Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold

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Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold

Shizan Fang

To cite this version:

Shizan Fang. Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold.

2019. �hal-02196468v2�

Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold

Shizan Fang^1∗

1I.M.B, Universit´e de Bourgogne, BP 47870, 21078 Dijon, France

September 20, 2019

Abstract

What is the suitable Laplace operator on vector fields for the Navier-Stokes equa- tion on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden’s Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved.

MSC 2010: 35Q30, 58J65

Keywords: Nash embedding, shape operator, vector valued Laplacians, Navier-Stokes equa- tions, stochastic representation

1 Introduction

The Navier-Stokes equation on a domain of Rⁿ or on a torusTⁿ, (∂tu+ (u· ∇)u−ν∆u+∇p= 0,

∇ ·u= 0, u|_t=0=u₀, (1.1)

describe the evolution of the velocity u of an incompressible viscous fluid with kinematic viscosity ν > 0, as well as the pressure p. Such equation attracts the attention of many researchers, with an enormous quantity of publications in the literature. The model of peri- odic boundary conditions is introduced to simplify mathematical considerations. There is no doubt on the importance of the Navier-Stokes equation on a Riemannian manifold, which is more suitable for models in aerodynamics, meteorology, and so on.

In a seminal paper [11], the Navier-Stokes equation has been considered on a compact Rie- mannian manifold M using the framework of the group of diffeomorphisms of M initiated by V. Arnold in [5]; the Laplace operator involved in the text of [11] is de Rham-Hodge Laplacian, however, the authors said in the note added in proof that the convenient Laplace operator comes from deformation tensor. Let’s give a brief explanation. Let ∇be the Levi- Civita connection on M, for a vector field A on M, the deformation tensor Def (A) is a symmetric tensor of type (0,2) defined by

(Def A)(X,Y) = 1 2

h∇_XA,Yi+h∇_YA,Xi

, X,Y ∈ X(M), (1.2)

∗Email: Shizan.Fang@u-bourgogne.fr.

where X(M) is the space of vector fields on M. Formula (1.2) says that Def A is the sym- metrized part of ∇A. Then Def : TM→S²T^∗M maps a vector field to a symmetric tensor of type (0,2). Let Def^∗ : S²T^∗M→ TM be the adjoint operator. Following [22] or [26], the Ebin-Marsden’s Laplacian ˆ is defined by

ˆ = 2Def^∗Def. (1.3)

The following formula holds (see [26] or [3]),

ˆA=−∆A−Ric(A)− ∇div(A), A∈ X(M) (1.4) where ∆A = Trace(∇²A). The Ebin-Marsden’s Laplacian ˆ has been used in [23] to solve Navier-Stokes equation on a compact Riemannian manifold; recently in [24] to the case of Riemannian manifolds with negative Ricci curvature : it is quite convenient with sign minus in formula (1.4). In [7], the authors gave severals arguments from physics in order to convive the relevance of ˆ.

On the other hand, the De Rham-Hodge Laplacian=dd^∗+d^∗dwas widely used in the field of Stochastic Analysis (see for example [21, 16, 6, 12, 13, 25]). The Navier-Stokes equation with on the sphereS² was considered by Temam and Wang in [28] , the case where Ricci tensor is positive. By Bochner-Weitzenb¨ock formula (see [21]):

=−∆ + Ric, (1.5)

examples of positive Ricci tensor are now confortable with . Besides, in [17], S. Kobayashi pleaded for the De Rham-Hodge Laplacianin the formalism of the Navier-Stokes equation on manifolds with curvature.

During last decade, a lot of works have been done in order to establish V. Arnold’s variational principle [5] for Navier-Stokes on manifolds (see for example [1, 2, 3, 9, 20, 4]). Connections between Navier-Stokes equations and stochastic evolution also have a quite long history: it can be traced back to a work of Chorin [8]. In [15], a representation formula using noisy flow paths for 3-dimensional Navier-Stokes equation was obtained. An achievement has been realized by Constantin and Iyer in [10] by using stochastic flows. We also refer to [10]

for a more complete description on the history of the developments. In [3], it was showed that the Ebin-Marsden’s Laplacian is naturally involved from the point of view of variational principle; while in [14], the De Rham-Hodge Laplacian was showed to be natural for obtaining probabilistic representations.

2 The case of S

For reader’s convenience, we first introduce some elements in Riemannian manifolds. Let M be a compact Riemannian manifold, and ϕ : M → M a C¹-diffeomorphism; the pull-back ϕ∗A by ϕof a vector field Ais defined by

(ϕ∗A)(x) =dϕ(ϕ⁻¹(x))A_ϕ⁻¹_(x), x∈M; the pull-backϕ^∗ω by ϕof a differential form ω is defined by

hϕ^∗ω, Ai_x =hω_ϕ(x), dϕ(x)A_ϕ(x)i x∈M.

