Sep 11, 2015 - equalities, we obtain d(Dtg · (gâ t â gt)) = d(Dtg) · (gâ t â gt) + (Dtg · Dtgâ â |Dtg|2)dt. It follows that. DT g · (gâ. T â gT ) â D0g · (gâ. 0 â g0). = â«.
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. 2015.
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Generalized stochastic Lagrangian paths for the Navier-Stokes equation Marc Arnaudona a b
Ana Bela Cruzeirob
and Shizan Fangc
Institut de Maths. de Bordeaux, Universit´e de Bordeaux I, 33405 Talence Cedex, France
GFMUL and Dep. de Mat. Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal c
Institut de Maths. de Bourgogne, Universit´e de Bourgogne, 21078 Dijon Cedex, France
September 11, 2015
Abstract The purpose of this work is to establish a counterpart for the Navier-Stokes equation of Brenier’s generalized flows for the Euler equation. We introduce a class of semimartingales on a compact Riemannian manifold which will be generalized solutions of the Navier-Stokes equation. We prove that these semimartingales are critical points to the corresponding kinetic energy and that 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 of Rd , or on a compact Riemannian manifold M without boundary, they read d ut + (ut · ∇)ut = −∇p, div(ut ) = 0. (1.1) dt 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 gt (x) = ut (gt (x)), dt
g0 (x) = x.
(1.2)
When (t, x) → ut (x) is smooth, the ODE (1.2) defines a flow of C ∞ -diffeomorphisms gt . From the position values, we get the velocity by ut (x) =
d −1 gt (gt (x)). dt
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 of M . In a famous work [6], V.I. Arnold proved that u is a solution to (1.1) if and only if t → gt is a geodesic on Diff(M ) equipped with the L2 metric. Equivalently, g· minimizes the action 1
1 S[ϕ] = 2
Z TZ 2 d dxdt ϕt (x) dt Tx M 0 M
(1.3)
on C([0, T ], Diff(M )), where dx denotes the normalized Lebesgue measure on U or the normalized Riemannian volume on M . In [7], Y. Brenier relaxed (1.3) by looking for probability measures η on the path space C([0, T ], M ), which minimize the kinetic energy S[η] =
1 2
Z
C([0,T ],M )
hZ
T 0
i 2 |γ(t)| ˙ dt dη(γ), Tγ(t) M
(1.4)
with constraints (et )∗ η = dx, where et : 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 µ on [0, T ] × T M by Z f (t, x, v) µ(dt, dx, dv) [0,T ]×T M Z TZ
=
1 T
0
f (t, γ(t), γ ′ (t))dη(γ) dt.
C([0,T ],M )
Then µ solves the Euler equation in generalized sense: Z h i v · w(x)α′ (t) + v · (∇w(x) · v)α(t) µ(dt, dx, dv) = 0 for any α ∈ C ∞ (]0, T [) and any smooth vector field w such that div(w) = 0. We also refer to [1] in which the authors used the theory of mass transportation. The purpose of this work is to establish the above Brenier’s approach for the Navier-Stokes equation d ut + (ut · ∇)ut − νut = −∇p, div(ut ) = 0, (1.5) dt where ν > 0 is the viscosity coefficient and is the De Rham-Hodge operator on vector fields. In this context, it is suitable to consider that the underlying Lagrangian trajectories are semimartingales ξt on the manifold M . In contrast to Brenier’s generalized flows, the paths t → ξt 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 [17] as well as in connection with quantum mechanics [25]; we mention also [18] for the relation of (stochastic) kinetic energy and entropy as well as [24], for its appereance in the study of the Navier-Stokes equation. In the recent years it has been used with success in various contexts (see for example [2, 3, 4, 5, 10, 11, 12, 14, 19, 22]). 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 of ν-Brownian incompressible semimartingales. We prove that such a semimartingale
2
is a critical point of the corresponding kenetic energy [12] if and only if it solves the NavierStokes equation in the sense of DiPerna-Majda [7, 8]. In section 3, we shall show, in the case of a torus Td , that a classical solution to Navier-Stokes equation gives rise to a νBrownian incompressible martingale which realizes the minimum of the kinetic energy in a convenient class.
