Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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396 16 Hydrodynamics

from dim M = 2 to higher dimensions. (See [3, Appendix 2] and [178]fora

detailed discussion of this matter.)

Consider an LHS of an ideal incompressible ﬂuid without external force on

a manifold M. Geometrically, the existence of trajectories on (−∞, ∞)means

that the weak Riemannian manifold D

(M) with the metric given by (5.1)is

geodesically complete. Note that here we have no analog of the Hopf-Rinow

theorem, i.e., geodesic completeness does not mean that any two points of

(M) can be connected by a geodesic. Later in this section we discuss the

problem of whether or not any element of D

(M) can be obtained as a ﬂow

of an incompressible ﬂuid with the initial condition e ∈D

(M).

Now let us turn to the regularity theorem, which is a very important result

in higher-dimensional hydrodynamics. This theorem claims that on the inter-

val where the ﬂow of the ﬂuid exists, the diﬀeomorphisms forming the ﬂow

are as smooth as the initial conditions. (In two-dimensional hydrodynamics

the regularity theorem follows from the existence and uniqueness theorem.)

Let M be a compact orientable Riemannian manifold, possibly with

boundary, and let n =dimM .Assumealsothats>n/2+1, q>0, and

the external force f

(t) ∈ T

s+q

(M) ⊂ T

(M) is continuous in t in the

topology of T

s+q

(M). Select the initial condition X

∈ T

(M)tobean

s+k

-smooth vector ﬁeld on M with 0 ≤ k ≤ q. Denote by η(t) ∈D

(M)

the ﬂow of an ideal ﬂuid on M with external force f

such that η(0) = e and

˙η(0) = X

For the case of M with boundary we need the following technical state-

ment. Let N be a compact orientable Riemannian manifold without bound-

ary, dim N =dimM , such that M is a compact submanifold of N.For

example, if M is a given Riemannian manifold, we may set N = double M

(see Section 1.1) and equip N with a Riemannian metric which coincides with

the original metric on one of the copies of M ⊂ N. Assume that s>n/2+1

and denote by D

(M)andD

(N) the groups of volume preserving diﬀeo-

morphisms of M and N, respectively. Also, let j : Vect N → Vect M be the

restriction morphism of vector ﬁelds, S the spray on T D

(N) (Theorem 5.9),

and π : T D

(N) →D

(N) the natural projection.

Theorem 16.17 [18, 19] There exists a C

∞

-smooth right-invariant sub-

bundle Ξ

of T D

(N) and a C

∞

-smooth ﬁber-wise right-invariant projection

R: T D

(N) → Ξ

with the following properties:

(i) The projection j : Ξ

→ T

(M) is an isomorphism. (Here Ξ

is the

ﬁber of Ξ

over e.)

(ii) The distribution Ξ

is non-holonomic. The ﬁbers of Ξ

have inﬁnite

dimension and inﬁnite codimension in the ﬁbers of TD

(N).

(iii) Denote by T

R: TTD

(N) → TΞ

the tangent map of

R.LetX(t) be

the integral curve of T

R ◦S with the initial condition X(0) = Y ∈ Ξ

Then the curve η(t)=πX(t) consists of diﬀeomorphisms which pre-

serve M,andη(t) |

is a curve in D

(M). This curve is a trajectory

16.3 The Regularity Theorem and a Review of Results on the Existence of Solutions 397

of the LHS of an ideal incompressible ﬂuid (without external force) on

M with the initial condition Y

= j(Y ).

A detailed proof of Theorem 16.17 is given in [106, Section 26].

Note that the family of diﬀeomorphisms η(t) does not correspond to the

motion of the ﬂuid on N \M, even though η(N \ M)=N \ M. We should

also point out that a “free” geodesic on D

(N) with initial condition Y is a

ﬂow of the ﬂuid on N that mixes M and N \ M.

Let f(t) ∈ Ξ

be a time-dependent external force and

f(t) the right-

invariant vector ﬁeld on D

(N) corresponding to f(t). To obtain the results

on the ﬂow of a ﬂuid with external force, we just need to slightly reﬁne (ii).