LetV be a vector field onM,ϕtthe group of diffeomorphisms associated toV, then the Lie derivativeL_VAof A with respect toV is defined by

L_VA= lim

t→0

(ϕ⁻¹_t )∗A−A

t ,

and the Lie derivative L_Vω by

L_Vω= lim

t→0

ϕ^∗_tω−ω t .

We have L_VA= [V, A]. For a vector field A on M, the divergence div(A) of Ais defined by Z

M

h∇f, Aidx=− Z

M

fdiv(A)dx,

wheredxis the Riemannian volume ofM. If∇denotes the Levi-Civita covariant derivative, then

div(A)(x) =

n

X

i=1

h∇_viA, v_ii_TxM

for any orthonormal basis {v₁, . . . , v_n} of T_xM. A vector field A on M is said to be of divergence free if div(A) = 0. It is important to emphasize that L_VA is of divergence free if V and Aare (see [14]).

In order to understand the geometry of diffusion processes on a manifold M, Elworthy, Le Jan and Li in [13] embedded M into a higher dimensional Riemannian manifold so that the De Rham-Hodge Laplacianon differential forms admits suitable decompositions as sum of squares of Lie derivative of a family of vector fields. More precisely, let {A₁, . . . , AN} be a family of vector fields on M, assume they satisfy

|v|²_T

xM =

N

X

i=1

hA_i(x), vi², v∈T_xM, x∈M, (2.1)

N

X

i=1

∇_AiA_i= 0, (2.2)

and N

X

i=1

Ai∧ ∇_VAi = 0, for any vector fieldV. (2.3) Then they obtained in [13], for any differential form ω on M,

ω=−

N

X

i=1

L²_A

iω. (2.4)

There is a one-to-one correspondence between the space of vector fields and that of differential 1-forms. Given a vector fieldA(resp. differential 1-formω), we shall denote byA^[(resp. ω^]) the corresponding differential 1-form (resp. vector field). The action of the de Rham–Hodge Laplacian on the vector field Ais defined as follows:

A:= (A^[)^].

Surprising enough it was remarked in [14] that the decomposition (2.4) is in general no more valid if ω is replaced by a vector field A. It is why probabilistic representation formulae for Navier-Stokes equations hold only, until now, on symmetric Riemannian manifolds.

Proposition 2.1. Let{A₁, . . . , A_N}be a family of vector fields satisfying (2.1)–(2.3). Define T₁(B) =

N

X

i=1

div(A_i)L_AiB, B∈ X(M). (2.5) Then T₁ is a tensor : X(M)→ X(M) such that

N

X

i=1

L²_A

iB =−B−2T₁(B). (2.6)

Proof. Conditions (2.2) and (2.3) imply the following identity (see [13, 14])

N

X

i=1

div(Ai)Ai= 0. (2.7)

By torsion free, L_AiB = [Ai, B] =∇_AiB− ∇_BAi. Then according to (2.7),

N

X

i=1

div(A_i)L_AiB =−

N

X

i=1

div(A_i)∇_BA_i.

It follows thatT₁(f B) =fT₁(B) for any smooth functionf onM. Concerning equality (2.6), it was already calculated in proof of Theorem 3.7 of [14] that

Z

M

∆(ω(B))dx=− Z

M

(ω)(B)dx− Z

M

ω(

N

X

i=1

L²_A

iB)dx−2 Z

M

ω(T₁(B))dx.

The left hand side of above equality vanishes and R

M(ω)(B)dx = R

Mω(B)dx, equality (2.6) follows.

The tensor T₁ has explicit expression onSⁿ (see [14]). For the sake of self-contained, we give here a short presentation. We denote byh,ithe canonical inner product ofRⁿ⁺¹. Letx∈Sⁿ, the tangent space TxSⁿ of Sⁿ at the point x is given by

TxSⁿ=

v∈Rⁿ⁺¹; hv, xi= 0 .

Then the orthogonal projection P_x :Rⁿ⁺¹ →T_xSⁿ has the expression:

Px(y) =y− hx, yix.

LetB={e₁,· · ·, en+1}be an orthonormal basis ofRⁿ⁺¹; then the vector fieldAi defined by A_i(x) = P_x(e_i) has the expression: A_i(x) =e_i− hx, e_iix for i= 1,· · ·, n+ 1. Let v∈T_xSⁿ such that |v|= 1, consider

γ(t) =x cost+v sint.