2
Generalized stochastic paths for the Navier-Stokes equation
In this section, M will denote a connected compact Riemannian manifold without boundary. Let (Ω, F, P ) be a probability space equipped with a filtration {Ft ; t ≥ 0} satisfying the usual conditions. A M -valued stochastic process ξt defined on (Ω, F, P ) is said to be a semimartingale on M if for any f ∈ C 2 (M ), f (ξt ) is a real valued semimartingale. This notion is independent of the chosen connection on M ; however, the corresponding local characteristics are dependent of the choice of connection. For a semimartingale (ξt ) starting from a point x ∈ M and given a connection ∇, the stochastic parallel translation //t along ξ· can be defined and Z t //−1 ζt = s ◦ dξs 0
is a Tx M -valued semimartingale. Then [9] there exist processes (ξ 0 (s), H1 (s), . . . , Hm (s)) which are adapted to Ft such that ξ 0 (s), H1 (s), . . . , Hm (s) ∈ Tξs M and ζt admits Itˆo form ζt = (wt1 , · · ·
Z
t 0
0 //−1 s ξ (s) ds +
m Z X i=1
t 0
i //−1 s Hi (s) dws
(2.1)
, wtm )
where wt = is a standard Brownian motion on Rm . For example, if the semimartingale ξt comes from a stochastic differential equation (SDE) on M : dξt = X0 (t, ξt )dt +
m X i=1
Xi (t, ξt ) ◦ dwti ,
then
ξ0 = x,
m
1X (∇Xi Xi )(t, ξt ). ξ (t) = X0 (t, ξt ) + 2 0
i=1
For simplicity, in what follows, we only consider the Levi-Civita connection ∇ on M . As in [3], [12], we consider the operator Dt ξ = //t lim E ε→0
ζ
− ζt Ft , ε
t+ε
(2.2)
which is well-defined and equals ξ 0 (t). For a semimartingale ξt given by (2.1), the Itˆo formula has the following form (see [9], p. 409)
3
Z t m 1X 0 h∇f (ξs ), ξ (s)i + f (ξt ) = f (ξ0 ) + h∇Hi (s) (∇f )(ξs ), Hi (s)i ds 2 0 i=1 m Z t X h∇f (ξs ), Hi (s)i dwsi . + i=1
(2.3)
0
Let {gt (x, ω); t ≥ 0, x ∈ M, ω ∈ Ω} be a family of continuous semimartingales with values in M . Let Pg denote the law of g in the continuous path space C([0, T ], M ), that is, for every cylindrical functional F , Z hZ Z i g F (gt1 (x), ..., gtn (x))dPgx dx F (γ(t1 ), ..., γ(tn ))dP (γ) = C([0,T ],M )
where Pg =
Pgx
M
⊗ dx and under
Pgx ,
C([0,T ],M )
the semimartingale gt starts from x.
We shall say that the semimartingale gt is incompressible if, for each t > 0, Z f (x)dx, for all f ∈ C(M ) EPg [f (gt )] =
(2.4)
M
the expectation being taken with respect to the law Pg of g. Let ν > 0; we shall say that gt is a ν-Brownian semimartingale if, under Pg , there exists a time-dependent adapted random vector field ut over gt such that Z t f ν∆f (gs ) + hus , ∇f (gs )i ds, (2.5) Mt = f (gt ) − f (g0 ) − 0
is a local continuous martingale with the quadratic variation given by Z t f1 f2 h∇f1 , ∇f2 i(gs )ds. hMt , Mt i = 2ν 0
For a semimartingale ξt given by (2.1), if {H1 (s), . . . , Hm (s)} is an orthogonal system m X hv, Hi (s)i2 = 2ν|v|2 , then it is a ν-Brownian such that for any vector v ∈ Tξs M , semimartingale.
i=1
Example 2.1. In the flat case Rd , such a semimartingale admits the following form √ dgt (w) = 2ν dwt + ut (w) dt, (2.6) where (wt ) is a Brownian motion on Rd and {ut ; t ≥ 0} is an adapted Rd -valued process RT such that 0 |ut (w)|2 dt < +∞ 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 (Vt )t∈[0,T ] be a family of C 1 vector fields such that the dependence t → Vt is C 1 . Denote by V˜t the horizontal lift of Vt to O(M ). Let div(Vt ) and div(V˜t ) be respectively the divergence operators on M and on O(M ); they are linked by (see [15], p. 595) div(V˜t ) = div(Vt ) ◦ π, 4
where π : O(M ) → M is the canonical projection. It follows that if div(Vt ) = 0, then div(V˜t ) = 0. Consider the horizontal diffusion rt on O(M ) defined by the SDE drt =
√
2ν
d X i=1
Hi (rt ) ◦ dwti + V˜t (rt ) dt,
r0 ∈ O(M )
(2.7)
where {H1 , · · · , Hd } are the canonical horizontal vector fields on O(M ). Let dr be the Liouville measure on O(M ), then the stochastic flow r0 → rt (r0 ) leaves dr invariant. Set ξ(t, x) = π(rt (r0 )),
r0 ∈ π −1 (x).
For any continuous function f on M , Z Z E(f (ξ(t, x)) dx = M
(2.8)
f (x) dx. M
Then ξ is an incompressible ν-Brownian diffusion, with Dt ξ(x) = Vt (ξ(t, x)). Remark 2.3. Let Pt be the semigroup associated to for any f ∈ C 2 (M ), d dt
Z
Pt f (x) dx =
Z
M
M
1 ∆M + Vt with div(Vt ) = 0; then 2
1 ∆M Pt f + Vt Pt f dx = 0. 2
It follows that for any continuous function f : M → R, Z Z f (x) dx. Pt f (x) dx = M
M
Therefore any SDE on M defining a Brownian motion with drift V gives rise to an incompressible ν-Brownian diffusion ξ with Dt ξ(x) = Vt (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 r r ν cos(k · θ) ⊥ ν sin(k · θ) ⊥ Ak (θ) = k , Bk (θ) = k , θ ∈ T2 , β ν0 |k| ν0 |k|β where β > 1 is some constant. ˜ 2 the subset of Z2 where we identify vectors k, k′ such that k + k′ = 0 and let Let Z 0 0 ν0 =
X
˜2 k∈Z 0
1 . 2|k|2β
The family {Ak , Bk : k ∈ Z20 } constitutes an orthogonal basis of the space of divergence free vector fields on T2 and satisfies X hAk , vi2 + hBk , vi2 = ν |v|2Tθ T2 ,
˜2 k∈Z 0
v ∈ Tθ T2 ,
and X
˜2 k∈Z 0
X
∇Ak Ak = 0,
˜2 k∈Z 0
5
∇Bk Bk = 0.