We replace the ﬁeld T

R ◦ S by the ﬁeld T

R ◦ (S +

f(t)

), where

f(t)

is the

vertical lift of

f(t)toT D

(N). With this modiﬁcation in mind, we have:

Corollary 16.18 The curve η(t) |

in D

(M) is a trajectory of the LHS

of an ideal incompressible ﬂuid on M with the external force f

(t)=jf(t).

Theorem 16.17 (suitably modiﬁed) still holds for η(t) as above.

It is clear from (i) that f(t)andY are entirely given by specifying f

(t)

and Y

, respectively.

Remark 16.19. It is clear by deﬁnition that the vector bundle Ξ

deﬁned

for all k ≥ s is the intersection of Ξ

with T D

(N) ⊂ T D

(N)and

R is a

vector bundle morphism T D

(N) → Ξ

Note that we have two equivalent descriptions of the ﬂow of an ideal in-

compressible ﬂuid on a manifold M with boundary. The ﬁrst one uses an

LHS on D

(M), while the second one is in terms of an LHS on D

(N) with

constraint Ξ. This means that the solutions of ﬂow equations in both descrip-

tions must exist for the same values of t simultaneously. It turns out that the

use of the second description sometimes simpliﬁes the argument. We apply

this in the proof of the following.

Theorem 16.20 (Regularity theorem) Let X

and f

belong to the space

s+k

(M) ⊂ T

(M) and η(t) be the ﬂow of an ideal incompressible ﬂuid

on M with external force f

such that η(0) = e and ˙η(0) = X

.ForM both

with and without boundary the diﬀeomorphism η(t) belongs to D

s+k

(M) for

all t for which the ﬂow exists in D

(M). Equivalently, the solution X(t),

X(0) = X

, of Euler’s equation with external force f

is an H

s+k

-smooth

vector ﬁeld on M (i.e., X(t) ∈ T

s+k

(M)) for all t such that X(t) exists

as an element of H

Proof. Following [15, 17, 18, 19], we ﬁrst analyze the more complex case where

∂M = ∅, and then conclude the proof by indicating what modiﬁcations can

be made when ∂M = ∅.

Let, as before, N be a closed Riemannian n-manifold and let M be isomet-

rically embedded in N via an embedding i. Recall that one may, for example,

398 16 Hydrodynamics

take the double of M as N. Then the metric on N is chosen to coincide

with the original metric on one of the copies of M ⊂ N. First, let us use

the constraint Ξ

introduced in Theorem 16.17.LetY

◦ i(X

)and

f =

◦ i(f

). By deﬁnition, we have (see Remark 16.19)

∈ Ξ

∩ T

s+k

(N)=Ξ

s+k

and f ∈ Ξ

∩ T

s+q

(N)=Ξ

s+q

Denote by

the right-invariant vector ﬁeld on D

(N) such that (

)

= Y

Clearly,

is a C

-section of Ξ

which is C

-smooth as a vector ﬁeld on

(N)(seeTheorem5.4).

Consider the mechanical system with the constraint Ξ

and the vector ﬁeld

R(S +

)onΞ

introduced above. (Recall that

is the vertical lift of the

right-invariant vector ﬁeld

f(t).) By deﬁnition, T

R(S +

) is right-invariant

on D

(N). This ﬁeld is also C

-smooth, since

f(t)isC

-smooth on D

(N)

and T

R and S are both C

∞

-smooth. Denote by φ

the ﬂow of T

R(S +

)on

. The local existence of φ

is guaranteed by the smoothness of the ﬁeld. In

other words, for any initial condition V ∈ Ξ

,themapt → φ

(V ) is deﬁned

for a suﬃciently small t. Since the ﬁeld is D

(N)-right-invariant, so is the ﬂow

.Foraﬁxedt, the diﬀeomorphism φ

is a C

-smooth map Ξ

→ Ξ

.Since

and φ

are both right-invariant, the domain of the function t → φ

(V )

is independent of V ∈

.Forthesamereason,theﬁeld

Y (t)=φ

(

)is

right-invariant for every t. Note also that because

is a C

-submanifold of

, the ﬁeld

Y (t)isC

-smooth on D

(N) for every t. Denote by Y

(t)the

vector of

Y (t) that belongs to Ξ

⊂ T

(N). The results of Chapter 5 yield

that Y

(t)isanH

s+k

-smooth vector ﬁeld on N for all t such that φ

(

)

exists, i.e., Y

(t) ∈ T

(N).