Then {γ(t); t ∈ [0,1]} is the geodesic on Sⁿ such that γ(0) = x, γ⁰(0) = v. We have A_i(γ(t)) =e_i− hγ(t), e_iiγ(t). Taking the derivative with respect tot and att= 0, we get

(∇_vA_i)(x) =P_x −hv, e_iix− hx, e_iiv

=−hx, e_iiv. (2.8) It follows that

div(A_i) =−nhx, e_ii. (2.9)

Hence in this case it is obvious that

n+1

X

i=1

div(Ai)Ai =−n

n+1

X

i=1

hx, e_iie_i− hx, e_ii²x

=−n(x−x) = 0. (2.10) Replacing v by A_i in (2.8), we have ∇_AiA_i =−hx, e_iie_i+hx, e_ii²x; therefore summing over i, we get

n+1

X

i=1

∇_AiAi= 0. (2.11)

Now let v∈TxSⁿ and a, b∈TxSⁿ, we have

hA_i∧ ∇_vA_i, a∧bi=hA_i, aih∇_vA_i, bi − hA_i, bih∇_vA_i, ai

=−ha, e_iihx, e_iihv, bi+hx, e_iihb, e_iihv, ai.

Summing over iyields

n+1

X

i=1

hA_i∧ ∇_vA_i, a∧bi=ha, xihv, bi − hx, bihv, ai= 0. (2.12) By (2.10) and (2.11), we see that the family of vector fields {A₁, . . . , A_n+1} satisfy above conditions (2.1)–(2.3). Let B be a vector field on Sⁿ; by (2.8), ∇_BA_i = −hx, e_iiB. Using L_AiB=∇_AiB− ∇_BAi and combining with (2.9) and (2.10), we get that

T₁(B) =

m

X

i=1

div(Ai)L_AiB=−nB. (2.13)

In what follows, we will compute

n+1

X

i=1

L²_A

iB directly on Sⁿ. Again by torsion free and using (2.8), we have

L_AiB =∇_AiB− ∇_BA_i=∇_AiB+hx, e_iiB, and L²_A

iB =∇_Ai(L_AiB)−(∇_LAiB)Ai. Using (2.8) for last term, taking∇_Ai on the two sides of above equality, we get

L²_A

iB =∇_Ai∇_AiB+∇_Ai(hx, e_iiB) +hx, e_iiL_AiB.

But ∇_Ai(hx, e_iiB) =hA_i, e_iiB+hx, e_ii∇_AiB; therefore L²_A

iB =∇_Ai∇_AiB+hA_i, e_iiB+ 2hx, e_ii∇_AiB− hx, e_ii∇_BA_i. We have, by (2.10)

n+1

X

i=1

hx, e_ii∇_Ai =−1 n

n+1

X

i=1

div(A_i)∇_AiB = 0.

Besides

n+1

X

i=1

hA_i, eii=

n+1

X

i=1

(1− he_i, xi²) =nand

n+1

X

i=1

hx, e_ii∇_BAi =

n+1

X

i=1

hx, e_ii²B =B.

According to (2.11),

∆B =

n+1

X

i=1

∇_Ai∇_AiB.

Therefore combining above calculations, we get

n+1

X

i=1

L²_A

iB = ∆B+ (n+ 1)B. (2.14)

Note that Ric(B) = (n−1)B on Sⁿ, and by Bochner-Weitzenb¨ock formula − = ∆−Ric, we see that (2.14) is compatible with (2.6) together with (2.13).

Proposition 2.2. Let ∆Bˆ =

n+1

X

i=1

L²_A

iB. Then the following dimension free identity holds

− Z

M

h∆B, Biˆ dx= 2 Z

M

|Def (B)|²dx−2 Z

M

|B|²dx, for any div(B) = 0.

Proof. By (1.4), ∆B =−ˆB+ Ric(B) if div(B) = 0,. Then ˆ∆B =−ˆB+ 2B, due to (2.14).

The result follows.

3 Nash embedding and sum of squares of Lie derivatives

In what follows, we will find links between above different objects when the compact Rie- mannian manifold M is isometrically embedded in an Euclidian spaceR^N. LetJ :M →R^N be a Nash embedding, that is, for anyx∈M,

|dJ(x)v|_RN =|v|_TxM, v∈T_xM. (3.1) Denote by (dJ(x))^∗ :R^N →T_xM the adjoint operator ofdJ(x):

h(dJ(x))^∗ξ, vi_TxM =hξ, dJ(x)vi

R^N, ξ ∈R^N, v∈T_xM.

For eachx∈M, we denote Ex =dJ(x)(TxM) andE^⊥_x the orthogonal inR^N toEx. Define Λx =dJ(x)(dJ(x))^∗.

We have

hξ−Λxξ, dJ(x)vi=hξ, dJ(x)vi − hdJ(x)(dJ(x))^∗ξ, dJ(x)vi= 0.

Therefore Λx:R^N →Ex is the orthogonal projection. Set Λ^⊥_x = Id−Λx. By polarization of (3.1), we have

(dJ(x))^∗dJ(x) = Id_TxM.

