Generalized stochastic Lagrangian paths for the Navier-Stokes equation

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Generalized stochastic Lagrangian paths for the Navier-Stokes equation

Marc Arnaudon, Ana Bela Cruzeiro, Shizan Fang

To cite this version:

Marc Arnaudon, Ana Bela Cruzeiro, Shizan Fang. Generalized stochastic Lagrangian paths for the Navier-Stokes equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2018, 18 (3), pp.1033-1060. �10.2422/2036-2145.201602_006�. �hal-01197123v2�

Generalized stochastic Lagrangian paths for the Navier-Stokes equation

Marc Arnaudon^a Ana Bela Cruzeiro^b and Shizan Fang^c

aInstitut de Maths. de Bordeaux, Universit´e de Bordeaux I, 33405 Talence Cedex, France

bGFMUL and Dep. de Mat. Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

cInstitut de Maths. de Bourgogne, Universit´e de Bourgogne, 21078 Dijon Cedex, France

February 18th, 2016

Abstract

In the note added in proof of the seminal paper [Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a compact Riemannian manifold. In spirit of Brenier’s generalized flows for the Euler equation, we introduce a class of semimartingales on a compact Riemannian manifold.

We prove that these semimartingales are critical points to the corresponding kinetic energy if and only if its drift term solves weakly the Navier-Stokes equation defined with Ebin-Marsden’s Laplacian. We also show that for the case of torus, classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.

1 Introduction

Euler equations describe the velocity of incompressible non-viscous fluids. Considering these equations on a bounded domain U ofR^d, or on a compact Riemannian manifoldM without boundary, they read

d

dtu_t+ (u_t· ∇)u_t=−∇p, div(u_t) = 0. (1.1) Lagrange’s point of view consists in describing the position of the particles: for a solution u to (1.1), it concerns solutions of the ordinary differential equation (ODE)

d

dtg_t(x) =u_t(g_t(x)), g₀(x) =x. (1.2) When (t, x) → u_t(x) is smooth, the ODE (1.2) defines a flow of C^∞-diffeomorphismsg_t. From the position values, we get the velocity by

ut(x) = d dtgt

(g_t⁻¹(x)).

In this case, the two points of view are equivalent. Throughout the whole paper we shall consider the interval of time [0, T]. Equation (1.2) defines a continuous map

g_·: [0, T]→Diff(M)

from [0, T] to the group of diffeomorphisms ofM.

In a famous work [6], V.I. Arnold gave a geometric interpretation to the incompressible Euler equation, saying that u is a solution to (1.1) if and only if t→ g_t is a geodesic on the submanifold of Diff(M) keeping the volume measure invariant, equipped with the L² metric. Equivalently, g_· minimizes the action

S[ϕ] = 1 2

Z _T

0

Z

M

d dtϕ_t(x)

2

TxMdxdt (1.3)

on C([0, T],Diff(M)), where dx denotes the normalized Lebesgue measure on U or the normalized Riemannian volume on M (see also [14]).

In [7], Y. Brenier gave a probabilistic interpretation to (1.1), by looking for probability measures η on the path spaceC([0, T], M), which minimize the kinetic energy

S[η] = 1 2

Z

C([0,T],M)

hZ T

0 |γ˙(t)|²Tγ(t)Mdti

dη(γ), (1.4)

with constraints (et)∗η=dx, whereet:γ →γ(t) denotes the evaluation map. Let X(γ, t) =γ(t).

Then under η, {X(·, t);t ≥0} is a M-valued stochastic process. Moreover, in [7] as well as in [8], Brenier proved that such a probability measure η gives rise to a weak solution of the Euler equation in the sense of Di Perna and Majda [13]. More precisely, define a probability measure µon [0, T]×T M by

Z

[0,T]×T M

f(t, x, v)µ(dt, dx, dv)

= 1 T

Z _T

0

Z

C([0,T],M)

f(t, γ(t), γ^′(t))dη(γ)dt.

Then µsolves the Euler equation in generalized sense:

Z h

v·w(x)α^′(t) +v·(∇w(x)·v)α(t)i

µ(dt, dx, dv) = 0

for anyα∈C^∞(]0, T[) and any smooth vector fieldwsuch that div(w) = 0. We also refer to [1] in which the authors used the theory of mass transportation.

In this work, we will deal with Navier-Stokes equations on a compact Riemannian manifold M. There are two natural ways to define the “Laplace” operator on vector fields. The first way is to use the de Rham-Hodge Laplace operator on differential 1-forms, that is =dd^∗+d^∗d. As usual, for a vector fieldA, we denote by A^♯ the associated differential 1-form; for a differential 1-form ω, we denote by ω^♭ the corresponding vector field. Then we define

A= (A^♯)^♭. The Weitzenb¨ock formula states that

−A= ∆A−Ric(A) (1.5)

where ∆A = Trace(∇²A) and Ric is the Ricci tensor. Another natural way, following [14], is to use the deformation tensor. More precisely, let A be a vector field on M, the deformation tensor DefA is a symmetric tensor of type (0,2) such that

(DefA)(X, Y) = 1 2

h∇XA, Yi+h∇YA, Xi

. (1.6)

Then Def :T M →S²T^∗M which sends a vector field to a symmetric tensor of type (0,2).

Let Def^∗ :S²T^∗M → T M be the adjoint operator. According to [25], as well as to [27], we define

ˆ = 2Def^∗Def. (1.7)

Then on vector fields of divergence free A, it holds true (see [25, 26])

−ˆA= ∆A+ Ric(A). (1.8)

Comparing (1.8) to (1.5), the sign of Ric is opposite.