Let ˜η(t) be the ﬂow of Y

(t)onN. It follows from the results of Chapter

5 that ˜η(t) ∈D

s+k

(N) as long as Y

(t) ∈ T

(N). Furthermore, ˜η(t)isa

-smooth curve on D

(N) (Section 5.1). In particular,

˜η(t)=Y

(t) ◦ ˜η(t)=

Y (t) |

˜η(t)

By Corollary 16.18,weseethatη(t)=˜η(t) |

is a trajectory of the LHS

of an ideal incompressible ﬂuid on M and X(t)=Y

(t) |

is a solution of

Euler’s equation. It is clear that η(t) ∈D

s+k

(M)andX(t) ∈ T

s+k

(M)for

all t for which η(t)andX(t) exist and belong to H

If ∂M = ∅, the proof is easier; there is no need to pass to Y

, f ,andthe

ﬁelds on Ξ

. Instead, working with X

, f

,andS +

on T D

(M), one can

apply the same argument. 

Corollary 16.21 Let the force f

(t) be a continuous curve in T

∞

(M)

and let X

∈ T

∞

(M).Thenη(t) ∈D

∞

(M) as long as η(t) ∈ T

(M).

Equivalently, X(t) is C

∞

-smooth as long as it is H

-smooth.

Remark 16.22. The idea of our proof of Theorem 16.20 was originally used

in [61] to prove the regularity theorem on a closed manifold or on M \∂M.Itis

16.3 The Regularity Theorem and a Review of Results on the Existence of Solutions 399

essential that in our method (developed in [17, 18]) we deal with vector ﬁelds

on Ξ

right-invariant under D

(N) and that the latter group can “move” the

boundary ∂M. On the other hand, the group D

(M)usedin[61] preserves

∂M. As a consequence, one cannot obtain regularity in the normal directions

to ∂M by working only with the ﬁelds on T D

(M).

Remark 16.23. In [151], the regularity for a manifold with boundary was

proved in the particular case of a potential external force. The proof of this

result can be reduced to the study of the ﬂow of a free ﬂuid. In our proof the

force is only assumed to be divergence-free. As pointed out in Remark 16.8,

the general case can be formally derived from the case we have analyzed. Note

also that the regularity theorem for a general external force on a bounded

domain in R

was announced in [219, 220]. However, as the author later

pointed out (see [221]), the proof was incomplete.

Let us now turn to the problem of whether or not two given elements of

(M) can be connected by a ﬂow of an ideal incompressible ﬂuid with-

out external force, i.e., by a geodesic of the weak Riemannian metric. For

dim M = 2 and dim M = 3, this problem was studied by Shnirelman [209]in

the following context. Let η ∈D

(M) and let there exist a piecewise smooth

curve η(t), t ∈ [0, 1], in D

(M) which joins id with η (i.e., η(0) = id and

η(1) = η). In other words, we assume that η belongs to the path-connected

component of id. Denote by l

(η(·)) |

the length of the curve η(t) evaluated

with respect to the H

-metric deﬁned by (5.1). Thus,

(η(·))







(˙η(t), ˙η(t)) dt.

Taking into account that the ﬂow of an ideal incompressible ﬂuid is a

geodesic of (5.1), we see that the question is whether or not there exists a

smooth extremal of l

with ﬁxed end-points id and η.

The main result of [209] (Theorem 1.1) is as follows. Let M be the three-

dimensional cube. Then there is a diﬀeomorphism η in the path-connected

component of id such that for any piecewise smooth path η(·) with η(0) = id

and η(1) = η,thereexistsapathη



(·) with the same end-points and strictly

smaller length: l

(η



(·)) |

(η(·)) |

. As a consequence, η cannot be joined

with id by a ﬂow of the ﬂuid.

For a two-dimensional manifold M , it is still unknown whether or not a

given diﬀeomorphism η from the connected component of id in D

(M)can

be connected with id by a ﬂow of the ideal ﬂuid. However, it was conjectured

in [209] that such a ﬂow always exists.