The mapping Λ :M →L(R^N,R^N) fromM to the space of linear maps ofR^N is smooth. For ξ ∈R^N given,v ∈TxM, consider a smooth curveγ on M such that γ(0) =x, γ⁰(0) =v, we denote

dΛ_x(v)ξ= d dt|t=0

Λ_γ(t)ξ

∈R^N.

Proposition 3.1. (see [25],ch.5) Let x∈M, for anyv ∈T_xM, it holds true:

(i) Λ_x◦dΛ_x =dΛ_x◦Λ^⊥_x, (ii) Λ^⊥_x ◦dΛ_x =dΛ_x◦Λ_x. (3.2)

Proof. We have Λ_x(Λ_xξ) = Λ_xξ. Taking the derivative with respect to x in direction v on two sides of this equality, we get

dΛ_x(v)(Λ_xξ) + Λ_x(dΛ_x(v)ξ) =dΛ_x(v)ξ,

which yields the first equality in (3.2). For (ii), it suffices to use Λ^⊥_x(Λ^⊥_xξ) = Λ^⊥_xξ and dΛ^⊥_x(v) =−dΛ_x(v).

Now let ξ ∈ Ex, then by (i) in (3.2), Λx(dΛx(v)ξ) = 0, which implies that dΛx(v)ξ ∈ E_x^⊥. This means that dΛx(v) sends Ex intoE_x^⊥. By (ii) in (3.2),dΛx(v) sends E_x^⊥ intoEx. Now we introduce the Levi-Civita covariant derivative on M: for B ∈ X(M) and v ∈TxM, set

(∇_vB)(x) = (dJ(x))^∗

∂_v(dJ(·)B·)(x)

, (3.3)

where ∂v denotes the derivative on the manifold M in direction of v. If Bx = (dJ(x))^∗ξ for a fix ξ, then

h∇_vB, ui_TxM =hdΛ_x(v)ξ, dJ(x)ui

R^N, u∈T_xM.

The second fundamental form α onM is defined as follows: foru, v∈TxM,

αx(u, v) =dΛx(u)·dJ(x)v. (3.4)

By Proposition 3.1, α_x(u, v) ∈E_x^⊥. For X, Y ∈ X(M), we define α(X, Y)(x) = α_x(X_x, Y_x).

It is clear that α(X, f Y) = f α(X, Y). In what follows, we will see that α is symmetric bilinear application: α(X, Y) =α(Y, X).

LetX be a vector field onM, thenx→dJ(x)Xx is a R^N-valued function; therefore there is a function ¯X:R^N →R^N such that ¯X(J(x)) =dJ(x)X_x forx∈M. LetY be another vector field on M, and ¯Y :R^N → R^N such that ¯Y(J(x)) =dJ(x)Y_x, x∈M. Taking the derivative of x→Y¯(J(x)) alongX:

∂_X Y¯ ◦J

(x) = ¯Y⁰(J(x))·dJ(x)X_x= ¯Y⁰(J(x)) ¯X(J(x)) = (DX¯Y¯)(J(x)), (3.5) where ¯Y⁰ denotes the differential of ¯Y and DX¯ the derivative on R^N with respect to ¯X. It follows that

[ ¯X,Y¯](J) =∂_X( ¯Y(J))−∂_Y( ¯X(J)). (3.6) Since ¯Y(J)(x)∈E_x, then Λ_x( ¯Y(J(x))) = ¯Y(J(x)). Taking the derivative with respect to X and according to definition (3.4), we get

∂_X( ¯Y(J))(x) =dΛ_x(X_x)·Y¯(J(x)) + Λ_x

∂_X( ¯Y(J))(x)

=αx(Xx, Yx) + Λx

∂_X( ¯Y(J))(x)

.

Combining this with (3.6), we get

[ ¯X,Y¯](J) =α(Y, X)−α(X, Y) + Λ· [ ¯X,Y¯](J) .

By property of embedding, [ ¯X,Y¯](J(x)) ∈ E_x so that Λ· [ ¯X,Y¯](J)

= [ ¯X,Y¯]. Then the symmetry ofα follows.

Now according to definition (3.3) and to (3.5), we get (∇_XY)(x) = (dJ(x))^∗(DX¯Y¯)(J(x)).

Therefore the following orthogonal decomposition holds

(DX¯Y¯)(J(x)) =dJ(x)(∇_XY)(x) +α_x(X_x, Y_x). (3.7) Now let x→Vx∈E^⊥_x be a field of normal vectors toM. Set, for X∈ X(M),

A(X, V)(x) =−(dJ(x))^∗(∂XV)(x). (3.8) Since x→ hdJ(x)Y_x, V_xi_RN = 0, taking the derivative with respect to X yields

h∂_X(dJ(·)Y)(x), V_xi

R^N +hdJ(x)Y_x,(∂_XV)(x)i

R^N = 0.