In this work, we will consider the following Navier-Stokes equation on M d

dtu_t+∇utu_t+νˆu_t=−∇p, div(u_t) = 0, (1.9) where ν >0 is the viscosity coefficient. Since div(u_t) = 0, we have R

Mh∇utu_t, u_tidx= 0.

Using the relation ˆ=−2Ric and equation (1.9), we get 1

2 d dt

Z

M|u_t|²dt+ν Z

M

|du^♯_t|²+|d^∗u^♯_t|²

dx−2ν Z

MhRicu_t, u_tidx= 0. (1.10) When Ric is negative, the above relation yields the existence of Leray’s weak solution (see for instance [26]). For the general case, the existence of Leray’s weak solution to (1.9) was proved in [27] (Theorem 4.6, p.498 and p.504).

In contrast to Euler equations, there is no geometrical interpretation for Navier-Stokes equations. The purpose of this work is to develop a probabilistic interpretation to equation (1.9). Note that in this context, it is suitable to consider that the underlying Lagrangian trajectories are semimartingalesξton the manifoldM. Comparing to Brenier’s generalized flows for Euler equations, the pathst→ξ_t are never of finite energy in the sense of (1.3).

Instead, we shall consider the mean kinetic energy (see definition (2.13) below). This functional first appeared in stochastic optimal control [18] as well as in connection with quantum mechanics [29]; we mention also [19] for the relation of (stochastic) kinetic energy and entropy as well as [28], for its appereance in the study of the Navier-Stokes equation.

Roughly speaking, the main result of this paper (see Theorem 2.10 below) says that the semi-martingale ξ_t in a suitable class is a critical point to the stochastic kinetic energy (2.13) if and only if its drift term u_t solves Navier-Stokes equation (1.9) in the sense of Di-Perna and Majda.

In the recent years the functional (2.13) has been used with success in various contexts (see for example [2, 3, 4, 5, 10, 11, 12, 15, 20, 23]). In comparison with [2, 3, 4, 5], we do not require, in the present work, that martingales have the flow property.

The organisation of the paper is as follows. In section 2, we shall introduce and study the class ofν-Brownian incompressible semimartingales. We prove that such a semimartingale is a critical point of the corresponding kenetic energy [12] if and only if it solves the Navier-Stokes equation in the sense of DiPerna-Majda [7, 8]. We also prove the existence

of a minimum under certain conditions. In section 3, we shall show, in the case of a torus T^d, that a classical solution to Navier-Stokes equation gives rise to a ν-Brownian incompressible martingale which realizes the minimum of the kinetic energy in a convenient class.

2 Generalized stochastic paths for the Navier-Stokes equa- tion

In this section,M will denote a connected compact Riemannian manifold without bound- ary. Let (Ω,F, P) be a probability space equipped with a filtration {Ft;t≥0} satisfying the usual conditions.

A M-valued stochastic process ξ_t defined on (Ω,F, P) is said to be a semimartingale on M if for any f ∈C²(M), f(ξ_t) is a real valued semimartingale. This notion is indepen- dent of the chosen connection onM; however, the corresponding local characteristics are dependent of the choice of connection. For a semimartingale (ξ_t) starting from a point x ∈ M and given a connection ∇, the stochastic parallel translation //_t along ξ_· can be defined and

ζ_t= Z t

0

//⁻¹_s ◦dξ_s

is a T_xM-valued semimartingale. Then there exist processes (ξ⁰(s), H₁(s), . . . , H_m(s)) which are adapted to Ft such that

ξ⁰(s), H₁(s), . . . , H_m(s)∈T_ξsM and ζ_t admits Itˆo form

ζ_t= Z _t

0

//⁻¹_s ξ⁰(s)ds+

m

X

i=1

Z _t

0

//⁻¹_s H_i(s)dwⁱ_s (2.1) wherew_t= (w¹_t,· · · , w^m_t ) is a standard Brownian motion onR^m (see for example [9]). For example, if the semimartingale ξ_t comes from a stochastic differential equation (SDE) on M:

dξ_t=X₀(t, ξ_t)dt+

m

X

i=1

X_i(t, ξ_t)◦dw_tⁱ, ξ₀=x, then

ξ⁰(t) =X₀(t, ξ_t) +1 2

m

X

i=1

(∇XiX_i)(t, ξ_t).

For simplicity, in what follows, we only consider the Levi-Civita connection ∇on M. As in [3], [12], we consider the operator

D_tξ =//_t lim

ε→0E

ζ_t+ε−ζ_t ε

Ft

, (2.2)

which is well-defined and equals ξ⁰(t). For a semimartingale ξ_t given by (2.1), the Itˆo formula has the following form (see [9], p. 409)

f(ξ_t) =f(ξ₀) + Z t

0

h∇f(ξ_s), ξ⁰(s)i+1 2

m

X

i=1

h∇Hi(s)(∇f)(ξ_s), H_i(s)i ds

+

m

X

i=1

Z t

0 h∇f(ξ_s), H_i(s)idwⁱ_s.

(2.3)

Let{g_t(x, ω); t≥0, x∈M, ω ∈Ω}be a family of continuous semimartingales with values in M. Let P^g denote the law of g in the continuous path space C([0, T], M), that is, for every cylindrical functional F,

Z

C([0,T],M)

F(γ(t₁), ..., γ(t_n))dP^g(γ) = Z

M

hZ

C([0,T],M)

F(g_t1(x), ..., g_tn(x))dP^gx

idx where P^g=P^gx⊗dxand underP^gx, the semimartingale g_t starts fromx.

We shall say that the semimartingale g_t isincompressible if, for eacht >0, EP^g[f(g_t)] =

Z

M

f(x)dx, for all f ∈C(M) (2.4) the expectation being taken with respect to the law P^g of g.