The proof of the main theorem of [209] is based on the following important,

though technical, results. Let dist(ξ

,ξ

) be the inﬁmum of the l

-lengths over

all curves in D

(M) which connect ξ

and ξ

. As in the ﬁnite-dimensional

case, dist is a metric (i.e., a Riemannian distance) on D

(M). This metric

induces the weak (i.e., H

-) topology on D

(M). (According to a result of

400 16 Hydrodynamics

[209], the closure of D

(M) with respect to dist contains no interior points.) It

is shown in [209] that the diameter of D

(M) with respect to dist is ﬁnite if M

is three-dimensional and contractible, and inﬁnite if M is a two-dimensional

domain. When M is three-dimensional, we have the following estimate for

dist. Let Δ

(x)=ρ (x, ξ(x)) be the distance from x to ξ(x)onM.Itisnot

hard to see that Δ

∈ L

(M).Asshownin[209], there exist constants α>0

and C>0, which depend on M only, such that for every ξ ∈D

(M)

dist(id,ξ) ≤ C (Δ



)

Let n =dimM and s>n/2 + 1. Recall that, by Theorem 5.14 (proved

originally in [61]), for any manifold M, there exists an H

-neighborhood W

of id ∈D

(M) such that every element of W belongs to a ﬂow of a free

ideal incompressible ﬂuid starting at id. It is shown in [13]thatW is also

ﬁlled out by ﬂows of the ﬂuid with an external force, provided that the

following smoothness hypothesis holds. We assume ﬁrst that s>n/2+1 and

either the external force f is independent of time and H

s+1

-smooth (i.e., f ∈

s+1

(M)) or, if f is time-dependent, then f(t)isH

s+1

-smooth for every t,

continuous in t in T

s+1

(M)andC

-smooth in t in T

(M). Furthermore,

if s>n/2 + 2, then these assumptions can be relaxed: f ∈ T

(M) for an

autonomous f; or, otherwise, f (t) is continuous in t in T

(M).

To prove the latter assertion one passes to the group D

s−1

(M)andthen

applies the regularity theorem (Theorem 16.20) together with the relative

version of the theory of topological degree [13].

Remark 16.24. As is shown in [14], the neighborhood W is also covered by

ﬂows of a viscous incompressible ﬂuid. The problem of determining the size

of W seems to be interesting and important. The aforementioned results of

[209] show that in the three-dimensional case W is strictly smaller than the

connected component of id in D

(M).

In conclusion, let us prove that the ﬂow of a free incompressible ﬂuid

has a ﬁrst integral. This result is analogous to the angular momentum con-

servation law for the motion of a rigid body with a stationary point. (See

Section 11.2 and, in particular, Remark 11.4.) The existence of such a ﬁrst

integral is known in hydrodynamics as the circulation conservation law [3].

Apparently, this integral was originally considered in [211] by means of the

Lagrangian approach for the group of C

∞

-diﬀeomorphisms. Similarly to the

ﬁnite-dimensional case, the existence of a ﬁrst integral follows from the fact

that the metric (the Lagrangian) is invariant with respect to the group

structure (Noether’s theorem). The proof below is obtained by adapting the

standard ﬁnite-dimensional argument to the inﬁnite-dimensional setting on

(M) (cf., [3]). An essentially new point in the proof is that now we apply

the regularity theorem (Theorem 16.20).

Let s>n/2 + 1. Consider the set β

s−1

of H

s−1

-smooth divergence-free

vector ﬁelds on M which are tangent to the boundary. The ﬁelds from β

s−1

16.3 The Regularity Theorem and a Review of Results on the Existence of Solutions 401

are continuous on M, but may not be C

-smooth. Thus, the vanishing of the

divergence means only that β

s−1

is orthogonal to the space of exact forms in

the Hodge decomposition. (See (5.5)and(5.9).)

Denote by

s−1

the bundle over D

(M) obtained by right-translations of

s−1

, i.e., with ﬁbers

s−1

= {X ◦ g | X ∈ β

s−1

,g∈D

(M)}.For

V ∈ β

s−1

,let

V be the left-invariant cross-section of

s−1

, i.e.,

= Tg◦V .

