, we are required to keep track of N

stochastic flows of diffeomorphisms, and not just the motion of N different particles,

as is the conventional approach.

We study both the behaviour as N → ∞ and t → ∞ of the system obtained. The

behaviour as N → ∞ is as expected: In two dimensions on any finite time interval

[0,T], the system converges as N → ∞ to the solution of the true Navier-Stokes

equations at rate roughly O(

√N). In three dimensions, we can only guarantee this if

we have certain apriori bounds on the solution (Theorem 4.1). These apriori bounds

are of course guaranteed for short time, but are unknown (in the 3-dimensional

setting) for long time [7,8].

1

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

100200 300 400 500

(a) N = 2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100200 300400 500

(b) N = 8

Figure 1. Graph of ?ωN

t?2

L2 vs time

At first glance, the behaviour as t → ∞ for fixed N is less intuitive. For the

2-dimensional problem, Figures 1(a) and 1(b) show a graph of ?ωN

N = 2 and N = 8 respectively1. A little reflection shows that this behavior is not

completely surprising. The dissipation occurs through the averaging of different

copies of the flow. With only N copies, one can only produce dissipation of order

1/N of the original energy. It is tempting, to speak of the inability to represent

the correct interaction of small scale structures with such small number of data.

However, we can not make this precise and since each of the objects being averaged

is an entire diffeomorphism with an infinite amount of information it is unclear

what this means.

t?2

L2vs time, with

1These computations were done using a 24 × 24 mesh on the periodic box with side length

2π. The initial vorticity was randomly chosen, and normalized with ?ω0?L2 = 1. The behaviour

depicted in these two figures is however characteristic, and insensitive to changes of the mesh size,

length, or diffusion coefficient

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A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES3

In Section 5 we obtain a sharp lower bound to this effect. We show (Theorem

5.2) that

E?∇ut?2

where L is a length scale. Further, we explicitly compute the t → ∞ behaviour in

the special case of shear flows and verify that our lower bound is sharp.

limsup

t→∞

L2 ?

1

NL2?u0?2

L2 ,

We remark that we considered the analogue of the system above for the one

dimensional Burgers equations. As is well known the viscous Burgers equations have

global strong solutions. However preliminary numerical simulations show that the

system above forms shocks almost surely, even for very large N. We are currently

working on understanding how to continue this system past these shocks, in a

manner analogous to the entropy solutions for the inviscid Burgers equations, and

studying its behaviour as t → ∞ and N → ∞.

We do not propose this particle system as an efficient particle method for nu-

merical computation. Though there may be special cases were it may be useful, in

general the computational cost of representing N entire diffeomorphisms is large.

Rather we see it as an interesting and novel regularization which might give useful

insight into the structure and role of dissipation in the system.

2. The particle system

In this section we construct a particle system for the Navier-Stokes equations

based on stochastic Lagrangian trajectories. We begin by describing a stochastic

Lagrangian formulation of the Navier-Stokes equations developed in [9,11].

Let W be a standard 2 or 3-dimensional Brownian motion, and u0some given

divergence free C2,αinitial data. Let E denote the expected value with respect

to the Wiener measure and P be the Leray-Hodge projection onto divergence free

vector fields. Consider the system of equations

√2ν dWt,(2.1)

ut= EP[(∇∗Yt)(u0◦ Yt)],

With a slight abuse of notation, we denote by Xtthe map from initial conditions

to the value at time t. Hence Xtis a stochastic flow of diffeomorphisms with X0

equal to the identity and Ytthe “spatial” inverse. In other words, Yt: Xt(x) ?→ x.

Also by ∇∗Yt we mean the transpose of the Jacobian of map Yt. Observe that

(∇∗Yt)(u0◦ Yt) can be viewed as a function of x where both the Jacobian and

the vector field u0◦ Ytto which it is applied are both evaluated at x. We impose

periodic boundary conditions on the displacement λt(y) = Xt(y) − y, and on the

Leray-Hodge projection P.

In [9,11] it was shown that the system (2.1)–(2.2) is equivalent to the Navier-

Stokes equations in the following sense: If the initial data is regular (C2,α), then the

pair X,u is a solution to the system (2.1)–(2.2) if and only if u is a (classical) solution

to the incompressible Navier-Stokes equations with periodic boundary conditions

and initial data u0.

We digress briefly and comment on the physical significance of (2.1)–(2.2). Note

first that equation (2.2) is algebraically equivalent to the equations

ut= −△−1∇ × ωt

ωt= E[(∇Xt)ω0] ◦ Yt

dXt(x) = ut(Xt(x))dt +

X0(x) = x,

Yt= X−1

t .(2.2)

(2.3)

(2.4)

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4GAUTAM IYER AND JONATHAN MATTINGLY

This follows by direct computation, and was shown [6] and [9] for instance. We

recall that (2.4) is the usual vorticity transport equation for the Euler equations,

and (2.3) is just the Biot-Savart formula.