Let ν >0; we shall say that g_t is a ν-Brownian semimartingale if, underP^g, there exists a time-dependent adapted random vector field u_t over g_tsuch that

M_t^f =f(g_t)−f(g₀)− Z t

0

ν∆f(g_s) +hu_s,∇f(g_s)i

ds, (2.5)

is a local continuous martingale with the quadratic variation given by hM_t^f1, M_t^f2i= 2ν

Z t

0 h∇f₁,∇f₂i(g_s)ds.

For a semimartingale ξ_t given by (2.1), if {H₁(s), . . . , H_m(s)} is an orthogonal system such that for any vector v ∈ T_ξsM,

m

X

i=1

hv, Hi(s)i² = 2ν|v|², then it is a ν-Brownian semimartingale.

Example 2.1. In the flat caseR^d, such a semimartingale admits the following form dg_t(w) =√

2ν dw_t+u_t(w)dt, (2.6)

where (w_t) is a Brownian motion on R^d and {u_t;t ≥0} is an adapted R^d-valued process such that RT

0 |u_t(w)|²dt <+∞ almost surely.

Example 2.2. For the general case of a compact Riemannian manifold M, we consider the bundle of orthonormal frames O(M). Let (V_t)_t∈[0,T] be a family of C¹ vector fields such that the dependence t→ V_t is C¹. Denote by ˜V_t the horizontal lift of V_t to O(M).

Let div(Vt) and div( ˜Vt) be respectively the divergence operators onM and onO(M); they are linked by (see [16], p. 595)

div( ˜V_t) = div(V_t)◦π,

where π : O(M) → M is the canonical projection. It follows that if div(Vt) = 0, then div( ˜V_t) = 0. Consider the horizontal diffusionr_t onO(M) defined by the SDE

dr_t=√ 2ν

d

X

i=1

H_i(r_t)◦dwⁱ_t+ ˜V_t(r_t)dt, r₀ ∈O(M) (2.7) where {H1,· · · , H_d} are the canonical horizontal vector fields on O(M). Let dr be the Liouville measure on O(M), then the stochastic flowr₀→r_t(r₀) leaves drinvariant. Set

ξ(t, x) =π(rt(r0)), r0 ∈π⁻¹(x). (2.8) For any continuous function f onM,

Z

ME(f(ξ(t, x))dx= Z

M

f(x)dx.

Then ξ is an incompressible ν-Brownian diffusion, withD_tξ(x) =V_t(ξ(t, x)).

Remark 2.3. Let Pt be the semigroup associated to 1

2∆M +Vt with div(Vt) = 0; then for any f ∈C²(M),

d dt

Z

M

P_tf(x)dx= Z

M

1

2∆_MP_tf +V_tP_tf

dx= 0.

It follows that for any continuous function f :M →R, Z

M

P_tf(x)dx= Z

M

f(x)dx.

Therefore any SDE on M defining a Brownian motion with drift V gives rise to an incom- pressible ν-Brownian diffusionξ withD_tξ(x) =V_t(ξ(t, x)).

Example 2.4. Let Z² be the set of two dimensional lattice points and define Z²₀ = Z²\ {(0,0)^∗}. For k∈Z²0, we consider the vector k^⊥= (k₂,−k₁)^∗ and the vector fields

A_k(θ) = rν

ν0

cos(k·θ)

|k|^β k^⊥, B_k(θ) = r ν

ν0

sin(k·θ)

|k|^β k^⊥, θ∈T², where β >1 is some constant.

Let ˜Z²0 the subset of Z²0 where we identify vectorsk, k^′such thatk+k^′ = 0 and let ν0 = X

k∈Z˜²₀

1 2|k|^2β.

The family {A_k, B_k :k ∈Z²₀} constitutes an orthogonal basis of the space of divergence free vector fields onT² and satisfies

X

k∈˜Z²₀

hA_k, vi²+hB_k, vi²

=ν|v|²TθT², v∈T_θT², and

X

k∈˜Z²₀

∇AkA_k = 0, X

k∈Z˜²₀

∇BkB_k= 0.

Consider the SDE on T², dξ_t= X

k∈Z˜²₀

A_k(ξ_t)◦dw^k_t +B_k(ξ_t)◦dw˜^k_t

+u(t, ξ_t)dt, θ₀=θ∈T² (2.9) where {w^k_t,w˜^k_t; k∈Z²₀} are independent standard Brownian motions on R, andu(t,·) is a family of divergence free vector fields in H¹(T²), such that,

Z _T

0

Z

T²

(|u|²+|∇u|²)dxdt <+∞.

Then by [12, 16], for β ≥ 3, the SDE (2.9) defines a stochastic flow of measurable maps which preserves the Haar measuredxonT². More precisely, for almost surely w, the map

x→ξ_t(x, w) solution to (2.9) with initial conditionx leaves dx invariant; this property is stronger than that of incompressibility.

In what follows, we shall denote by S the set of incompressible semimartingales, by Sν

the set of incompressibleν-Brownian semimartingales and byD^ν the set of incompressible ν-Brownian diffusions. Clearly we have

Dν ⊂ Sν ⊂ S. Proposition 2.5. Let g∈ Sν, then for any f ∈C²(M),

EP^g(h∇f(g_t), u_ti) = 0. (2.10) Proof. Taking the expectation with respect toP^g in (2.5), we have

EP^g(f(g_t))−EP^g(f(g₀)) =ν Z t

0 EP^g(∆f(g_s))ds+ Z t

0 EP^g(h∇f(g_s), u_si)ds.

It follows that ν

Z t 0

Z

M

∆f(x)dx ds+ Z t

0 EP^g(h∇f(g_s), u_si)ds= 0.

Since R

M∆f(x)dx= 0, we get the result.