By (3.7) and (3.8), we get

hα(X, Y), Vi=hA(X, V), Yi. (3.9) From above expression, A(X, f V) =fA(X, V). Ais called shape operator ofM.

Proposition 3.2. Let B ∈ X(M) defined byB_x= (dJ(x))^∗ξ for a fixed ξ ∈R^N. Then

∇_XB =A(X,Λ^⊥_· ξ), X∈ X(M). (3.10) Proof. By (3.3), (∇_vB)(x) = (dJ(x))^∗ ∂v(Λ·ξ)

which is equal to −(dJ(x))^∗ ∂v(Λ^⊥_· ξ) as Λxξ+ Λ^⊥_x =ξ. Now definition (3.8) gives the result.

For the sake of self-contained, we use (3.10) to check properties (2.2), (2.3) and (2.7). Let B={e₁, . . . , eN} be an orthonormal basis of R^N, we define

A_i(x) = (dJ(x))^∗e_i, i= 1, . . . , N.

Proposition 3.3. We have

N

X

i=1

∇_AiA_i= 0.

Proof. Letu∈T_xM, then by (3.10) and (3.9) respectively,

h(∇_AiA_i)(x), ui=hA(A_i,Λ^⊥_· ξ_i), ui=hα_x((dJ(x))^∗e_i, u), e_ii.

The sum fromi= 1 toN of above terms is basis independent. Then taking{e₁, . . . , e_n} ⊂E_x and {e_n+1, . . . , e_N} ⊂E_x^⊥, and remarking (dJ(x))^∗e_j = 0 for j > n, we have

h

N

X

i=1

(∇_AiA_i)(x), ui=

n

X

i=1

hα_x((dJ(x))^∗e_i, u), e_ii= 0, due to the orthogonality. The result follows.

Proposition 3.4. We have

N

X

i=1

A_i∧ ∇_VA_i= 0, V ∈ X(M).

Proof. Leta, b∈T_xM, we have

hA_i∧ ∇_VA_i, a∧bi=hA_i, aih∇_VA_i, bi − hA_i, bih∇_VA_i, ai, which is equal to, by (3.10),

hA_i, aihA(V_x,Λ^⊥_xei), bi − hA_i, bihA(V_x,Λ^⊥_xei), ai, which is equal to, by (3.9),

he_i, dJ(x)aihα(V_x, b), e_ii − he_i, dJ(x)bihα(V_x, a), e_ii.

Therefore

N

X

i=1

hA_i∧ ∇_VA_i, a∧bi=hα(V_x, b), dJ(x)ai − hα(V_x, a), dJ(x)bi= 0.

Proposition 3.5. Let B(x) = (dJ(x))^∗ξ for a fixed ξ ∈R^N. Then div(B) =hTrace(α), ξi

R^N. (3.11)

Proof. Let{v₁, . . . , vn}be an orthonormal basis of TxM; by (3.10) and (3.9), div(B) =

n

X

i=1

hA(v_i,Λ^⊥_xξ), vii=

n

X

i=1

hα(v_i, vi), ξi=hTrace(α), ξi

R^N.

Theorem 3.6. LetB={e₁, . . . , e_N}be an orthonormal basis ofR^N, andAi(x) = (dJ(x))^∗ei. Define

T₁(B) =

N

X

i=1

div(A_i)L_AiB. (3.12)

Then

T₁(B) =−A(B,Trace(α)). (3.13)

Proof. We first see, by (3.11), that

N

X

i=1

div(A_i)(x)A_i(x) =

N

X

i=1

hTrace(α_x), e_ii(dJ(x))^∗e_i= (dJ(x))^∗(Trace(α_x)) = 0,

since Trace(αx) ∈ E_x^⊥. Therefore term (3.12) becomes T₁(B) =

N

X

i=1

div(Ai)∇_BAi. Again using (3.10),

T₁(B) =−

N

X

i=1

hTrace(α), e_iiA(B,Λ^⊥_· e_i) =−A(B,Trace(α)).

Remark: In the case of Sⁿ, for x ∈ Sⁿ, the outer normal unit vector n(x) = x; then for v∈T_xM and γ(t) =xcos(t) +vsin(t), we see that d

dt|t=0

n(γ(t)) =v∈T_xM. Therefore

A(v,n) =v.

Using (3.13) gives T₁(B) =−nB.

In what follows, we will give a link between the Ricci tensor and T₁. Let R(B, Y)Z = [∇_B,∇_Y]Z− ∇_[B,Y]Z be the curvature tensor onM. Using second fundamental form,R can be expressed by (see [18]):

hR(B, Y)Z, Wi=hα(Y, Z), α(B, W)i − hα(B, Z), α(Y, W)i. (3.14) Let Ai be defined as above. Define

ψ(B, W) =

N

X

i=1

hα(B, A_i), α(W, A_i)i,

which is basis independent, symmetric bilinear form. Using (3.9), we get ψ(B, W) =

N

X

i=1

hA(B,Λ^⊥_· e_i),A(W,Λ^⊥_· e_i)i.