Thus in particular when ν = 0, the system (2.1)–(2.2) is exactly the incompress-

ible Euler equations. Hence the system (2.1)–(2.2) essentially does the following:

We add Brownian motion to Lagrangian trajectories. Then recover the velocity u

in the same manner as for the Euler equations, but additionally average out the

noise.

We remark that the system (2.1)–(2.2) is non-linear in the sense of McKean

[20]. The drift of the flow X depends on its distribution. However in this case,

the law of X alone is not enough to compute the drift u. This is because of the

presence of the ∇∗Y term in (2.2), which requires knowledge of spatial covariances,

in addition to the law of X. In other words, one needs that law of the entire flow

of diffeomorphism and not just the law of the one-point motions.

We now motivate our particle system. For the formulation (2.1)–(2.2) above,

the natural numerical scheme would be to use the law of large numbers to compute

the expected value. Let (Wi) be a sequence of independent Wiener processes, and

consider the system

dt +√2ν dWi

(2.5)

dXi,N

t

= uN

t

?

N

?

Xi,N

t

?

(∇∗Yi,N

t,Yi,N

t

=

?

Xi,N

t

?−1

uN

t=

1

N

i=1

P

?

t

)(u0◦ Yi,N

t

)

?

(2.6)

with initial data X0(x) = x. We impose again periodic boundary conditions on the

initial data u0, the displacement λt(x) = Xt(x)−x, and the Leray-Hodge projection

P.

We remark that the algebraic equivalence of (2.2) and (2.3)–(2.4) is still valid

in this setting. Thus the system (2.5)–(2.6) could equivalently be formulated by

replacing equation (2.6) with the more familiar equations

(2.7)ωN

t =

1

N

N

?

i=1

?

(∇Xi,N

t

)ω0

?

◦ Yi,N

t

and

(2.8)uN

t = −△−1∇ × ωN

t.

Finally we clarify our previous remark, stating that the above formulation re-

quires us to keep track of N stochastic flows, and the knowledge of the one point

motions of Xi,N

t

alone is not sufficient. The standard method of obtaining a solu-

tion to the heat equation (assuming the drift u is time independent) would be to

consider the process (2.1), and read off the solution θ by

θt(a) = Eθ0◦ Xt(a)

where θ0is the given initial temperature distribution. Thus knowing the trajecto-

ries (and distribution) of the process X starting at one particular point a will be

sufficient to determine the solution θtat that point.

This however is not the case for our representation. The reason is twofold:

First, the representation (2.1)–(2.2) involves a non-local singular integral operator.

Second our representation involves composing with the spatial inverse of the flow

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A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES5

Xt, and then averaging. If we for a moment ignore the non-locality of the Leray-

Hodge projection, determining utat one fixed point a one would need the law of

Yt(a), for which the knowledge of Xt(a) alone is not enough. One needs the entire

(spatial) map Xtto compute the spatial inverse Yt(a).

The above is not a serious impediment to a numerical implementation. Given an

initial mesh ∆, we first compute Xi,N

t

on this mesh. By definition of the inverse, one

knows Yi,N

t

on the (non-uniform) mesh Xi,N

and find Yi,N

t

on the mesh ∆. In two spatial dimensions, global existence and

regularity (Theorem 3.5) together with incompressibility will show that this mesh

does not degenerate in finite time.

This surprisingly is not the case for the (one dimensional) Burgers equations.

Numerical computations indicate the mesh ∆ almost surely degenerates in finite

time for non-monotone initial data, and the solution ‘shocks’ almost surely. Thus,

while the system (2.5)–(2.6) appears natural, and convergence as N → ∞ is to

be expected, caution is to be exercised. We suspect that the results (existence,

convergence, etc.) proven in this paper for the system (2.5)–(2.6) are in fact false

for the Burgers equations. This is indeed puzzling as global existence, and regularity

for the viscous Burgers equations is well known. It further underlines the fact that

the finite N approximation modifies the dissipation in a different way then other

approximation such as a spectral approximation.

t

(∆), after which one can interpolate

In the next section, we show the existence of global solutions to (2.5)–(2.6) in

two dimensions. In section 4, we show that the solution to (2.5)–(2.6) converges

to the solution of the Navier-Stokes equations as N → ∞. Finally in Section 5 we

study the behaviour of the system (2.5)–(2.6) as t → ∞ (for fixed N), and partially

explain the behaviour shown in Figures 1(a) and 1(b).