Proposition 2.6. Let g_t be a semimartingale on M satisfying dg_t(x) =

m

X

i=1

A_i(g_t(x))◦dw_tⁱ+u_t(w, x)dx,

where A₁,· · · , A_m are C² divergence free vector fields on M and u_t(w, x) ∈ T_gt(x)M is adapted such that R

M Ex(RT

0 |u_t(w, x)|²dt)dx < +∞; if g is incompressible, then for any f ∈C²(M)

EP^g(h∇f(g_t), u_ti) = 0.

Proof. Letf ∈C²(M); then by Itˆo formula (2.3), f(g_t) =f(g₀) +M_t^f +1

2

m

X

i=1

Z t 0

h∇Ai(∇f), A_ii+h∇f,∇AiA_ii ds+

Z t

0 h∇f(g_s), u_sids, where M_t^f is the martingale part. Note that h∇^Ai(∇f), Aii+h∇f,∇^AiAii = L^AiL^Aif whereLAdenotes the Lie derivative with respect toA; then taking the expectation under EP, we get

1 2

m

X

i=1

Z

MLAiLAif dx

+EP^g(h∇f(g_t), u_ti) = 0.

Since for each i,R

ML^AiL^Aif dx= 0, the result follows.

In general it is not clear whether the incompressibility condition implies the relation (2.10).

However, the following is true:

Proposition 2.7. Let A₁,· · · , A_m be C^2+α vector fields on M and A₀ be a C^1+α vector field with some α >0; consider

dξ_t(x) =

m

X

i=1

A_i(ξ_t(x))◦dw_tⁱ+A₀(ξ_t(x))dt, ξ₀=x. (2.11) Then for almost allw, the mapx→ξ_t(x)preserves the measure dxif and only ifdiv(A_i) = 0 for i= 0,1,· · · , m.

Proof. We give a sketch of proof (see [17] for more discussions). By [21], ;x→ξ_t(x) is a diffeomorphism ofM and the push forward measure (ξ_t⁻¹)_#(dx) ofdxby the inverse map of ξt admits the densityKt which is given by (see [22]):

K_t(x) = exp

−

m

X

i=1

Z t 0

div(A_i)(ξ_s(x))◦dwⁱ_t− Z t

0

div(A₀)(ξ_s(x))ds

. (2.12)

If div(A_i) = 0 for i = 0,1,· · · , m, it is clear that K_t = 1 and x → ξ_t(x) preserves dx.

Conversely, K_t(x) = 1 for any x∈M and t≥0 implies that,

m

X

i=1

Z t 0

div(A_i)(ξ_s(x))◦dwⁱ_t+ Z t

0

div(A₀)(ξ_s(x))ds= 0;

or in Itˆo form:

m

X

i=1

Z t 0

div(A_i)(ξ_s(x))dwⁱ_t+ Z t

0

h1 2

X

i=1^m

LAidiv(A_i) + div(A₀)i

(ξ_s(x))ds= 0.

The first term of above equality is of finite quadratic variation, while the second one is of finite variation; so that for each i= 1,· · · , m, div(A_i)(ξ_s(x)) = 0 and also

h1 2

X

i=1^m

LAidiv(A_i) + div(A₀)i

(ξ_s(x)) = 0.

It follows that, almost everywhere,

div(A_i)(ξ_s(x)) = 0 for i= 0,1,· · · , m;

so that div(A_i) = 0 for i= 0,1,· · · , m.

According to [12], as well as [4, 20, 15], we introduce the following action functional on semimartingales.

Definition 2.8. Let

S(g) = 1 2EP^g

Z T

0 |D_tg|²dt

. (2.13)

We say that g hasfinite energy if S(g)<∞.

In what follows, we shall denote more precisely D_tg(x) for D_tg under the law P^gx. Then the action defined in (2.13) can be rewritten in the following form:

S(g) = 1 2

Z

ME_P^g_x Z T

0 |Dtg(x)|²dt

dx. (2.14)

We first recall briefly known results about the calculus of stochastic variation (see [12, 4, 10]). Let u_t(x) be a smooth vector field on a compact manifold (or on R^d) which, for every t, is of divergence zero. Consider an incompressible diffusion g_t(x) with covariance a such that a(x, x) = 2µg⁻¹(x) where g is the metric tensor and time-dependent drift u(t,·). It defines a flow of diffeomorphisms preserving the volume measure. We have:

D_tg(x) =u_t(g_t(x)) and

S(g) = 1 2

Z

T^d

E_P^g_x Z _T

0 |u_t(g_t(x))|²dt dx.

There are two manners to perform the perturbation.

First perturbation of identity:

Letwbe a smooth divergence free vector field andα∈C¹(]0, T[). Consider, for forε >0, the ODE,

dΦ^ε_t(x)

dt =ε α^′(t)w(Φ^ε_t(x)),Φ₀(x) =x. (2.15) For each t > 0, Φ^ε_t is a perturbation of the identity map id. By Itˆo’s formula, for each fixed ε > 0, t → Φ^ε_t(g_t(x)) is a semimartingale starting from x. Note that g and Φ^ε(g) are defined on the same probability space. It was proved in [12, 4] that u is a weak solution to Navier-Stokes equation if and only if g is a critical point of S. More precisely,

d

dεS(Φ^ε(g))_|ε=0 = 0 if and only if Z

T^d

Z _T

0 hu_t, α^′(t)w+α(t)∇w·u_t−ν α(t)widtdx= 0. (2.16) Second perturbation of identity:

Note that in [20], the perturbation of the identity was defined in a different way. For each fixed t >0, the author of [20] considered the ODE

dΨ^t_s

ds =α(t)w(Ψ^t_s), Ψ^t₀(x) =x. (2.17) Set Ψ(g)^ε_t(x) = Ψ^t_ε(gt(x)). Then d

dεS(Ψ(g)^ε)_|ε=0 = 0 if and only if the equation (2.16) holds.