This quantity is independent of the dimension of M, but the co-dimension of M in R^N. Define the tensorT₂ by

hT₂(B), Wi=ψ(B, W), W ∈ X(M).

The tensor T₂ is directly related to the manner of the embeddingM intoR^N. Proposition 3.7. It holds true

T₁ =−Ric− T₂. (3.15)

Proof. We have hRic(B), Wi =

N

X

i=1

hR(B, A_i)A_i, Wi. The relation (3.14) leads hRic(B), Wi to

N

X

i=1

hα(A_i, Ai), α(B, W)i −

N

X

i=1

hα(B, A_i), α(W, Ai)i

=hA(B,Trace(α)), Wi −ψ(B, W) =−hT₁(B), Wi − hT₂(B), Wi.

The result (3.15) follows.

Theorem 3.8. Let B be a vector field on M of divergence free, then

N

X

i=1

L²_A

iB =−ˆB+ 2T₂(B). (3.16)

Proof. Equality (3.16) follows from (2.6) and (3.15).

4 Probabilistic representation formula for the Navier-Stokes equation

Let’s first state Constantin-Iyer’s probabilistic representation formula for Navier-Stokes equa- tion (1.1) on Tⁿ.

Theorem 4.1 (Constantin–Iyer). Letν >0, W be ann-dimensional Wiener process,k≥1, and u₀ ∈C^k+1,α a given deterministic divergence-free vector field. Let the pair (X, u) satisfy the stochastic system

(dX_t=√

2νdW_t+u_t(X_t) dt, ut=EP

∇X_t⁻¹∗

u0◦X_t⁻¹

, (4.1)

where P is the Leray–Hodge projection and the star ∗ denotes the transposed matrix. Then u satisfies the incompressible Navier–Stokes equations (1.1).

For a given sufficiently regular initial velocity u₀, there is a T > 0 such that system (4.1) admits a unique solution over [0, T] and X_t is a diffeomorphism of Tⁿ (see [10]). Using the terminology of pull-back by diffeomorphism of vector fields, the following intrinsic formulation to the second identity in (4.1) was given in [14]:

Z

Tⁿ

hu_t, vidx=E Z

Tⁿ

u0,(X_t⁻¹)∗v dx

, div(v) = 0, ∀t≥0,

which means that the evolution ofutin the directionvis equal to the average of the evolution of v under the inverse flow X_t⁻¹ in the initial direction u0. The generalization of Theorem 4.1 to a Riemannian manifold gave rise to a problem how to decompose the De Rham-Hodge Laplacian on vector fields as sum of squares of Lie derivatives : this has been achieved in [14]

on Riemannian symmetric spaces.

The purpose of this section is to derive a Feymann-Kac type representation for solutions to the following Navier-Stokes equation onM:

(∂_tu+∇_uu+νu+∇p= 0, div(u) = 0, u|_t=0=u0

(4.2) where ν >0.

Consider a Nash embedding J : M → R^N as in Section 3; let B = {e₁, . . . , e_N} be an orthonormal basis of R^N and Ai(x) = (dJ(x))^∗ei, x ∈ M. Let {u_t, t ∈ [0, T]} be a time- dependent vector fields in C^1,β withβ >0, then the following Stratanovich SDE on M

dX_t=√ 2ν

N

X

i=1

A_i(X_t)◦dW_tⁱ+u_t(X_t)dt, X₀ =x∈M (4.3) admits a unique solution {X_t, t ∈[0, T]} which defines a flow of diffeomorphisms of M (see [6, 16, 12]), where {W_tⁱ;i= 1, . . . , N} is aN-dimensional standard Brownian motion.

The space X(M) of vector fields onM, equipped with uniform norm||B||_∞= sup

x∈M

|B_x|_TxM, is a Banach space. We equip L(X,X), the space of linear map from X(M) into X(M), the norm of operator:

|||Q|||= sup

||B||∞≤1

||QB||_∞.

For a diffeomorphism ϕ:M →M, the pull-backϕ∗ sends X(M) into X(M), which is in the spaceL(X,X) with|||ϕ_∗||| ≤ ||ϕ||_C1. For a tensorT which sendsX(M) toX(M), we denote (T_∗B)(x) =T(x)B_x, then |||T_∗||| ≤ ||T||_∞. Now we consider the following linear differential equation on L(X,X):

dQt

dt =ν Qt·(X_t⁻¹)∗ T∗ (Xt)∗, Q0= identity (4.4) where (Xt) is solution to SDE (4.3).