3. Global existence of the particle system in two dimensions.

In this section we prove that the particle system (2.5)–(2.6) has global solutions

in two dimensions. Once we are guaranteed global in time solutions, we are able

to study the behaviour as t → ∞, which we do in Section 5. We remark also that

as a consequence of Theorem 3.5 (proved here), our convergence result as N → ∞

(Theorem 4.1) applies on any finite time interval [0,T] in the two dimensional

situation.

We first establish some notational convention: We let L > 0 be a length scale,

and assume work with the spatial domain is [0,L]d, where d ? 2 is the spatial

dimension. We define the non-dimensional Lpand H¨ older norms by

?u?Lp =

?

1

Ld

?

sup

[0,L]d|u|p

?1

p

?u?L∞ =

x∈[0,L]d|u(x)|

|u(x) − u(y)|

|x − y|α

Lk|Dmu|α+

|u|α= Lα

sup

x?=y∈[0,L]d

?u?k,α=

?

|m|=k

?

|m|<k

L|m|?Dmu?L∞

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6GAUTAM IYER AND JONATHAN MATTINGLY

We denote the H¨ older space Ck,αand Lebesgue space Lpto be the space of functions

u which are periodic on [0,L]dwith ?u?k,α< ∞ and ?u?Lp< ∞ respectively.

Let W1,...,WNbe N independent (2 dimensional) Wiener processes, with fil-

tration Ft.

Proposition 3.1. Let t0? 0, and u1

functions such that the norms ?ui

T = T(α,?u1

?t

1

N

i=1

has a solution on the interval [t0,T].

U(α,?ui

(3.3) sup

t∈[t0,T]

almost surely. Consequently, u ∈ C([t0,T],C1,α), and ?u?1,α? U.

Proposition 3.1 is proved in Appendix A

Definition 3.2. We call Xi, u in Proposition 3.1 the solution of the system (3.1)−

−(3.2) with initial data u1

The functions Xi,Yiare not periodic themselves, but have periodic displace-

ments: Namely, if we define

λi

(3.4)

t0,...,uN

t0?1,αare almost surely bounded. Then there exist

t0?1,α) such that the system

t0,s(x))ds +√2ν(Wi

t0be Ft0-measurable, periodic mean 0

t0?1,α,...,?uN

Xi

t0,t(x) = x +

t0

us(Xi

t− Wi

t0),Yi

t0,t= (Xi

t0,t)−1

(3.1)

ut(x) =

N

?

P?(∇∗Yi

t0,t)(ui

t0◦ Yi

t0,t)?(x)

Further there exists deterministic U =

(3.2)

t0?1,α) such that

??P?(∇∗Yi

t0,t)ui

t0◦ Yi

t0,t

???

1,α? U

t0,...,uN

t0.

t(y) = Xi

t(x) = Yi

t(y) − y

t(x) − x ,µi

(3.5)

then µi,λiare periodic.

We remark that if t0 = 0 and the ωi

(3.2) reduces to (2.5)–(2.6). However when formulated as above, solutions can be

continued past time T by restarting the flows Xi, as in the Lemma below.

Lemma 3.3. Say t0 < t1 < t2, and Xi

initial data ui

t0. For any t > t0define

ui

t0’s are all equal, then the system (3.1)–

t0,s, us solve (3.1)–(3.2) on [t0,t2], with

t= P?(∇∗Yi

t0,t)ui

t0◦ Yi

t0,t

?.

Let˜ Xi

we have ˜ ui

t1,s, ˜ ussolve (3.1)–(3.2) on [t1,t2] with initial data ui

s= ui

Xi

almost surely.

t1. Then for all s ∈ [t1,t2]

sand

t0,s=˜ Xi

t1,s◦ Xi

t0,t1

Proof. The proof is identical to the proof of Proposition 3.3.1 in [11], and we do

not provide it here.

?

For the remainder of this section, we assume without loss of generality that

t0= 0 (we allow of course F0to be non-trivial). For ease of notation, we use Xs

to denote X0,s. We now prove that the system (3.1)–(3.2) has global solutions in

two dimensions. This essentially follows from a Beale-Kato-Majda type condition

[1], and the two dimensional vorticity transport.

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A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES7

Lemma 3.4. If u is divergence free and periodic in Rd, then for any α ∈ (0,1),

there exists a constant c = c(α,d) such that

?∇u?L∞ ? c?ω?L∞

?

1 + ln+

?

|ω|α

?ω?L∞

??

where ω = ∇ × u.

The lemma is a standard fact about singular integral operators, and we provide

a proof in Appendix B for completeness.

Theorem 3.5. In 2 dimensions, the system (3.1)–(3.2) has time global solutions

provided the initial data u1

almost surely. In particular we have global solutions to (3.1)–(3.2) in two dimen-

sions, if u1

0= u0is deterministic, H¨ older 1 + α and periodic.