Now we deal with the general case of compact Riemannian manifolds.

Definition 2.9. LetM be a compact Riemannian manifold,g a semimartingale onM of finite energy. Define the probability measure µon [0, T]×T M by

Z

[0,T]×T M

f(t, x, v)µ(dt, dx, dv) = 1 TEP^g

hZ _T

0

f t, g(t), D_tg dti

(2.18) where f : [0, T]×T M →R is any continuous function.

We have the following result,

Theorem 2.10. Suppose that g ∈ Sν. Then g is a critical point of S with variations defined in (2.17) if and ony if µ is a solution to the Navier-Stokes equation in the sense of DiPerna-Majda, that is,

Z T 0

Z

T M

α^′(t)v·w+α(t)v· ∇vw−να(t)v·ˆw

dµ(t, x, v) = 0 (2.19) for all α∈C_c¹(]0, T[) and all smooth vector fields w such that div(w) = 0.

Proof. Let Ψ^t_ε be the perturbation of identity defined in (2.17). Set η_t^ε = Ψ^t_ε(g_t(x)).

Then {η_t^ε, t≥0} is a semimartingale on M. We denote by (ξ⁰(s), H₁(s), . . . , H_m(s)) the local characteristics of g_t(x). By Itˆo’s formula (see [9], p. 408), the drift term in local characteristics of η_t^ε is given by

D_tΨ^t_ε(g_t(x)) = ∂

∂tΨ^t_ε(g_t(x)) +dΨ^t_ε(g_t(x))·ξ_t⁰+1 2

m

X

i=1

∇(dΨ^t_ε)(g_t(x))(H_i(t), H_i(t)), (2.20) where dΨ^t_ε(g_t(x)) denotes the differential of Ψ^t_ε at g_t(x). Let ϕ(ε, t) = D_tΨ^t_ε(g_t(x)) ∈ Tη_t^εM; then

S(Ψ_ε(g)) = 1 2EP^g

Z T 0

ϕ(ε, t)

2dt . We have: ϕ(0, t) =D_tg(x). Let

ϕ₁(ε, t) = ∂

∂tΨ^t_ε(g_t(x)), ϕ2(ε, t) =dΨ^t_ε(gt(x))·ξ_t⁰, ϕ₃(ε, t) = 1

2

m

X

i=1

∇(dΨ^t_ε)(g_t(x))(H_i(t), H_i(t)).

Since the torsion is free, we have D

dεϕ₁(ε, t)_|ε=0 = D dε

d

dtΨ^t_ε(g_t(x))_|ε=0 = D dt

d dε|ε=0

Ψ^t_ε(g_t(x)) =α^′(t)w(x).

In order to compute the derivative of ϕ2, consider a smooth curve β(s) ∈ M such that β(0) =g_t(x), β^′(0) =D_tg(x). Then

dΨ^t_ε(g_t(x))·ξ_t⁰ = d ds|s=0

Ψ^t_ε(β(s)).

Therefore

D dε|ε=0

ϕ₂(ε, t) = D ds|s=0

d dε|ε=0

Ψ^t_ε(β(s)) = D ds|s=0

hα(t)w(β(s))i

=α(t) (∇w)(g_t(x))·D_tg(x).

For computingϕ3, we shall use another description given in [9] (p. 405). For the moment, consider a C² mapf :M →M. Letx∈M and two tangent vectorsu, v ∈T_xM be given.

Let x(t) ∈M be a smooth curve such thatx(0) =x, x^′(0) =u, andY_t∈T_xtM such that Y₀=v. Define Q(f)(x) :T_xM ×T_xM →T_f(x)M by

Q(f)(x)(u, v) = d dt|t=0

h

//⁻¹_t (df(x_t)·Y_t)i

−df(x)· ∇uv. (2.21) Then ϕ₃ can be expressed by

ϕ₃(ε, t) = 1 2

m

X

i=1

Q(Ψ^t_ε(g_t(x)))(H_i(t), H_i(t)).

Letβ(s)∈M be a smooth curve such thatβ(0) =g_t(x) andβ^′(0) =H_i(t) and{Y_s;s≥0} be a family of tangent vectors along {β(s);s≥0} such thatY0=Hi(t). Set

γ(ε, s) = Ψ^t_ε(β(s)) and X(ε, s) =dΨ^t_ε(β(s))·Y_s.

If R denotes be the curvature tensor onM, the following commutation relation holds, D

dε D

dsX(ε, s) = D ds

D

dεX(ε, s) +R ∂γ

∂ε,∂γ

∂s

X(ε, s).

We have X(0,0) =H_i(t),∂γ

∂ε(0,0) =α(t)w(x),∂γ

∂s(0,0) =H_i(t); therefore h

R ∂γ

∂ε,∂γ

∂s

X(ε, s)i

|ε=0,s=0

=α(t)R(w(g_t(x)), H_i(t))H_i(t).

Now let c(τ)∈M be a smooth curve such that c(0) =β(s), c^′(0) =Y_s. We have D

dε|ε=0

X(ε, s) =hD dτ

d

dεΨ^t_ε(c(τ))i (0,0)

=α(t)D dτ|τ=0

w(c(τ)) =α(t) (∇Ysw)(β(s)),

and D

ds|s=0

(∇Ysw)(β(s)) =h∇Hi(t)∇w, H_i(t)i+h∇w,∇Hi(t)H_i(t)i. Note that

D dε|ε=0

dΨ^t_ε(g_t(x))· ∇Hi(t)H_i(t) =α(t)h∇w,∇Hi(t)H_i(t)i. Using (2.21), we finally get

D dε|ε=0

ϕ₃(ε, t) = 1 2α(t)

m

X

i=1

hh∇Hi(t)∇w, H_i(t)i+R(w, H_i(t))H_i(t)i .