Consider the SDE on T2 , X dξt = Ak (ξt ) ◦ dwtk + Bk (ξt ) ◦ dw ˜tk + u(t, ξt ) dt, ˜2 k∈Z 0
θ0 = θ ∈ T2
(2.9)
˜tk ; k ∈ Z20 } are independent standard Brownian motions on R, and u(t, ·) is where {wtk , w a family of divergence free vector fields in H 1 (T2 ), such that, Z TZ 0
T2
(|u|2 + |∇u|2 ) dxdt < +∞.
Then by [12, 15], for β ≥ 3, the SDE (2.9) defines a stochastic flow of measurable maps which preserves the Haar measure dx on T2 . More precisely, for almost surely w, the map x → ξt (x, w) solution to (2.9) with initial condition x 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 incompressible ν-Brownian semimartingales and by Dν the set of incompressible ν-Brownian diffusions. Clearly we have Dν ⊂ Sν ⊂ S. Proposition 2.5. Let g ∈ Sν , then for any f ∈ C 2 (M ), EPg (h∇f (gt ), ut i) = 0.
(2.10)
Proof. Taking the expectation with respect to Pg in (2.5), we have EPg (f (gt )) − EPg (f (g0 )) = ν
Z
t 0
EPg (∆f (gs )) ds +
Z
t 0
EPg (h∇f (gs ), us i) ds.
It follows that ν
Z tZ 0
Since
R
M
∆f (x) dx ds +
t
Z
0
M
EPg (h∇f (gs ), us i) ds = 0.
∆f (x) dx = 0, we get the result.
Proposition 2.6. Let gt be a semimartingale on M satisfying dgt (x) =
m X i=1
Ai (gt (x)) ◦ dwti + ut (w, x) dx,
where A1 , · · · , Am are C 2 divergence free vector fields on M and ut (w, x) ∈ Tgt (x) M is R RT adapted such that M Ex ( 0 |ut (w, x)|2 dt) dx < +∞; if g is incompressible, then for any f ∈ C 2 (M ) EPg (h∇f (gt ), ut i) = 0.
6
Proof. Let f ∈ C 2 (M ); then by Itˆo formula (2.3), Z t m Z 1 X t f h∇f (gs ), us i ds, h∇Ai (∇f ), Ai i + h∇f, ∇Ai Ai i ds + f (gt ) = f (g0 ) + Mt + 2 0 0 i=1
where Mtf is the martingale part. Note that h∇Ai (∇f ), Ai i + h∇f, ∇Ai Ai i = LAi LAi f where LA denotes the Lie derivative with respect to A ; then taking the expectation under EP , we get m Z 1 X LAi LAi f dx + EPg (h∇f (gt ), ut i) = 0. 2 M i=1 R Since for each i, M LAi LAi f 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 , · · · , Am be C 2+α vector fields on M and A0 be a C 1+α vector field with some α > 0; consider dξt (x) =
m X i=1
Ai (ξt (x)) ◦ dwti + A0 (ξt (x)) dt,
ξ0 = x.
(2.11)
Then for almost all w, the map x → ξt (x) preserves the measure dx if and only if div(Ai ) = 0 for i = 0, 1, · · · , m. Proof. We give a sketch of proof (see [16] for more discussions). By [20], ; x → ξt (x) is a diffeomorphism of M and the push forward measure (ξt−1 )# (dx) of dx by the inverse map of ξt admits the density Kt which is given by (see [21]): Z t m Z t X i div(A0 )(ξs (x)) ds . div(Ai )(ξs (x)) ◦ dwt − Kt (x) = exp − i=1
(2.12)
0
0
If div(Ai ) = 0 for i = 0, 1, · · · , m, it is clear that Kt = 1 and x → ξt (x) preserves dx. Conversely, Kt (x) = 1 for any x ∈ M and t ≥ 0 implies that, Z t m Z t X div(A0 )(ξs (x)) ds = 0; div(Ai )(ξs (x)) ◦ dwti + i=1
0
0
or in Itˆo form: Z th X m Z t i X 1 i div(Ai )(ξs (x))dwt + LAi div(Ai ) + div(A0 ) (ξs (x)) ds = 0. 0 2 0 m i=1
i=1
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(Ai )(ξs (x)) = 0 and also h1 X i LAi div(Ai ) + div(A0 ) (ξs (x)) = 0. 2 m i=1
It follows that, almost everywhere, div(Ai )(ξs (x)) = 0 for
i = 0, 1, · · · , m;
so that div(Ai ) = 0 for i = 0, 1, · · · , m. According to [12], as well as [4, 19, 14], we introduce the following action functional on semimartingales. 7
Definition 2.8. Let 1 S(g) = EPg 2
Z
T 0
2
|Dt g| dt .
(2.13)
We say that g has finite energy if S(g) < ∞.