0,...uN

0is periodic, F0-measurable with ?ui

0?1,αbounded

0= ··· = uN

Proof. Taking the curl of (3.2) gives the familiar Cauchy formula [6,9,10]

(3.6)ωt=

1

N

N

?

i=1

?(∇Xi

t)ωi

0

?◦ Yi

t,

where ωt= ∇ × ut. In two dimensions reduces to

(3.7)ωt=

1

N

N

?

i=1

ωi

0◦ Yi

t.

Taking H¨ older norms gives

(3.8)?ωt?α?

c

N

N

?

i=1

??ωi

0

??

α(1 +??∇Yi??α

L∞)a.s.

Now differentiating (3.1) gives

∇Xi

t= I +

?t

0

(∇us◦ Xi

s)∇Xi

sds a.s.

Taking the L∞norm, and applying Gronwall’s Lemma shows

??∇Xi

t

??

L∞ ? exp

?

c

?t

0

?∇us?L∞ ds

?

a.s.

Recall ∇ · u = 0, and hence det(∇Xi) = 1 almost surely. Thus the entries of ∇Y

are a polynomial (of degree 1) in the entries of ∇X. This immediately gives

??∇Yi

almost surely. Combining this with (3.8) gives us the apriori bound

N

?

Applying Lemma 3.4 gives us

?

?

N

i=1

(3.9)

t

??

L∞ ? exp

?

c

?t

0

?∇us?L∞ ds

?

(3.10)?ωt?α?

c

N

i=1

??ωi

0

??

αexp

?

c

?t

0

?∇us?L∞ ds

?

?∇u?L∞ ? c?ω?L∞

1 + ln+

?

?

|ω|α

?ω?L∞

??

|ω|α

? c?ω?L∞

1 + ln+

1

?N

??ωi

0

??

L∞

?

+ ln+

?

1

N

?N

i=1

?ω?L∞

??ωi

0

??

L∞

??

Page 8

8GAUTAM IYER AND JONATHAN MATTINGLY

Note that the function xln+ 1

bounded above by some constant c0. For the remainder of the proof, we let c0=

c0(?ωi

to line. Thus

?

xis bounded, so the last term on the right can be

0?α,α) denote a constant (with dimensions that of ω) which changes from line

?∇ut?L∞ ? c0+ c?ωt?L∞

1 + ln+

?

|ωt|α

1

N

?N

i=1

??ωi

0

??

L∞

??

? c0+ c?ωt?L∞

?

1 +

?t

0

?∇us?L∞ ds

?

and hence

?∇ut?L∞

?

1 +

?t

0

?∇us?L∞ ds

?−1

? c0+ c?ωt?L∞ .

Integrating gives us the apriori bound

?t

(3.11)

0

?∇ut?L∞ ? exp

?

c0t + c

?t

0

?ωt?L∞

?

− 1

? c0tec0t

since (3.7) implies ?ωt?L∞?

Now if we set ωi

1

N

??ωi

??

0?L∞.

t= ωi

0◦ Yi

t, then (3.9) and the apriori bound (3.11) gives

??ωi

α? c??ωi

t?1,α, which in conjunction with local existence (Proposition

3.1), and Lemma 3.3 concludes the proof.

(3.12)

t

α?

??ωi

??

0

??

αexp?c0tec0t?

If ui

tis as in Lemma 3.3, then (3.12) shows

(3.13)

??∇ui

gives a bound on ?ui

t

??

t

α? c?ω0?αexp?c0tec0t?.

Since the mean velocity is a conserved quantity, a bound on ?∇ui

t?αimmediately

?

4. Convergence as N → ∞

In this section, we fix a time interval [0,T], and show that the particle system

(2.5)–(2.6) converges to the solution to the Navier-Stokes equations as N → ∞.

The rate of convergence is O(

√N), which is comparable to the convergence rate of

the random vortex method [3,16]. As mentioned earlier, the system is intrinsically

non-local, and propagation of chaos [20] type estimates are not easy to obtain.

Consequently convergence results based on spatially averaged norms are easier to

obtain, and we present one such result in this section, under assumptions which are

immediately guaranteed by local existence.

1

Theorem 4.1. For each i,N, let Xi,N, uNbe a solution to the particle system

(2.5)–(2.6) with initial data u0 on some time interval [0,T]. Let u be a solution

to the Navier-Stokes equations (with the same initial data) on the interval [0,T].

Suppose U is such that

sup

t∈[0,T]?∇ut?L2 ?U

L

and sup

t∈[0,T]

???∇ui,N

??

t

???

L2?U

L

a.s.