When gt is aν-Brownian semimartingale, the right hand side of above equality is equal to να(t) (∆w+ Ricw)(g_t(x)),

which, due to (1.8), is equal to

να(t) (−ˆw)(g_t(x)).

In conclusion, d

dεS(Ψ_ε(g))_|ε=0 = 0 yields EP^g

Z _T

0

h

α^′(t)w(g_t)·D_tg+α(t) (∇Dtgw)(g_t)·D_tg−να(t) ˆw(g_t)·D_tgi

dt= 0. (2.22) According to (2.18), the above equation is nothing but (2.19).

As a consequence of this result, we obtain

Theorem 2.11. Let (ut)_t∈[0,T] be a family of divergence free vector fields on M, which belong to the Sobolev space D²₁ and are such that

Z

M

Z T 0

|u_t(x)|²+|∇u_t(x)|²

dtdx <+∞; (2.23) then equations (2.7),(2.8)define an incompressible ν-Brownian diffusionξ onM, which is a critical point of the action functional S if and only ifu_t solves weakly the Navier-Stokes equation, that is,

Z

M

Z _T

0 hu_t, α^′(t)w+α(t)∇w·u_t−ν α(t) ˆwidtdx= 0 (2.24) for all α∈C_c¹(]0, T[) and all smooth vector fields wsuch that div(w) = 0.

Proof. First we notice that in Proposition 4.3 in [16], the condition q > 2 insures the tightness of a family of probability measures; this condition can be relaxed to q = 2 using Meyer-Zheng tightness results (see the proof of Theorem 2.13 below). Therefore by Theorem 6.4 in [16], equations (2.7) and (2.8) define a diffusion process ξ, which is, a fortiori, inSν. Therefore by the above computations (see (2.22)),ξ is a critical point toS if and only if

EP^g

Z _T

0

h

α^′(t)w(ξ_t)·u_t(ξ_t) +α(t) (∇ut(ξt)w)(ξ_t)·u_t(ξ_t)−να(t) ˆw(ξ_t)·u_t(ξ_t)i

dt = 0, which yields the result.

Remark 2.12. It has been proved in [27] (see Theorem 4.6, p. 498) that for any u₀ ∈ L²(M, dx), there exists {u_t, t ∈ [0, T]} solution to (2.24), satisfying Condition (2.23).

Therefore equations (2.7), (2.8) define an incompressible ν-Brownian diffusion ξ on M, which is a critical point of the action functional S.

Note that in Theorem 3.2 of [4], a variational principle was established by using the first type of perturbations of identity, defined by (2.15); on the other hand the manifoldM was supposed there to be a symmetric space in order to insure the existence of semimartingales

with the desired properties. A variational principe on a quite general Lie groups framework was derived in [3] (c.f. also [10]).

In [7], generalized flows with prescribed initial and final configuration were introduced.

It is quite difficult to construct incompressible semimartingales with given prescriptions.

In order to emphasize the contrast with the situation in [7], let’s see the example of a Brownian bridgeg_t^x,y on Rover [0,1]. It is known that fort <1,g^x,y_t solves the following SDE

dg_t^x,y =dw_t−g_t^x,y−y

1−t dt, g^x,y₀ =x. (2.25)

Then g^x,y_t →y ast→1 and we have EZ 1

0 |D_tg^x,y|²dt

= +∞. (2.26)

Let η be a probability measure onM×M having dxas two marginals; we shall say that the incompressible semimartingale {g_t} hasη as final configuration if

EP^g(f(g₀, g_T)) = Z

M×M

f(x, y)dη(x, y), f ∈C(M×M). (2.27) This means that the joint law of (g₀, g_T) isη. Ifg_t is as in Example 2.2, then

EP^g(f(g₀, g_T)) = Z

M×M

f(x, y)p_T(x, y)dxdy,

where pt(x, y) is the heat kernel associated to (gt). Conversely if (ρt(x, y)) is solution to the following Fokker-Planck equation

d

dtρ_t(x, y) =ν∆_xρ_t(x, y) +hu_t(x),∇xρ_t(x, y)i, with lim

t→0ρ_t =δ_x, for someu ∈ L²([0, T],D²₁(M)) with div(u_t) = 0, we can construct an incompressibleν-Brownian semimartingale which hasρ_T(x, y)dxdy as final configuration.

In any case, we have the following result:

Theorem 2.13. Let η be a probability measure as above. If there exists an incompressible ν-Brownian semimartingalegon M of finite energyS(g) such thatη is its final configura- tion, then there exists one that minimizes the energy among all incompressible ν-Brownian semimartingales having η as final configuration.

Proof. LetJ :M →R^Nbe an isometric embedding; thendJ(x) : T_xM →R^Nis such that for each x∈M and v ∈T_xM,|dJ(x)·v|R^N =|v|TxM. Denote by (dJ(x))^∗ :R^N → T_xM the adjoint operator ofdJ(x), that is,

h(dJ(x))^∗a, viTxM =hdJ(x)v, aiR^N, a∈R^N, v∈T_xM.

Let {ε₁, . . . , ε_N}be an orthonormal basis of R^N and set A_i(x) = (dJ(x))^∗ε_i, i= 1, . . . , N.

Then it is well-known that the vector fields {A₁, . . . , A_N}enjoy the following properties:

(i) For anyv∈T_xM,|v|²TxM =

N

X

i=1

hA_i(x), vi²TxM. (ii)

N

X

i=1

∇AiA_i = 0.

Combining (i) and (ii) gives that ∆_Mf =

N

X

i=1

L²Aif for any f ∈ C²(M). On the other hand, let J(x) = (J₁(x), . . . , J_N(x)); then

hdJ(x)v, ε_ii=dJ_i(x)·v=h∇J_i(x), viTxM, for any v∈T_xM.