In what follows, we shall denote more precisely Dt g(x) for Dt g under the law Pgx . Then the action defined in (2.13) can be rewritten in the following form: Z T Z 1 2 S(g) = E g |Dt g(x)| dt dx. (2.14) 2 M Px 0
We first recall briefly known results about the calculus of stochastic variation (see [12, 4, 10]). Let ut (x) be a smooth vector field on a compact manifold (or on Rd ) which, for every t, is of divergence zero. Consider an incompressible diffusion gt (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: Dt g(x) = ut (gt (x)) and 1 S(g) = 2
Z
Td
E
Pgx
Z
T 0
2
|ut (gt (x))| dt dx.
There are two manners to perform the perturbation. First perturbation of identity: Let w be a smooth divergence free vector field and α ∈ C 1 (]0, T [). Consider, for for ε > 0, the ODE, dΦεt (x) = ε α′ (t) w(Φεt (x)), Φ0 (x) = x. (2.15) dt For each t > 0, Φεt is a perturbation of the identity map id. By Itˆo’s formula, for each fixed ε > 0, t → Φεt (gt (x)) is a semimartingale starting from x. Note that g and Φε (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 S(Φε (g))|ε=0 = 0 if and only if dε Z Z T hut , α′ (t)w + α(t) ∇w · ut + ν α(t)wi dtdx = 0. (2.16) Td 0
Second perturbation of identity: Note that in [19], the perturbation of the identity was defined in a different way. For each fixed t > 0, the author of [19] considered the ODE dΨts = α(t)w(Ψts ), ds
Ψt0 (x) = x.
(2.17)
d Set Ψ(g)εt (x) = Ψtε (gt (x)). Then S(Ψ(g)ε )|ε=0 = 0 if and only if the equation (2.16) dε holds. Now we deal with the general case of compact Riemannian manifolds.
8
Definition 2.9. Let M be a compact Riemannian manifold, g a semimartingale on M of finite energy. Define the probability measure µ on [0, T ] × T M by Z
f (t, x, v) µ(dt, dx, dv) =
[0,T ]×T M
h 1 EPg T
Z
T
0
i f t, g(t), Dt g dt
(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
TM
α′ (t) v · w + α(t) v · ∇v w + να(t) v · w dµ(t, x, v) = 0
(2.19)
for all α ∈ Cc1 (]0, T [) and all smooth vector fields w such that div(w) = 0.
Proof. Let Ψtε be the perturbation of identity defined in (2.17). Set ηtε = Ψtε (gt (x)). Then {ηtε , t ≥ 0} is a semimartingale on M . We denote by (ξ 0 (s), H1 (s), . . . , Hm (s)) the local characteristics of gt (x). By Itˆo’s formula (see [9], p. 408), the drift term in local characteristics of ηtε is given by m
Dt Ψtε (gt (x)) =
1X ∂ t Ψε (gt (x)) + dΨtε (gt (x)) · ξt0 + ∇(dΨtε )(gt (x))(Hi (t), Hi (t)), (2.20) ∂t 2 i=1
where dΨtε (gt (x)) denotes the differential of Ψtε at gt (x). Let ϕ(ε, t) = Dt Ψtε (gt (x)) ∈ Tηtε M ; then 2 Z T 1 S(Ψε (g)) = EPg ϕ(ε, t) dt . 2 0
We have: ϕ(0, t) = Dt g(x). Let
ϕ1 (ε, t) =
∂ t Ψ (gt (x)), ∂t ε
ϕ2 (ε, t) = dΨtε (gt (x)) · ξt0 , m
ϕ3 (ε, t) =
1X ∇(dΨtε )(gt (x))(Hi (t), Hi (t)). 2 i=1
Since the torsion is free, we have D D d t D d ϕ1 (ε, t)|ε=0 = Ψε (gt (x))|ε=0 = Ψt (gt (x)) = α′ (t)w(x). dε dε dt dt dε |ε=0 ε In order to compute the derivative of ϕ2 , consider a smooth curve β(s) ∈ M such that β(0) = gt (x), β ′ (0) = Dt g(x). Then dΨtε (gt (x)) · ξt0 = 9
d Ψt (β(s)). ds |s=0 ε
Therefore h i D D d D ϕ2 (ε, t) = Ψtε (β(s)) = α(t)w(β(s)) dε |ε=0 ds |s=0 dε |ε=0 ds |s=0 = α(t) (∇w)(gt (x)) · Dt g(x). For computing ϕ3 , we shall use another description given in [9] (p. 405). For the moment, consider a C 2 map f : M → M . Let x ∈ M and two tangent vectors u, v ∈ Tx M be given. Let x(t) ∈ M be a smooth curve such that x(0) = x, x′ (0) = u, and Yt ∈ Txt M such that Y0 = v. Define Q(f )(x) : Tx M × Tx M → Tf (x) M by Q(f )(x)(u, v) =
i h d (df (x ) · Y ) − df (x) · ∇u v. //−1 t t t dt |t=0
(2.21)
Then ϕ3 can be expressed by m
ϕ3 (ε, t) =
1X Q(Ψtε (gt (x)))(Hi (t), Hi (t)). 2 i=1
Let β(s) ∈ M be a smooth curve such that β(0) = gt (x) and β ′ (0) = Hi (t) and {Ys ; s ≥ 0} be a family of tangent vectors along {β(s); s ≥ 0} such that Y0 = Hi (t). Set γ(ε, s) = Ψtε (β(s))
and X(ε, s) = dΨtε (β(s)) · Ys .