Observe that the ﬁbers of

s−1

inherit the right-invariant inner product given

by (5.1). Since T

(M) ⊂

s−1

, the inner product (

)

,whereX

∈

(M), is well-deﬁned.

We emphasize that the metric given by (5.1) is simply the restriction of

the metric on the ﬁbers of

s−1

to the ﬁbers of T D

(M).

Let X

∈ T

(M)andletη(t) be the geodesic on D

(M) (the ﬂow of

ﬂuid) with initial conditions η(0) = id and ˙η(0) = X

Theorem 16.25 For any V ∈ β

s−1

, the inner product (

V, ˙η) is constant

along η.

Proof. First, let us assume that

V ∈ β = T

(M)andX

∈ T

s+1

(M) ⊂ T

(M).

Then by Theorem 16.20,wehaveη(t) ∈D

s+1

(M) as long as η(t)existsas

an element of D

(M). Hence,

= Tg ◦ V ∈ T

(M). Denote the ﬂow of

V by g(τ). In what follows we regard g(τ) as a one-parameter subgroup of

(M) and consider the right action R

g(τ)

on D

(M). Since η ∈D

(M), we

have

dτ

g(τ)

◦ η =

dτ



η ◦g(τ)



= Tη◦

dτ

g(τ)=Tη◦ V =

Thus,

V is the generator of R

g(τ)

.SincethesprayS on T D

(M) is right-

invariant under the action of D

(M) (see Section 5.1), the curve t → R

g(τ)

η(t)

is a geodesic for every ﬁxed τ.

Let

η(t, τ )=R

g(τ)

η(t)and ˙η(t, τ)=

η(t, τ ).

By deﬁnition, we have

dτ

η(t, τ )=

V . It is easy to see that the ﬁelds ˙η(t, τ)

and

V commute on the intersection of their domains, i.e.,



˙η(t, τ),



=0.

Clearly,

V (˙η(t, τ ), ˙η(t, τ )) =

dτ

(˙η(t, τ ), ˙η(t, τ )) = 0,

where

V (˙η(t, τ), ˙η(t, τ )) is the derivative of the scalar function ( ˙η(t, τ), ˙η(t, τ ))

in the direction of

V . Because ˙η(t, τ)and

V commute and the Levi-Civit´a

connection is torsion-free, we have

∇

˙η(t,τ )

V =

∇

˙η(t, τ).

402 16 Hydrodynamics

Observe that

∇

˙η(t,τ )

˙η(t, τ)=

˙η(t, τ)=0.

Then, by the deﬁnition of the Riemannian connection (see formula (2.29)),

V (˙η, ˙η)=2



∇

˙η, ˙η





∇

˙η

V, ˙η





∇

˙η

V, ˙η





∇

˙η

˙η,





V, ˙η(t, τ)



along η(t, τ). Therefore,



V, ˙η(t)



= 0 along the geodesic η(t)=η(t, 0).

This completes the proof of the particular case.

Now let us turn to the general case where V ∈ β

s−1

and X

∈ T

(M).

Recall that β = T

(M)isdenseinβ

s−1

and T

s+1

(M)isdensein

(M). Thus, there exists a sequence V

∈ β

s−1

converging to V in the

s−1

-topology and a sequence X

∈ T

s+1

(M) converging to X

in the

-topology. We pass to the limit as follows.

Let η

(t) be the geodesic in D

(M) with ˙η

(0) = X

. Since the solution

of a diﬀerential equation depends continuously on its initial condition, η

(t)

converges to η(t) uniformly on every ﬁnite interval. Note that for η

(t), X

and

, the theorem has already been proved.

It is not hard to see that the linear map V → (V,·), where (·, ·)isthe

weak inner product (5.1), is a continuous embedding of β into T

∗

(M). The

convergence of the vector ﬁelds in the H

s−1

-topology implies the convergence

in the space H

−s

. Recall that in the theory of Sobolev spaces the latter space

is identiﬁed with the dual to H

by means of the H

-inner product. It is clear

that the sequence (V

, ˙η

) converges to (V, ˙η)asi, j →∞. Since (V

, ˙η

)is

constant along η

, the function (V, ˙η) must also be constant along η. 