Theorem 4.2. Let −˜ =

N

X

i=1

L²_A

i +T_∗. Then if ˜ preserves the space of vector fields of divergence free, u_t∈C^2,β is solution to

(∂_tu_t+∇_utu_t+ν˜u_t+∇p= 0, div(ut) = 0, u|_t=0=u0

(4.5) if and only if, for each v∈ X(M) with div(v) = 0,

Z

M

hu_t, vidx=E Z

M

u0, Qt(X_t⁻¹)∗v dx

, ∀t∈[0, T]. (4.6) Moreover, u_t has the following more geometric expression

u_t=E h

P

ρ_t·(X_t⁻¹)^∗( ˜Q_tu₀)^[#i

(4.7) where Q˜t is the adjoint of Qt in the sense that R

MhQ˜tu, vidx = R

Mhu, Q_tvidx, ρt is the density of (X_t)_#(dx) with respect to dxand P is the Leray-Hodge projection on the space of vector fields of divergence free.

Proof. Suppose (4.6) holds. By Itˆo formula [19, p.265, Theorem 2.1], we have d(X_t⁻¹)∗(v) =

N

X

i=1

(X_t⁻¹)∗(L_Aiv)dW_tⁱ+ν

N

X

i=1

(X_t⁻¹)∗(L²_A

iv)dt+ (X_t⁻¹)∗(L_utv)dt.

We have, according to (4.4) and above formula, dh

Q_t·(X_t⁻¹)∗vi

=dQ_t·(X_t⁻¹)∗v+Q_t·d(X_t⁻¹)∗v

=ν Q_t·(X_t⁻¹)∗T_∗v dt+

N

X

i=1

Q_t·(X_t⁻¹)∗(L_Aiv)dW_tⁱ

+ν

N

X

i=1

Qt·(X_t⁻¹)∗(L²_A

iv)dt+Qt·(X_t⁻¹)∗(L_utv)dt, since (X_t)∗(X_t⁻¹)∗= identity. Now using definition of ˜, we get

Z

M

hu_t, vidx= Z

M

hu₀, vidx−ν Z t

0

E Z

M

u₀, Q_s·(X_s⁻¹)∗( ˜v) dx

ds

+ Z t

0

E Z

M

u0, Qs·(X_s⁻¹)∗(L_usv) dx

ds.

Since ˜v and L_usv are of divergence free, to them we apply (4.6) to get E

Z

M

u0, Qs·(X_s⁻¹)∗( ˜v) dx

= Z

M

hu_s,˜vidx, and

E Z

M

u0, Qs·(X_s⁻¹)∗(L_usv) dx

= Z

M

hu_s,L_usvidx,

Therefore we obtain the following equality:

Z

M

hu_t, vidx= Z

M

hu₀, vidx−ν Z t

0

Z

M

hu_s,˜vidxds+ Z t

0

Z

M

hu_s,L_usvidxds.

But the last term can be changed Z

M

hu_s,L_usvidx= Z

M

hu_s,∇_usvidx− Z

M

hu_s,∇_vu_sidx

= Z

M

hu_s,∇_usvidx−1 2

Z

M

v(|u_s|²)dx= Z

M

hu_s,∇_usvidx.

Finally Z

M

hu_t, vidx= Z

M

hu₀, vidx−ν Z _t

0

Z

M

hu_s,˜vidxds+ Z _t

0

Z

M

hu_s,∇_usvidxds.

It follows that for a.e. t≥0, d

dt Z

M

hu_t, vidx=ν Z

M

hu_t,˜vidx+ Z

M

hu_t,∇_utvidx.

Multiplying both sides by γ ∈C_c¹([0,∞)) and integrating by parts on [0,∞), we get γ(0)

Z

M

hu₀, vidx+ Z ∞

0

Z

M

γ⁰(t)hu_t, vi+γ(t)hu_t,∇_utvi+ν γ(t)hu_t,˜vi

dxdt= 0.

The above equation is the weak formulation of the Navier–Stokes (4.5) on the manifoldM. To prove (4.7), we note that

Z

M

ρt

(X_t⁻¹)^∗( ˜Qtu0)^[, v dx=

Z

M

ρt

( ˜Qtu0)^[,(X_t⁻¹)∗v

X_t⁻¹dx

= Z

M

ρ_t(X_t) ˜ρ_t

u^[₀, Q_t·(X_t⁻¹)∗v dx

= Z

M

u^[₀, Q_t·(X_t⁻¹)∗v dx=

Z

M

u₀, Q_t·(X_t⁻¹)∗v

TxMdx

where ˜ρ_t is the density of (X_t⁻¹)_#(dx) with respect todx. Now by (4.6), we have:

Z

M

hu_t, vidx=E Z

M

ρ_t

(X_t⁻¹)^∗( ˜Q_tu₀)^[, v dx

, div(v) = 0.