Then (uN) → u in the following sense:

lim

N→∞

sup

t∈[0,T]

E??uN

t− ut

L2 = 0

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A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES9

We remark that given C1,αinitial data, local existence (Proposition 3.1) guaran-

tees that the conditions of this theorem are satisfied on some small interval [0,T]. In

two dimensions, Theorem 3.5 shows that the conditions of this theorem are satisfied

on any interval finite [0,T].

The proof will follow almost immediately from the following Lemma.

Lemma 4.2. Let ui,N

ui,Nsatisfies the SPDE

t

= P[(∇∗Yi,N

t

)u0◦Yi,N

t

] be the ithsummand in (2.6). Then

(4.1)dui,N

t

+

?

(uN

t· ∇)ui,N

t

− ν△ui,N

t

+ (∇∗uN

t)ui,N

t

+ ∇pi,N

t

?

dt+

+

√2ν∇ui,N

t

dWi

t= 0

and uNsatisfies the SPDE

(4.2) duN

t+?(uN

t· ∇)uN

t− ν△uN

t+ ∇pN

t

?dt +

√2ν

N

N

?

i=1

∇ui,N

t

dWi

t= 0

Remark. We draw attention to the fact that the pressure term in (4.2) has bounded

variation in time.

Proof. We first recall a fact from [9,11] (see also [12]). If X is the stochastic flow

dXt= utdt +√2ν dWt

and Yt = X−1

t

is the spatial inverse. Then the process θt = f(Yt) satisfies the

SPDE

dθt+ (ut· ∇)θtdt − ν△θtdt +√2ν∇θtdWt= 0

This immediately shows that Yi,N

t

and vi,N

(4.3). For notational convenience, we momentarily drop the N as a superscript and

use the notation vi,jto denote the jthcomponent of vi.

Now we set wi

tand apply Itˆ o’s formula:

(4.3)

t

= u0◦ Yi,N

t

both satisfy the SPDE

t= (∇∗Yi

= d(∂jYi

t)vi

dwi,j

tt) · vi

t) ·?−(ut· ∇)vi

+ vi

−√2νvi

=?−(ut· ∇)wi

t+ (∂jYi

t) · dvi

t+ ν△vi

t− (ut· ∇)∂jYi

t·?∇∂jYi

t+ d

?

∂jYi,k

t

√2ν∂jYi

,vi,k

t

?

t·?∇vi

∂lvi,k

t

dt

?dt −

+ ∇qi,N

= (∂jYi

t

?dt −

tdWi

?dt−

t

?+

t·?−((∂jut) · ∇)Yi

t+ ν△∂jYi

t

tdWi

t

?+ 2ν∂2

jlYi,k

t

t+ ν△wi

t− (∇∗ut) · wi

t

√2ν∇wi

tdWi

t

Restoring the dependence on N to our notation, since ui,N= Pwi,Nwe know that

ui,N

t

= wi,N

tt

for some function qi,N

t

. Thus

dui,N

t

= dwi,N

t

+ d(∇qi,N

t· ∇)wi,N

−(uN

+

?

t

)

=

?

?

−(uN

t

+ ν△wi,N

+ ν△ui,N

t

− (∇∗uN

− (∇∗uN

) + ν△(∇qi,N

t)wi,N

t

?

dt −

dt −

t)(∇qi,N

√2ν∇wi,N

√2ν∇ui,N

t

dWi

t

=

t· ∇)ui,N

−(uN

ttt)ui,N

t

?

t

dWi

t+

t· ∇)(∇qi,N

tt

) − (∇∗uN

t

)

?

dt−

Page 10

10 GAUTAM IYER AND JONATHAN MATTINGLY

−

√2ν∇(∇qi,N

by

t

)dWi

t+ d(∇qi,N

t

).

If we define Pi,N

t

Pi,N

t

=

?t

0

?(uN

s· ∇)qi,N

s

− ν△qi,N

s

?ds +

?t

0

√2ν∇qi,N

s

· dWi

s+ qi,N

t

then we have

(4.4) dui,N

t

+

?

(uN

t· ∇)ui,N

t

− ν△ui,N

t

+ (∇∗uN

t)ui,N

t

?

dt

+ d(∇Pi,N

t

) +

√2ν∇ui,N

t

dWi

t= 0.

Notice that ui,Nis divergence free by definition, and thus ∇ui,NdWiis also diver-

gence free. Thus d(∇Pi,N), the only other term with possibly non-zero quadratic

variation, must have a divergence free martingale part. Since the martingale part

of d(∇Pi,N) is also a gradient, it must be 0. Thus

d(∇Pi,N

for some function pi,N

t

, which proves (4.1).

The identity (4.2) now follows by summing (4.1) in i, dividing by N, and defining

pN

t by

t

) = ∇pi,N

t

dt

pN

t=1

2∇??uN

t

??2+1

N

N

?

i=1

pi,N

t

?