If R denotes be the curvature tensor on M , the following commutation relation holds, ∂γ ∂γ DD DD X(ε, s) = X(ε, s) + R , X(ε, s). dε ds ds dε ∂ε ∂s ∂γ ∂γ We have X(0, 0) = Hi (t), (0, 0) = α(t)w(x), (0, 0) = Hi (t); therefore ∂ε ∂s h ∂γ ∂γ i R , X(ε, s) = α(t) R(w(gt (x)), Hi (t))Hi (t). ∂ε ∂s |ε=0,s=0 Now let c(τ ) ∈ M be a smooth curve such that c(0) = β(s), c′ (0) = Ys . We have hD d i D X(ε, s) = Ψtε (c(τ )) (0, 0) dε |ε=0 dτ dε D w(c(τ )) = α(t) (∇Ys w)(β(s)), = α(t) dτ |τ =0 and
D (∇Ys w)(β(s)) = h∇Hi (t) ∇w, Hi (t)i + h∇w, ∇Hi (t) Hi (t)i. ds |s=0
Note that D dΨt (gt (x)) · ∇Hi (t) Hi (t) = α(t) h∇w, ∇Hi (t) Hi (t)i. dε |ε=0 ε Using (2.21), we finally get m h i X D 1 h∇Hi (t) ∇w, Hi (t)i + R(w, Hi (t))Hi (t) . ϕ3 (ε, t) = α(t) dε |ε=0 2 i=1
10
When gt is a ν-Brownian semimartingale, the right hand side of above equality is equal to να(t) (∆w + Ric w)(gt (x)), which, due to Weitzenb¨ ock formula, is equal to να(t) w(gt (x)). In conclusion,
EPg
Z
0
Th
d S(Ψε (g))|ε=0 = 0 yields dε
i α′ (t) w(gt ) · Dt g + α(t) (∇Dt g w)(gt ) · Dt g + να(t) w(gt ) · Dt g 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 Z
M 0
T
|ut (x)|2 + |∇ut (x)|2 dtdx < +∞;
(2.23)
then equations (2.7), (2.8) define an incompressible ν-Brownian diffusion ξ on M , which is a critical point of the action functional S if and only if ut solves weakly the Navier-Stokes equation, that is, Z Z T hut , α′ (t)w + α(t) ∇w · ut + ν α(t)wi dtdx = 0 M 0
for all α ∈ Cc1 (]0, T [) and all smooth vector fields w such that div(w) = 0. Proof. First we notice that in Proposition 4.3 in [15], 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 2.12 below). Therefore by Theorem 6.4 in [15], equations (2.7) and (2.8) define a diffusion process ξ, which is, a fortiori, in Sν . Therefore by the above computations (see (2.22)), ξ is a critical point to S if and only if
E
Pg
Z
0
Th
i α′ (t) w(ξt ) · ut (ξt ) + α(t) (∇ut (ξt ) w)(ξt ) · ut (ξt ) + να(t) w(ξt ) · ut (ξt ) dt = 0,
which yields the result. 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 manifold M 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 11
Brownian bridge gtx,y on R over [0, 1]. It is known that for t < 1, gtx,y solves the following SDE dgtx,y = dwt − Then gtx,y → y as t → 1 and we have E
Z
1
0
gtx,y − y dt, 1−t
g0x,y = x.
|Dt gx,y |2 dt = +∞.
(2.24)
(2.25)
Let η be a probability measure on M × M having dx as two marginals; we shall say that the incompressible semimartingale {gt } has η as final configuration if Z f (x, y) dη(x, y), f ∈ C(M × M ). (2.26) EPg (f (g0 , gT )) = M ×M
This means that the joint law of (g0 , gT ) is η. If gt is as in Example 2.2, then Z f (x, y)pT (x, y) dxdy, EPg (f (g0 , gT )) = M ×M
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 ρt (x, y) = ν ∆x ρt (x, y) + hut (x), ∇x ρt (x, y)i, dt with lim ρt = δx , for some u ∈ L2 ([0, T ], D21 (M )) with div(ut ) = 0, we can construct an t→0
incompressible ν-Brownian semimartingale which has ρT (x, y)dxdy as final configuration. In any case, we have the following result: Theorem 2.12. Let η be a probability measure as above. If there exists an incompressible ν-Brownian semimartingale g on M of finite energy S(g) such that η is its final configuration, then there exists one that minimizes the energy among all incompressible ν-Brownian semimartingales having η as final configuration.
Proof. Let J : M → RN be an isometric embedding; then dJ(x) : Tx M → RN is such that for each x ∈ M and v ∈ Tx M , |dJ(x) · v|RN = |v|Tx M . Denote by (dJ(x))∗ : RN → Tx M the adjoint operator of dJ(x), that is, h(dJ(x))∗ a, viTx M = hdJ(x)v, aiRN ,
a ∈ RN , v ∈ Tx M.
Let {ε1 , . . . , εN } be an orthonormal basis of RN and set Ai (x) = (dJ(x))∗ εi ,
i = 1, . . . , N.