It follows that

A_i =∇J_i, i= 1,· · ·, N. (2.28) Let f ∈C²(M); then there exists ¯f ∈C²(R^N) such thatf(x) = ¯f(J(x)). We have

LAif =

N

X

j=1

∂f¯

∂x_j(J(x))h∇J_j(x), A_i(x)i

=

N

X

j=1

∂f¯

∂x_j(J(x))hA_j(x), A_i(x)i.

(2.29)

Therefore

∆_Mf =

N

X

i=1 N

X

j,k=1

∂²f¯

∂x_j∂x_k(J(x))hA_j, A_iihA_k, A_ii +

N

X

i=1 N

X

j=1

∂f¯

∂x_j(J(x))LAihA_j, A_ii. Notice that

N

X

i=1

LAihA_j, A_ii= div(A_j) = ∆_MJ_j, and according to property (i),

N

X

i=1

hA_j, A_iihA_k, A_ii=hA_j, A_ki. Finally the Laplacian ∆_M on M can be expressed by

∆_Mf =

N

X

j,k=1

∂²f¯

∂x_j∂x_k(J(x))hA_j, A_ki+

N

X

j=1

∂f¯

∂x_j(J(x)) ∆_MJ_j. (2.30) Having these preparations, we prove now the existence of a g ∈ S^ν such that the mini- mum of action functinal S is attained at g in the class of those in Sν having η as final configuration. Let

K = inf

g∈Sν

S(g).

There is a minimizing sequencegⁿ∈ S^ν, that is, lim

n→+∞S(gⁿ) =K. Consider the canonical decomposition:

J(gⁿ_t) =J(gⁿ₀) +M_tⁿ+ Z _t

0

bⁿ(s)ds.

Let M_tⁿ= (M_t^n,1,· · · , M_t^n,N); then

hM_t^n,i, M_t^n,ji= 2ν Z _t

0 h∇Ji,∇Jji(g_sⁿ)ds. (2.31) By Itˆo formula, we have

bⁿ(t) =dJ(gⁿ_t)·D_tgⁿ+ν∆J(g_tⁿ). (2.32) It follows that

E Z T

0 |bⁿ(t)|²dt

≤2S(gⁿ) + 2T ν||∆J||^∞. Therefore

Z _T

0 |bⁿ(t)|²dtis bounded inL². We can use Theorem 3 in [30] to conclude that the joint law ˆP_n of

(J(g_·ⁿ), M_·ⁿ, B_·ⁿ, U_·ⁿ)

inC([0, T],R^N)×C([0, T],R^N)×C([0, T],R^N)×C([0, T],R^N×N) is a tight family, where Bⁿ_t =

Z t 0

bⁿ(s)ds, U_tⁿ= (hM_t^n,i, M_t^n,ji)_1≤i,j≤N.

Let ˆP be a limit point; up to a subsequence, we suppose that ˆP_n converges weakly to ˆP. Again by Theorem 3 in [30], under ˆP, the coordinate process

(X_t, M_t, B_t, U_t) has the following properties:

(i)M₀ =B₀= 0, U₀ = 0,

(ii) (M_t) is a local martingale such that U_t= (hM_tⁱ, M_t^ji)_1≤i,j≤N and (iii)B_t=

Z t 0

b(s)dswith Z T

0 |b(s)|²ds <+∞almost surely.

Since J(M) is closed inR^N, we see that X_t∈J(M). Let X_t=J(g_t).

For any f ∈C²(M), by (2.30), we see thatf(gt) is a real valued semimartingale. In other words,{g_t; t≥0}is a semimartingale onM. Letf ∈C(M), the mapf◦J⁻¹ :J(M)→R can be extended as a bounded continuous function on R^N; therefore letting n→ ∞, we get

Z

M

f(x)dx=E(f(gⁿ(t))) =E(f◦J⁻¹(J(gⁿ_t)))→E(f◦J⁻¹(X_t)) =E(f(g_t)).

In the same way, forf ∈C(M ×M), we have Z

M×M

f(x, y)dη(x, y) =E(f(gⁿ(0), gⁿ(T))) =E(f(J⁻¹J(gⁿ(0)), J⁻¹J(gⁿ(T))))

which goes to, as n→+∞,

E(f(g(0), g(T))).

So g is incompressible and has η as final configuration.

Besides, by (2.31), we have

(hM_tⁱ, M_t^ji)_1≤i,j≤N = 2ν Z _t

0 h∇J_i,∇J_ji(g_s)ds. (2.33) Let f ∈C²(M); denote by M_t^f the martingale part of f(g_t). Then by Itˆo formula,

dM_t^f =

N

X

j=1

∂f¯

∂x_j(X_t)dM_t^j. Therefore for f₁, f₂ ∈C²(M), according to (2.33), we have

hdM_t^f1, dM_t^f2i=

N

X

j,k=1

∂f¯₁

∂x_j(X_t)∂f¯₂

∂x_k(X_t) 2νhA_j, A_kigtdt.

On the other hand, using relation (2.29) and property (i), we have h∇f1,∇f2i=

N

X

α=1

L^Aαf1L^Aαf2=

N

X

j,k=1

∂f¯₁

∂x_j

∂f¯₂

∂x_khAj, A_ki. Combinant above two equalities, we finally get

hdM_t^f1, dM_t^f2i= 2νh∇f1,∇f2i^gtdt. (2.34) Since X_t=J(g_t), we have

dB_t=dJ(g_t)·D_tg dt+1

2HessJ(g_t)dg_t⊗dg_t.