Proof of Theorem 4.1. Let u be a solution of the Navier-Stokes equations, with

initial data u0, and set vN= uN− u. Then vNsatisfies the SPDE

dvN

t+ (vN

t· ∇)utdt + (ut· ∇)vN

tdt − ν△vN

tdt + ∇(pN

t− pt)dt+

√2ν

N

i=1

+

N

?

∇ui,N

t

dWi

t= 0,

where p is the pressure in the Navier-Stokes equations, and pNthe pressure term

in (4.2).

Thus by Itˆ o’s formula,

1

2d??vN

t

??2

L2 +?vN

t,(vN

t· ∇)ut

?dt + ν??∇vN

+2ν

N

i=1

t

??2

t

L2 dt+

N

?

vN

t· (∇ui,N

dWi,k

t ) =

ν

N2

N

?

i=1

???∇ui,N

t

???

2

L2dt.

Here the notation (f,g) denotes the L2innerproduct of f and g. Taking expected

values gives us

1

2∂tE??vN

??2

concluding the proof.

t

??2

L2 ?U

LE??vN??2

L2 ?2νU2

L2Nte

L2 +νU2

NL2

and by Gronwall’s lemma we have

E??vN

t

Ut

L

?

Page 11

A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES11

5. Convergence as t → ∞

In this section, we fix N, and consider the behaviour of the system (2.5)–(2.6)

as t → ∞. We show that the system (2.5)–(2.6) does not dissipate all its energy as

t → ∞. Roughly speaking we show

limsup

t→∞

with constants independent of viscosity. This is in contrast to the true (unforced)

Navier-Stokes equations, which dissipate all of its energy as t → ∞ (provided of

course the solutions are defined globally in time).

In general we are unable to compute exact asymptotic behaviour of the system

(2.5)–(2.6) as t → ∞. But in the special case of shear flows, we compute this

exactly, and show that the system eventually converges to a constant, retaining

exactly

Ntimes its initial energy.

For the remainder of this section, we pick a fixed N ∈ N and for notational conve-

nience we omit the superscript N. We begin by computing exactly the asymptotic

behaviour of the system (2.5)–(2.6) in the special case of shear flows.

E?∇ut?2

L2 ? O?1

N

?,

1

Proposition 5.1. Suppose the initial data u0(x) = (φ0(x2),0) for some C1,αpe-

riodic function φ. If u is the velocity field that solves the system (2.5)–(2.6) with

initial data u0, then

(5.1) lim

t→∞Eωt(x)2=

1

N?ω0?2

L2

where ω = ∇ × u is the vorticity.

Proof. Let Xi, Yibe the flows in the system (2.5)–(2.6), and as before define uito

be the ithsummand in (2.6), and ωi= ω0◦ Yi.

First note that the SPDE’s for uiand u (equations (4.1) and (4.2)) are all

translation invariant. Thus since the initial data is independent of x1, the same

must be true for all time. Since ui, u are divergence free, the second coordinate

must necessarily be 0, and the form of the initial data is preserved. Namely,

ut(x) = (φt(x2),0) andui

t(x) = (φi

t(x2),0)

for some C([0,∞),C1,α) periodic functions φi,φ.

Now the definition of Xishows that

Xi

t(y) =

?y1+ λi,1

t(y2)

y2+√2νWi,2

t

?

and hence

Yi

t(x) =

?x1+ µi,1

t(x2)

x2−√2νWi,2

t

?

.

Recall λi, µiare as in (3.4), (3.5), and here the notation λi,1to denotes the first

coordinate of λi. This immediately shows

ωi

t(x) = ω0◦ Yi

= −∂2φ0(x2−

t(x)

√2νWi,2

t )

where Wi,2again denotes the second coordinate of the Brownian motion Wi.

Page 12

12 GAUTAM IYER AND JONATHAN MATTINGLY

Now using standard mixing properties of Brownian motion [13, Section 1.3], (or

explicitly computing in this case) we know that for every x ∈ [0,L]2

(5.2) lim

t→∞E

and

?

= lim

t→∞

?

L2

= 0 (5.3)

t→∞Eωi

t(x)2= lim

?

−∂2φ0(x2−

√2νWi,2

t )

?2

= ?ω0?2

L2

lim

t→∞Eωi

t(x)ωj

t(x) = lim

t→∞E

∂2φ0(x2−

E∂2φ0(x2−√2νWi,2

√2νWi,2

t )∂2φ0(x2−

? ?

√2νWj,2

t )

?

t )

?

t )

E∂2φ0(x2−√2νWj,2

?

=

1

?

[0,L]2∂2φ0

?2

when i ?= j.

Now by two dimensional Cauchy formula (3.7)

ωt=

1

N

N

?

i=1

ω0◦ Yi

t.