16.4 Description of Deterministic Viscous

Hydrodynamics Via a Stochastic Version of

Newton’s Law on Groups of Diﬀeomorphisms

Everywhere in this section we deal with ﬂuids moving in a ﬂat n-dimensional

torus T

. Recall that T

is the quotient space of R

with respect the integral

lattice where the Riemannian metric is inherited from R

(see Sections 5.2

and 10.2).

16.4.1 General construction

This section is devoted to the approach to hydrodynamics in terms of the

geometry of groups of diﬀeomorphisms, suggested for perfect ﬂuids by Arnold

16.4 Viscous hydrodynamics 403

[2] and Ebin and Marsden [61] (see the previous sections). It turns out that

to give an adequate description of viscous ﬂuids in this language requires the

involvement of stochastic processes (see, e.g., [106]and[107]). In particular,

Newton’s second law on the groups of diﬀeomorphisms, used in the case of

perfect ﬂuids, is replaced by its special stochastic analog in terms of Nelson’s

mean derivatives.

Here we use the material from Chapters 5 and 10. We shall deal with Itˆo

type equations on D

). Since the connection on D

) is generated by

the ﬂat connection on the torus, the corresponding exponential map is like

that on a linear space. So, without loss of generality we use a notation that is

typical for Itˆo equations in linear spaces (see Remark 10.4). Below we consider

a certain equation on the manifold

β (a sub-bundle of T D

) introduced

in Section 16.2) in general Belopolskaya-Daletskii form with respect to the

exponential map of some special connection. The equations on D

)are

considered as those on D

) subjected to a constraint

β as described in

Remark 10.4.

The main idea of the description of viscous hydrodynamics in the language

of mean derivatives is as follows.

Letarandomﬂowξ(t, m) with initial data ξ(0,m)=m ∈T

be given

on a ﬂat n-dimensional torus T

. Suppose that it is a general solution of a

stochastic diﬀerential equation of the type

ξ(t, m)=m +



a(s, ξ(s, m))ds + σw(t) (16.14)

modulo factorization with respect to the integral lattice, with real constant

σ>0. Let D

∗

ξ(t, m)=v(t, ξ(t, m)), where v(t, m)isaC

-smooth (in t)and

a C

-smooth (in m ∈T

) vector ﬁeld on T

.Letξ(t, m) satisfy the relation

∗

ξ(t, m)=F (t, m), (16.15)

where F (t, m) is a vector ﬁeld on T

. Taking into account formula (8.25), we

obtain

∗

ξ(t, m)=



∂

∂t

v +(v, ∇)v −

∇



∂

∂t

v +(v, ∇)v −

∇

(16.16)

Thus (16.15)meansthatv(t, m) satisﬁes the equality

∂

∂t

v +(v, ∇)v −

∇

v = F(t, m), (16.17)

which is the Burgers equation with viscosity

and external force F (t, m).

We interpret (16.15) as a stochastic analog of Newton’s second law on the

group of Sobolev diﬀeomorphisms D

404 16 Hydrodynamics

The case of viscous incompressible ﬂuids requires some additional con-

structions. Consider the vector space T

) of all Sobolev H

-vector

ﬁelds (s>

+1) on T

with the L

-inner product introduced by formula

(5.1)whereM is replaced by T

. Consider the subspace β = T

)

consisting of all divergence-free vector ﬁelds and the orthogonal projector

: T

) → T

)=β introduced in Section 5.1. Recall that by

formula (5.7) for any Y ∈ T

)wehaveP

(Y )=Y − gradp where p is

a certain H

s+1

function on T

that is unique to within an additive constant

for given Y .

Letarandomﬂowξ(t) be given on T

. Suppose that ξ(t)isageneral

solution of a stochastic diﬀerential equation of the type (16.14). Let D

∗

ξ(t)=

u(t, ξ(t)), where u(t, x) is a divergence-free vector ﬁeld on T

, C

-smooth in

t and C

-smooth in m ∈T

,andthatξ(t, x) satisﬁes the relation

∗

ξ(t)=F (t, ξ(t)), (16.18)

where F (t, x) is a divergence-free vector ﬁeld on T

. Taking into account

formulae (8.25)and(5.7), we obtain

∗

ξ(t, x)=P



∂

∂t

u +(u, ∇)u −

∇



∂

∂t

u +(u, ∇)u −

∇

u −gradp. (16.19)

Thus (16.18) means that the divergence-free vector ﬁeld u(t, x) satisﬁes the

relation

∂

∂t

u +(u, ∇)u −

∇

u −gradp = F, (16.20)

which is the Navier-Stokes equation with viscosity

and external force

F (t, x).