Then it is well-known that the vector fields {A1 , . . . , AN } enjoy the following properties: (i) For any v ∈ Tx M , (ii)
N X i=1
|v|2Tx M
N X hAi (x), vi2Tx M . = i=1
∇Ai Ai = 0. 12
N X
Combining (i) and (ii) gives that ∆M f =
i=1
hand, let J(x) = (J1 (x), . . . , JN (x)); then
L2Ai f for any f ∈ C 2 (M ). On the other
hdJ(x)v, εi i = dJi (x) · v = h∇Ji (x), viTx M ,
for any v ∈ Tx M.
It follows that Ai = ∇Ji ,
i = 1, · · · , N.
(2.27)
Let f ∈ C 2 (M ); then there exists f¯ ∈ C 2 (RN ) such that f (x) = f¯(J(x)). We have N X ∂ f¯ (J(x)) h∇Jj (x), Ai (x)i ∂xj
L Ai f =
j=1
N X ∂ f¯ = (J(x)) hAj (x), Ai (x)i. ∂xj
(2.28)
j=1
Therefore
∆M f =
N N X X
i=1 j,k=1
∂ 2 f¯ (J(x)) hAj , Ai ihAk , Ai i ∂xj ∂xk
N N X X ∂ f¯ (J(x))LAi hAj , Ai i. + ∂xj i=1 j=1
Notice that
N X i=1
LAi hAj , Ai i = div(Aj ) = ∆M Jj ,
and according to property (i), N X hAj , Ai ihAk , Ai i = hAj , Ak i. i=1
Finally the Laplacian ∆M on M can be expressed by ∆M f =
N X
j,k=1
N X ∂ f¯ ∂ 2 f¯ (J(x)) hAj , Ak i + (J(x)) ∆M Jj . ∂xj ∂xk ∂xj
(2.29)
j=1
Having these preparations, we prove now the existence of a g ∈ Sν such that the minimum of action functinal S is attained at g in the class of those in Sν having η as final configuration. Let K = inf S(g). g∈Sν
There is a minimizing sequence
gn
decomposition: J(gtn )
∈ Sν , that is, lim S(g n ) = K. Consider the canonical
=
n→+∞
J(g0n ) +
Mtn
13
+
Z
t 0
bn (s) ds.
Let Mtn = (Mtn,1 , · · · , Mtn,N ); then hMtn,i , Mtn,j i
= 2ν
Z
t 0
h∇Ji , ∇Jj i(gsn ) ds.
(2.30)
By Itˆo formula, we have bn (t) = dJ(gtn ) · Dt gn + ν ∆J(gtn ). It follows that E Therefore
Z
0
T
Z
T 0
(2.31)
|bn (t)|2 dt ≤ 2S(g n ) + 2T ν ||∆J||∞ .
|bn (t)|2 dt is bounded in L2 . We can use Theorem 3 in [26] to conclude that
the joint law Pˆn of
(J(g·n ), M·n , B·n , U·n ) in C([0, T ], RN ) × C([0, T ], RN ) × C([0, T ], RN ) × C([0, T ], RN ×N ) is a tight family, where Z t n bn (s) ds, Utn = (hMtn,i , Mtn,j i)1≤i,j≤N . Bt = 0
Let Pˆ be a limit point; up to a subsequence, we suppose that Pˆn converges weakly to Pˆ . Again by Theorem 3 in [26], under Pˆ , the coordinate process (Xt , Mt , Bt , Ut ) has the following properties: (i) M0 = B0 = 0, U0 = 0, (ii) (Mt ) is a local martingale such that Ut = (hMti , Mtj i)1≤i,j≤N and Z T Z t |b(s)|2 ds < +∞ almost surely. b(s) ds with (iii) Bt = 0
0
Since J(M ) is closed in
RN ,
we see that Xt ∈ J(M ). Let Xt = J(gt ).
For any f ∈ C 2 (M ), by (2.29), we see that f (gt ) is a real valued semimartingale. In other words, {gt ; t ≥ 0} is a semimartingale on M . Let f ∈ C(M ), the map f ◦J −1 : J(M ) → R can be extended as a bounded continuous function on RN ; therefore letting n → ∞, we get Z f (x) dx = E(f (g n (t))) = E(f ◦ J −1 (J(gtn ))) → E(f ◦ J −1 (Xt )) = E(f (gt )). M
In the same way, for f ∈ C(M × M ), we have Z f (x, y) dη(x, y) = E(f (g n (0), g n (T ))) = E(f (J −1 J(gn (0)), J −1 J(gn (T )))) M ×M
which goes to, as n → +∞,
E(f (g(0), g(T ))).
So g is incompressible and has η as final configuration. 14
Besides, by (2.30), we have (hMti , Mtj i)1≤i,j≤N
= 2ν
Z
t 0
h∇Ji , ∇Jj i(gs ) ds.
(2.32)
Let f ∈ C 2 (M ); denote by Mtf the martingale part of f (gt ). Then by Itˆo formula, dMtf =
N X ∂ f¯ (Xt ) dMtj . ∂xj j=1
Therefore for f1 , f2 ∈
C 2 (M ),
according to (2.32), we have N X ∂ f¯1 ∂ f¯2 = (Xt ) (Xt ) 2νhAj , Ak igt dt. ∂xj ∂xk
hdMtf1 , dMtf2 i
j,k=1
On the other hand, using relation (2.28) and property (i), we have h∇f1 , ∇f2 i =
N X
α=1
N X ∂ f¯1 ∂ f¯2 hAj , Ak i. L Aα f 1 L Aα f 2 = ∂xj ∂xk j,k=1
Combinant above two equalities, we finally get
Since Xt = J(gt ), we have
hdMtf1 , dMtf2 i = 2ν h∇f1 , ∇f2 igt dt.