(since in our case, ω1

0= ··· = ωN

1

N2

i=1

0= ω0). Thus

(5.4)

Eωt(x)2=

N

?

Eω0◦ Yi

t(x)2+

2

N2

N

?

i=2

i−1

?

j=1

Eω0◦ Yi

t(x)ω0◦ Yj

t(x)

and using (5.2) and (5.3) the proof is complete.

?

We remark that all we need for (5.1) to hold is the identities (5.2) and (5.3).

Equality (5.2) is guaranteed provided reasonable ergodic properties of the flow Xi

are known. Equality (5.3) is guaranteed provided the flows Xi

decorrelate.

While we are unable to guarantee these properties for a more general class of

flows, we conclude this section by proving a weaker version of (5.1) for two dimen-

sional flows with general initial data.

Theorem 5.2. Let Xi,u be a solution to the system (2.5)–(2.6) with (spatial)

mean zero initial data u0∈ C1,αand periodic boundary conditions. Suppose further

u ∈ C([0,∞),C1,α). Then

(5.5)limsup

t→∞

Note that the assumption u ∈ C([0,∞),C1,α) is satisfied in the two dimensional

situation with C1,αinitial data (Theorem 3.5). The proof we provide below will

also work in the three dimensional situation, as long as global existence and well-

posedness of (2.5)–(2.6) is known.

As is standard with the Navier-Stokes equations, the condition that u0is (spa-

tially) mean zero is not a restriction. By changing coordinates to a frame moving

with the mean of the initial velocity, we can arrange that the initial data (in the

new frame) has spatial mean 0.

Finally we remark that the lower bound in inequality (5.5) is sharp, since in the

special case of shear flows we have the equality (5.1). However we are unable to

obtain a bound on liminf E?∇ut?2

t

tand Xj

teventually

E?∇ut?2

L2 ?

1

NL2?u0?2

L2

L2.

Page 13

A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES13

Proof of Theorem 5.2. As before let ui

and Itˆ o’s formula we have

t= P?(∇∗Yi

t)u0◦ Yi

t

?. Using Lemma 4.2

(5.6)

1

2d??ui??2+ ui·?(u · ∇)ui− ν△ui+ (∇∗u)ui+ ∇pi?dt+

Note that

?

Thus integrating (5.6) in space, and using the fact that uiis divergence free gives

+√2νui· (∇uidWi) = ν??∇ui??2dt

((∇u)ui) · ui=ui· ((∇∗u)ui) =

??

ui· (u · ∇)ui= 0

d??ui??2

L2 = 0

and hence ?ui

Now suppose that for some ε > 0, there exists t0such that for all t > t0

t?L2 = ?ui

0?L2 almost surely.

E?∇ut?2

L2 ?

1

NL2?u0?2

L2 − ε.

Using Itˆ o’s formula and (4.2) gives

1

2d|u|2+ u · [(u · ∇)u − ν△u + ∇p] dt +

√2ν

N

N

?

i=1

u · (∇uidWi)

=

ν

N2

N

?

i=1

??∇ui??2dt

Integrating in space, and taking expected values gives

1

2ν∂tE?ut?2

L2 = E

?

1

N2

N

?

i=1

??∇ui

N

?

?u0?2

t

??2

??2

L2 − E?∇ut?2

L2 − ?∇ut?2

L2

?

? E

?

1

N2L2

i=1

??ui

t

L2 − ?∇ut?2

L2

?

=

1

N2L2

N

?

i=1

L2

=

1

NL2?u0?2

? ε

L2 − E?∇ut?2

L2

for t ? t0. Here we used the Poincar´ e inequality to obtain the second inequality

above. Note that we have assumed that the initial data has (spatial) mean 0. Since

the (spatial) mean is conserved by the system (2.5)–(2.6), utalso has (spatial) mean

zero, and our application of the Poincar´ e inequality is valid.

Now, the above inequality immediately implies E?ut?2

as t → ∞. This is a contradiction because

?????

L2becomes arbitrarily large

?ut?L2 =

1

N

N

?

i=1

ui

t

?????

L2

?

1

N

N

?

i=1

??ui

t

??

L2 = ?u0?L2

holds almost surely.

?

Page 14

14 GAUTAM IYER AND JONATHAN MATTINGLY

Appendix A. Local existence.

In this appendix we provide the proof of Proposition 3.1. A similar proof ap-

peared in [10] (see also [6]), and the proof provided here is based on similar ideas.

We present the proof here because we require local existence for C1,αinitial data

(the proof in [10] used C2,α), and to ensure that bounds and existence time therein

are independent of N.

Without loss, we assume t0= 0, and u1

a large constant and T a small time, both of which will be specified later.