We interpret (16.18) as a stochastic analog of Newton’s second law on the

group D

), subject to the mechanical constraint

β that is introduced in

Section 16.2. In spite of the fact that the constraint is holonomic (i.e., inte-

grable), we do not restrict attention to its integral manifolds. This allows us

to apply both the ﬁnite and inﬁnite-dimensional language to the investigation

more easily.

Now we are in a position to describe this approach in detail. We present it

for the case of a viscous incompressible ﬂuid. The compressible case (leading

to the Burgers equation as above) can be investigated by a simpliﬁcation of

the incompressible arguments and we leave it for the reader.

For simplicity of presentation, we suppose s>

+ 2. This means that the

vector ﬁelds on T

are at least C

The deﬁnition of mean derivatives for processes on D

) is analogous to

that on R

and on T

. In order to distinguish the derivatives on D

)and

16.4 Viscous hydrodynamics 405

on T

we denote the former by

∗

and

while D, D

∗

and D

remain

valid for T

Let a(t, x) be a divergence-free H

vector ﬁeld on T

.Denoteby¯a(t, f)

the corresponding right-invariant vector ﬁeld on D

). Consider also the

right-invariant ﬁeld of the linear operators

A introduced in Section 10.2,

→ T

). The ﬂow on T

generated by equation (16.14) is a solution

of the equation

dξ(t)=¯a(t, ξ(t))dt + σ

A(ξ(t))dw(t) (16.21)

on D

). One can easily see that for the quadratic mean derivative for

such ξ(t) we obtain the equality

ξ(t)=σ

I (16.22)

where

I is the ﬁeld of unit operators in tangent spaces to T

.Itisalsoevident

that if ξ(t) is described by an equation in Itˆo form and satisﬁes (16.22), the

diﬀusion term of the equation is σ

A(ξ(t))dw(t).

The mechanical interpetation of the sub-bundle

β introduced in Sec-

tion 16.2 is a constraint. According to the ideology of the geometric de-

scription of constraints from Section 11.6, we give the following deﬁnition.

Deﬁnition 16.26. A stochastic process ξ(t) is said to be forward admissible

to the constraint

β if Dξ(t) ∈

ξ(t)

a.s. for all t.

A stochastic process ξ(t) is called backward admissible to the constraint

if D

∗

ξ(t) ∈

ξ(t)

a.s. for all t.

A vector ﬁeld X is called admissible if X

∈ β

at any f ∈D

Following general ideas of mechanics with constraints we can introduce

the notions of covariant mean derivatives with respect to a constraint. By

we denote the right-invariant ﬁeld of projectors introduced by formula (5.6).

Deﬁnition 16.27. For an admissible vector ﬁeld X and forward admissible

process ξ(t) the expression

DX(t, ξ(t)) is called the covariant forward mean

derivative with respect to the constraint

β.

For an admissible vector ﬁeld X and backward admissible process ξ(t)the

expression

∗

X(t, ξ(t)) is called the covariant backward mean derivative

with respect to the constraint

β.

Let η(t) be a backward admissible process. Then, according to Deﬁnition

16.27, we can consider the covariant backward mean derivative

∗

ξ(t).

Let F (t, x) be a divergence-free H

-vector ﬁeld on T

, i.e., F (t, x)canbe

considered as a time-dependent vector F (t) ∈

. Denote by

F (t, f )the

right-invariant vector ﬁeld on D

) generated by F (t).

Theorem 16.28 Letforaprocessξ(t) on D

) the relation

∗

ξ(t)=

¯u(t, ξ(t)) holds where ¯u(t, f) is a right-invariant vector ﬁeld on D

), gen-

erated by a divergence-free H

-vector ﬁeld u(t, x) on T

.Ifξ(t) satisﬁes

(16.22) and the constrained Newton law