Relation (2.34) implies that ¹₂HessJ(g_t)dg_t⊗dg_t=ν∆_MJ(g_t)dt. Therefore we get B_t=

Z _t

0

dJ(g_s)·D_sg ds+ν Z _t

0

∆_MJ(g_s)ds. (2.35) In conclusion {g_t; t≥0}is a ν-Brownian semimartingale on M or g∈ Sν.

We want to see that K =S(g). Firstly using the relation (2.32), for anyt∈[0, T], Z _t

0

dJ(gⁿ_s)·D_sgⁿds=B_tⁿ−ν Z _t

0

∆J(g_sⁿ)ds.

Letφ:C([0, T],R^N)→Rbe a bounded continuous function, consider ϕ:C([0, T],R^N)× C([0, T],R^N)→Rdefined by

ϕ(B, g) =φ B_·−ν

Z · 0

∆J(g_s)ds .

Then ϕis a bounded continuous function onC([0, T],R^N)×C([0, T],R^N). It follows that Z _·

0

dJ(gⁿ_s)·D_sgⁿds converges in law to Z _·

0

dJ(g_s)·D_sg ds. Letε >0; forn big enough,

E Z _T

0 |dJ(g_sⁿ)·D_sgⁿ|²ds

≤K+ε.

Now by Theorem 10 in [24], E

Z _T

0 |dJ(g_s)·D_sg|²ds

≤K+ε,

or EZ T

0 |D_sg|²ds

≤K+ε. Lettingε→0 gives S(g)≤K. SoS(g) =K.

3 Classical solutions and generalized paths

In this section, M will be a torus: M =T^d. Let g∈ Dν be the solution of the following SDE on T^d

dg_t=√

2ν dw_t−u(T −t, g_t)dt, g₀ ∈T^d (3.1) where g₀ is a random variable having dx as law, w_t is the standard Brownian motion on R^d, and {u(t, x);t∈[0, T]} is a family ofC² vector fields onT^d, identified to vector fields on R^d which are 2π-periodic with respect to each space component. Suppose that u is a strong solution to the Navier-Stokes equation

∂

∂tu(t, x) +∇u(t, x)·u(t, x)−ν∆u(t, x) =−∇p(T −t, x).

By Itˆo ’s formula,

du(T −t, g_t) =−(∂u

∂t)(T−t, g_t)− ∇u(T−t, g_t)·u(T−t, g_t) +ν∆u(T −t, g_t) +√

2ν∇u(T−t, g_t)·dw_t

=∇p(t, g_t)dt+√

2ν∇u(T −t, g_t)·dw_t

(3.2)

According to definition (2.2),D_tg=−u(T−t, g_t) and

DtDtg=−∇p(t, gt). (3.3) In what follows, we shall consider

G=

g^∗ ∈ Sν; dg^∗_t =√

2ν dw_t+D_tg^∗dt, g^∗(0) =g(0), g^∗(T) =g(T) . (3.4) Note that semimartingales in G are defined on a same probability space.

Example 3.1. Letα be a real continuous function on R^d and set β(w, t) = sin(πt

T) Z _t

0

α(w_s)ds, c(w, t) = d

dtβ(w, t).

Let a∈R^d be fixed. Considerv(w, t) =c(w, t)a; then v is an adapted vector field on T^d. Define

g^∗_t =gt+ Z t

0

v(w, s)ds.

Then g^∗ ∈ G.

We have the following result

Theorem 3.2. Let g∈ D^ν be given in (3.1). Assume that the processg is associated with the Navier-Stokes equation in the sense that

D_tD_tg=−∇p(t, g_t)

a.s. for a regular pression psuch that∇²p(t, x)≤RId, withRT² ≤π². Thengminimizes the energy S in the class G.

Proof. We define the following:

B(g) = 1 2

Z _T

0 |D_tg|²dt− Z _T

0

p(t, g(t))dt (3.5)

Notice that the function b(x, y) defined in [7] (p. 243) has no meaning in our setting (c.f.

(2.25) and (2.26)). Let g^∗ ∈ G; we shall prove that

E(B(g))≤E(B(g^∗)). (3.6)

Consider the function

φ(t, x) = R

2|x|²−p(t, x).

For each t ≥ 0, the function x → φ(t, x) is convex on R^d as ∇²p(t, x) ≤ RId. By Itˆo formula

d Dtg·gt

=d(Dtg)·gt+√

2ν Dtg·dwt+|Dtg|²dt+d(Dtg)·dgt. Analogously,

d(D_tg·g^∗_t) =d(D_tg)·g^∗_t +√

2ν D_tg·dw_t+D_tg·D_tg^∗+d(D_tg)·dg^∗_t.

Remarking thatd(D_tg)·dg_t=d(D_tg)·dg_t^∗, and making the substraction of the above two equalities, we obtain

d D_tg·(g^∗_t −g_t)

=d(D_tg)·(g^∗_t −g_t) + D_tg·D_tg^∗− |D_tg|² dt.

It follows that

D_Tg·(g_T^∗ −g_T)−D₀g·(g₀^∗−g₀)

= Z _T

0

d(Dtg)·(g^∗_t −gt) + Z _T

0

Dtg·Dtg^∗− |Dtg|² dt.

Notice that g^∗₀ =g₀, g_T^∗ =g_T, and using (3.1), we have Z _T

0 −D_tg·D_tg^∗+|D_tg|² dt=

Z _T

0

d(D_tg)·(g_t−g_t^∗)

= Z T

0

(g_t^∗−gt)· −√

2ν∇u(T−t, gt)dwt− ∇p(t, gt)dt .

(3.7)

Using the convexity, of φ, we have

φ(t, g^∗_t)−φ(t, g_t)≥ Rg_t− ∇p(t, g_t)

·(g^∗_t −g_t). (3.8)