The formula (4.7) follows.

For proving the converse, we use the idea in [30, Theorem 2.3]. Let ut∈C² be a solution to (4.5), then

Z

M

hu_t, vi dx= Z

M

hu₀, vidx−ν Z t

0

Z

M

hu_s,˜vidxds+ Z t

0

Z

M

hu_s,L_usvidxds.

Define

˜ u_t=E

h P

ρ_t·(X_t⁻¹)^∗( ˜Q_tu₀)^[#i .

Then calculations as above lead to Z

M

h˜u_t, vidx= Z

M

hu₀, vidx−ν Z t

0

Z

M

hu˜_s,˜vidxds+ Z t

0

Z

M

h˜u_s,L_usvidxds.

Let z_t=u_t−u˜_t; we have Z

M

hz_t, vidx=−ν Z t

0

Z

M

hz_s,˜vidxds+ Z t

0

Z

M

hz_s,L_usvidxds.

It follows that (z_t) solves the following heat equation onM dzt

dt =−ν˜zt− L^∗_u

tzt, z0 = 0,

whereL^∗_ut is the adjoint operator. By uniqueness of solutions, we get thatzt= 0 for allt≥0.

Thus ut= ˜ut. Then (4.7) follows. The proof of Theorem 4.2 is complete.

Remark 4.3. By Proposition 2.1,−B =

N

X

i=1

L²_A

iB+ 2T₁(B); TakingT = 2T₁ in Theorem 4.2, we obtain a probabilistic representation formula for Navier-Stokes equation (4.2) on a general compact Riemnnian manifold, sincepreserves the space of vector fields of divergence free. According to Theorem 3.8, the Ebin-Marsden Laplacian ˆhas the expression −ˆB =

N

X

i=1

L²_A

iB−2T₂(B). However, by (1.4) and (1.5),

ˆB=B+ 2Ric(B) for div(B) = 0;

therefore ˆdoes notpreserve the space of vector fields of divergence free, except for the case where Ric =kId, that is to say that M is a Einstein manifold.

Remark 4.4. Letg be the Riemannian metric ofM. A vector fieldA on M is said to be a Killing vector field if

L_Ag= 0.

Let X, Y ∈ X(M), we have, using Lie derivatives,

∂_A g(X, Y)

= (L_Ag)(X, Y) +g(L_AX, Y) +g(X,L_AY), (i) and using covariant derivatives,

∂_A g(X, Y)

=g(∇_AX, Y) +g(X,∇_AY). (ii) Since ∇_AX− L_AX =∇_XA, combining (i) and (ii) yields

(L_Ag)(X, Y) =g(∇_XA, Y) +g(X,∇_YA) = 2Def (A)(X,Y).

That is to say thatAis a Killing vector field if and only it Def (A) = 0, which implies ˆA= 0.

Conversely if div(A) = 0 and ˆA= 0, then Ais a Killing vector field on M (see [29]). Does is it why Ebin and Marsden said that ˆis more convenient in [11] ?

Proposition 4.5. Letρ_t be given in Theorem 4.2; then the following stochastic partial equa- tion (SPDE) holds

dρ_t=−√ 2ν

dJ(x)∇ρ_t+ Trace(α)ρ_t, dW_t

+ ν∆ρ_t+L_utρ_t

dt. (4.8)

Proof. Letf be a C²-function onM, by Itˆo formula, we have d f(Xt) =

√ 2ν

N

X

i=1

(L_Aif)(Xt)dW_tⁱ+ν

N

X

i=1

(L²_A

if)(Xt)dt+ (L_utf)(Xt)dt.

Since ∆f =

N

X

i=1

L²_A

if and Z

M

(L_Aif)(Xt) dx= Z

M

L_Aif ρt dx=− Z

M

f div(Ai)ρt+L_Aiρt

dx,

Z

M

(∆f)(Xt)dx= Z

M

∆f ρt dx= Z

M

f∆ρt dx, Z

M

L_ut(X_t) dx= Z

M

L_utf ρ_t dx= Z

M

f L_utρ_tdx,

and according to R

Mf(Xt)dx=R

Mf ρtdx, we get that ρtsatisfies the following SPDE dρt=−√

2ν

N

X

i=1

div(Ai)ρt+L_Aiρt

dW_tⁱ+

ν∆ρt+L_utρt

dt. (4.9)

Using (3.11), we have

N

X

i=1

div(Ai)ρtdW_tⁱ =hTrace(α)ρ_t, dWti.

Besides,

N

X

i=1

h(dJ(x))^∗ei,∇ρ_tidW_tⁱ =

N

X

i=1

he_i, dJ(x)∇ρ_tidW_tⁱ = hdJ(x)∇ρ_t, dWti. Therefore (4.8) follows from (4.9).

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