Define U = U(T,U) be the set of all time continuous Ft-adapted C1,αvalued

divergence free and spatially periodic processes u such that

0, ..., uN

0to be F0measurable. Let U be

u0=

1

N

N

?

i=1

ui

0

andsup

t∈[0,T]

?ut?1,α? U

hold almost surely. Also, we define M = M(T) to be the set of all time continuous

Ft-adapted C1,αvalued spatially periodic processes µ such that

µ0= 0and sup

t∈[0,T]

hold almost surely.

Now given u ∈ U let Xi,ube the flow solving the SDE

dXi,u

t

= ut(Xi,u

with initial data Xi,u

?∇µt?α?1

2

t )dt +

√2ν dWi

t

0(y) = y. As before, define Yi,u

λi,u

µi,u

t

= (Xi,u

t )−1, and define

t (y) = Xi,u

t(x) = Yi,u

t (y) − y

(x) − x

t

to be the Eulerian and Lagrangian displacements respectively.

Finally define the (non-linear) operator W by

1

N

W(u)t=

N

?

i=1

P

??

∇∗Yi,u

t

? ?

ui

0◦ Yi,u

t

??

Clearly a fixed point of W will produce a solution to the system (3.1)–(3.2). Thus

the proof will be complete if we show that for an appropriate choice of T and U,

W maps U into itself, and is a contraction with respect to the weaker norm

?u?U= sup

t∈[0,T]?ut?α

We first show W maps U into itself, using the two lemmas below.

Lemma A.1. There exists c = c(α) such that

?W(u)?1,α? c

?

max

1?i?N

?1 +??∇µi,u??

α

?2+α?1

N

N

?

i=1

??ui

0

??

1,α

a.s.

Proof. First recall P vanishes on gradients. Thus

(A.1)

P[(∇∗Y )v] = −P[(∇∗v)Y ].

Now

∂iP[(∇∗Y )v] = P[(∇∗Y )∂iv + (∇∗∂iY )v]

Page 15

A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR NAVIER-STOKES15

= P[(∇∗Y )∂iv − (∇∗v)∂iY ]

where we used (A.1) for the second term. Note that the right hand side involves only

first order derivatives. Since P is a standard Calder´ on-Zygmund singular integral

operator, which is bounded on H¨ older spaces, we obtain the estimate

?P[(∇∗Y )v]?1,α? c?∇∗Y ?α?v?1,α

for some constant c = c(α).

Applying this estimate to W, we have

(A.2)?W(u)t?1,α?

c

N

N

?

i=1

???∇∗Yi,u

t

???

α

???ui

0◦ Yi,u

t

???

1,α

a.s.

from which the Lemma follows.

?

Lemma A.2. There exists T = T(U,α) such that λi,u,µi,u∈ M(T).

We note that the diffusion coefficient is spatially constant, and thus we get the

desired (almost sure) control on ∇λ. Since ∇ · u = 0, det(∇Xi,u) = 1, giving the

desired control on ∇µ. The details are standard, and we do not provide the them

here (see for instance [10,13]).

Now choosing U = c(3

Lemma A.1 shows that W maps U(U,T) into itself. Note that each summand on

the right of (A.2) is bounded by U, which will prove the bound (3.3). Further,

given a uniform (in i) bound on ?ui

of N.

It remains to show that W is a contraction. By definition of W we have

1

N

i=1

2)2+α 1

N

??ui

0?1,α, and then choosing T as in Lemma A.2,

0?1,α, our choice of U can be made independent

W(u)t− W(v)t=

N

?

N

?

+1

P

?

(∇∗Yi,u

t

)ui

0◦ Yi,u

t

− (∇∗Yi,v

t

)ui

0◦ Yi,v

t

?

=

1

N

i=1

P

?

(∇∗Yi,u

t

)

?

ui

0◦ Yi,u

t

− ui

0◦ Yi,v

t

??

+

N

N

?

?

N

?

i=1

P

??

∇∗Yi,u

t

− ∇∗Yi,v

t

?

ui

0◦ Yi,v

t

?

=

1

N

N

?

−1

i=1

P

(∇∗Yi,u

t

)

?

ui

0◦ Yi,u

t

− ui

0◦ Yi,v

t

??

−

N

i=1

P

?

∇∗(ui

0◦ Yi,v

t

)

?

Yi,u

t

− Yi,v

t

??

where we used the identity (A.1) to obtain the last equality. Now we recall that

µi,u,µi,v∈ M, and take Cαnorms. This gives

c

LN

i=1

Now from the definition of Yi,uand Yi,vwe have

?t

(A.3)?W(u)t− W(v)t?α?

N

?

??ui

0

??

1,α

???Yi,u

t

− Yi,v

t

???

α

a.s.

Yi,u

t

− Yi,v

t

=

0

?us(Yi,u

s

) − vs(Yi,v

s )?ds.