(2.33)
1 dBt = dJ(gt ) · Dt g dt + HessJ(gt ) dgt ⊗ dgt . 2 Relation (2.33) implies that 21 HessJ(gt ) dgt ⊗ dgt = ν∆M J(gt ) dt. Therefore we get Z t Z t ∆M J(gs ) ds. (2.34) dJ(gs ) · Ds g ds + ν Bt = 0
0
In conclusion {gt ; t ≥ 0} is a ν-Brownian semimartingale on M or g ∈ Sν . We want to see that K = S(g). Firstly using the relation (2.31), for any t ∈ [0, T ], Z t Z t ∆J(gsn ) ds. dJ(gsn ) · Ds gn ds = Btn − ν 0
0
C([0, T ], RN )
Let φ : → R be a bounded continuous function, consider ϕ : C([0, T ], RN ) × N C([0, T ], R ) → R defined by Z · ∆J(gs ) ds . ϕ(B, g) = φ B· − ν 0
Then C([0, T ], RN ) × C([0, T ], RN ). It follows that Z · ϕ is a bounded continuous functionZ on · dJ(gsn ) · Ds gn ds converges in law to dJ(gs ) · Ds g ds. Let ε > 0; for n big enough, 0
0
E
Z
T 0
|dJ(gsn ) · Ds gn |2 ds ≤ K + ε.
Now by Theorem 10 in [23], E or E
Z
0
T
Z
T 0
|dJ(gs ) · Ds g|2 ds ≤ K + ε,
|Ds g|2 ds ≤ K + ε. Letting ε → 0 gives S(g) ≤ K. So S(g) = K. 15
3
Classical solutions and generalized paths
In this section, M will be a torus: M = Td . Let g ∈ Dν be the solution of the following SDE on Td √ (3.1) dgt = 2ν dwt − u(T − t, gt ) dt, g0 ∈ Td where g0 is a random variable having dx as law, wt is the standard Brownian motion on Rd , and {u(t, x); t ∈ [0, T ]} is a family of C 2 vector fields on Td , identified to vector fields on Rd which are 2π-periodic with respect to each space component. Suppose that u is a strong solution to the Navier-Stokes equation ∂ u(t, x) + ∇u(t, x) · u(t, x) − ν ∆u(t, x) = −∇p(T − t, x). ∂t By Itˆo ’s formula, ∂u )(T − t, gt ) − ∇u(T − t, gt ) · u(T − t, gt ) ∂t √ + ν ∆u(T − t, gt ) + 2ν ∇u(T − t, gt ) · dwt √ = ∇p(t, gt ) dt + 2ν ∇u(T − t, gt ) · dwt
du(T − t, gt ) = − (
(3.2)
According to definition (2.2), Dt g = −u(T − t, gt ) and Dt Dt g = −∇p(t, gt ).
(3.3)
√ G = g∗ ∈ Sν ; dgt∗ = 2ν dwt + Dt g∗ dt, g ∗ (0) = g(0), g ∗ (T ) = g(T ) .
(3.4)
In what follows, we shall consider
Note that semimartingales in G are defined on a same probability space. Example 3.1. Let α be a real continuous function on Rd and set Z t πt d β(w, t) = sin( ) α(ws ) ds, c(w, t) = β(w, t). T 0 dt
Let a ∈ Rd be fixed. Consider v(w, t) = c(w, t)a; then v is an adapted vector field on Td . Define Z gt∗ = gt +
t
v(w, s) ds.
0
Then g ∗ ∈ G. We have the following result Theorem 3.2. Let g ∈ Dν be given in (3.1). Assume that the process g is associated with the Navier-Stokes equation in the sense that Dt Dt g = −∇p(t, gt ) a.s. for a regular pression p such that ∇2 p(t, x) ≤ R Id, with RT 2 ≤ π 2 . Then g minimizes the energy S in the class G.
16
Proof. We define the following: 1 B(g) = 2
Z
T 0
2
|Dt g| dt −
Z
T
p(t, g(t))dt
(3.5)
0
Notice that the function b(x, y) defined in [7] (p. 243) has no meaning in our setting (c.f. (2.24) and (2.25)). Let g ∗ ∈ G; we shall prove that E(B(g)) ≤ E(B(g ∗ )).
(3.6)
Consider the function
R 2 |x| − p(t, x). 2 For each t ≥ 0, the function x → φ(t, x) is convex on Rd as ∇2 p(t, x) ≤ R Id. By Itˆo formula φ(t, x) =
√ d Dt g · gt = d(Dt g) · gt + 2ν Dt g · dwt + |Dt g|2 dt + d(Dt g) · dgt .
Analogously,
d(Dt g · gt∗ ) = d(Dt g) · gt∗ +
√
2ν Dt g · dwt + Dt g · Dt g∗ + d(Dt g) · dgt∗ .
Remarking that d(Dt g) · dgt = d(Dt g) · dgt∗ , and making the substraction of the above two equalities, we obtain d Dt g · (gt∗ − gt ) = d(Dt g) · (gt∗ − gt ) + Dt g · Dt g ∗ − |Dt g|2 dt.
It follows that
DT g · (gT∗ − gT ) − D0 g · (g0∗ − g0 ) Z T Z T d(Dt g) · (gt∗ − gt ) + Dt g · Dt g∗ − |Dt g|2 dt. = 0
0
Notice that
g0∗
=
g0 , gT∗ Z
T
∗
2
−Dt g · Dt g + |Dt g|
0
=
= gT , and using (3.1), we have
Z
T
0
dt =
Z
0
T
d(Dt g) · (gt − gt∗ )
√ (gt∗ − gt ) · − 2ν ∇u(T − t, gt )dwt − ∇p(t, gt )dt .
(3.7)
Using the convexity, of φ, we have φ(t, gt∗ ) − φ(t, gt ) ≥ Rgt − ∇p(t, gt ) · (gt∗ − gt ).
From (3.7) and (3.8), we get Z T −Dt g · Dt g∗ + |Dt g|2 + Rgt · (gt∗ − gt ) dt 0 Z T √ Z T ∗ ≤ − 2ν (gt − gt ) · ∇u(T − t, gt )dwt + φ(t, gt∗ ) − φ(t, gt ) dt.
(3.9)
0
0
We have gt∗ − gt =
(3.8)
Z
0
t
(Ds g∗ − Ds g) ds. Since g0∗ − g0 = gT∗ − gT = 0, by Poincar´e ’s
inequaliy on the circle to get
17
Z
T 0
|gt∗
T − gt | dt ≤ ( )2 π
Z
T
1 − gt | dt ≤ 2
Z
T
2
0
|Dt g∗ − Dt g|2 dt.
1 T Since ( )2 ≤ , we have π R R 2
Z
T 0
|gt∗
2
0
|Dt g ∗ − Dt g|2 dt.
(3.10)
Remark that the inequality, for x, y, a, b ∈ R 1 R R 1 x2 − xy − Rb2 + Rab ≥ x2 − y 2 − b2 + a2 2 2 2 2 holds if and only if
1 R (x − y)2 ≥ (b − a)2 . 2 2
Therefore by (3.10), we have Z T |Dt g|2 − Dt g · Dt g ∗ − R|gt |2 + Rgt · gt∗ dt 0 Z T 1 R R 1 |Dt g|2 − |Dt g ∗ |2 − |gt |2 + |gt∗ |2 dt. ≥ 2 2 2 2 0
(3.11)
Combining (3.9) and (3.11), we get Z T 1 1 R R |Dt g|2 − |Dt g∗ |2 − |gt |2 + |gt∗ |2 dt 2 2 2 2 0 Z T √ Z T ∗ φ(t, gt∗ ) − φ(t, gt ) dt, (gt − gt ) · ∇u(T − t, gt )dwt + ≤ − 2ν 0
0
from which we deduce Z T 1 R |Dt g|2 − |gt |2 + φ(t, gt ) dt 2 2 0 Z Z T √ ∗ ≤ − 2ν (gt − gt ) · ∇u(T − t, gt )dwt +
0
0
or
T
R 1 |Dt g∗ |2 − |gt∗ |2 + φ(t, gt∗ ) dt, 2 2
T
1 |Dt g|2 − p(t, gt ) dt 2 0 Z Z T √ ∗ ≤ − 2ν (gt − gt ) · ∇u(T − t, gt )dwt + Z
0
T 0
1 |Dt g ∗ |2 − p(t, gt∗ ) dt. 2
Using definition (3.5), √ Z B(g) ≤ − 2ν
0
T
(gt∗ − gt ) · ∇u(T − t, gt )dwt + B(g ∗ ).
Taking the expectation of this inequality, we obtain (3.6). Notice that RT ∗ ∗ 0 E(p(t, gt )) dt; then (3.6) yields E(S(g)) ≤ E(S(g )).
RT 0
E(p(t, gt )) dt =
The following result provides a perturbation in a natural way and illustrates Theorem 3.2. 18
Proposition 3.3. Let v(w, t) be the vector field constructed in Example 3.1. Consider the following perturbation of gt given by (3.1): √ dgtε = 2νdwt − u(T − t, gt ) dt + ε v(w, t) dt, g0ε = x. Then we have
d S(g ε )|ε=0 = 0. dε
Proof. We see that {g ε ; ε ≥ 0} ⊂ G. We have Z 1 T |u(T − t, gt ) − ε v(w, t)|2 dt . S(gε ) = E 2 0 Therefore
Let Vt =
Z T d S(g ε )|ε=0 = −E hu(T − t, gt ), v(w, t)i dt . dε 0
Rt 0
vs ds. By construction of v, VT = 0. Now by integration by parts, −
Z
0
T
hu(T − t, gt ), V˙ (w, t)i dt =
which is equal to, using (3.2), d S(gε )|ε=0 = dε
RT 0
Z
0
T
hd(u(T − t, gt )), V (w, t)i dt
h∇p(t, gt ), V (w, t)i dt. Therefore
T
E
0
=
Z
Z
0
Z
T
h∇p(t, gt (x)), β(w, t)ai dx dt Td Z h∇p(t, x), ai dx dt = 0. E(β(w, t)) Td
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