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

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386 15 The Newton-Nelson Equation

Let the force ﬁeld ¯α(m, X)=¯α

(m)X on M

be such that the tensor

¯α

(m) and its covariant derivative have bounded absolute values of compo-

nents with respect to the frames, parallel to b along R

W (τ). The simplest

example of such an ¯α is ¯α = 0, it holds when no external forces but gravitation

alone are under consideration.

Make the parallel translation of ¯α

W (τ)) and tr∇¯α

(¯α

) along

(W (·)) to m

. Introduce the Euclidean inner product in T

by consid-

ering b as an orthonormal frame in the Euclidean sense. Fix some t

∈ (0,l).

Now, having omitted all summands with a



Ric term in equations (15.31),

(15.33), etc., one can apply the construction of Section 15.2.2 and easily ob-

tain the existence of the corresponding ζ(τ ) and its development ξ(τ ).

Theorem 15.50 For τ ∈ (t

,l) the process ξ(τ) satisﬁes (15.58).

This is a simple corollary of (15.57) and the equality v

(τ)=¯v

(τ).

Remark 15.51. It should be pointed out that ξ(τ) does not satisfy (15.58)

for τ ∈ (0,t

)sinceheret

(τ)=

and E

(Γ

τ,0

−1

ζ(τ )

τ·t

) = E

(Γ

τ,0

−1

ζ(τ )

Thus ξ(τ) can be interpreted only as a trajectory of a stochastic mechanical

system beginning at the instant t

of proper time from a random conﬁguration

ξ(t

) with initial mean forward derivative E

(Γ

τ,0

a(t

)). It is clear that t

may be chosen arbitrarily close to zero, and so we can bring the initial values

to m

and a

as closely as desired. However, we cannot set t

=0,since

the integral



dW (s) does not exist (indeed,



ds diverges, see, e.g.,

[162]), i.e., when t

= 0 the analogs of equations (15.31)and(15.33) are ill-

posed. We suggest the hypothesis that this situation may be thought of as a

description of the ‘big bang’, the initial point for all the trajectories. Indeed,

all the equations of physical laws are ill-posed at the ‘big bang’; we can set

initial conditions for any time greater than the instant of the ‘big bang’, but

cannot do this for the ‘big bang’ itself.

The following analog of Theorem 15.31 holds.

Theorem 15.52 Let the Levi-Civit´a connection on L(M

) be stochastically

complete and the (1, 1)-tensor ﬁeld α(m) be continuous and have compact

support. Then for η as in Theorem 15.31 and for a continuous vector ﬁeld

a(m) with compact support there exists a solution ξ(τ ) of equation (15.58)

with initial conditions ξ(0) = η and

ξ(0) = a(η).

The analogous statement with initial condition for the forward mean

derivative Dξ(0) = a(η) is also true.

Taking into account formula (15.57) and the fact that ¯v

(τ)=v

(τ)(see

above) it is easy to see that Theorem 15.52 is a simple generalization of

Theorem 15.31 (here we use another bundle and another connection). Note

that the hypothesis of Theorem 15.52 has been expressed in a form that is

easy to verify. It can be replaced by the corresponding analogs of Conditions

15.20 and 15.21.

Chapter 16

Hydrodynamics

16.1 The Lagrangian Formalism of the Hydrodynamics

of an Ideal Barotropic Fluid

Following the lines of Chapter 11, we can apply the results of Section 5.1 to

study mechanical systems on the conﬁguration spaces D

(M), H

(M,M),

or H

(M,N) with kinetic energy given by the (weak) Riemannian metric.

Here we analyze those systems which are naturally related to certain prob-

lems of hydrodynamics. Note that according to the Lagrangian formalism, a

trajectory of such a system gives the ﬂow of a ﬂuid.

16.1.1 Diﬀuse matter

In what follows, we use the notation and the hypotheses of Section 5.1.In

particular, M denotes a compact Riemannian manifold without boundary

and ·, · is its Riemannian metric.

Consider a mechanical system on H

(M,M) with zero potential energy

and kinetic energy K(X)=

(X,X)

,where(·, ·) is given by (5.1). Then New-

ton’s equation for the system is

˙g(t)=0, (16.1)

where

is deﬁned as in Section 5.1.

Deﬁnition 16.1. The mechanical system deﬁned above by (16.1) is called a

Lagrangian hydrodynamical system (LHS ) of diﬀuse matter (with zero exter-

nal force).

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

16, © Springer-Verlag London Limited 2011

387

388 16 Hydrodynamics

It is clear by (16.1) that the trajectory of every particle of diﬀuse matter

is a geodesic of ·, · on M. In other words, the trajectory of the LHS with

initial condition e = id is given by the vector ﬁeld X ∈ T

(M) of the initial

velocities and the metric ·, ·.

The kinetic energy is constant along a trajectory of the LHS on H

(M,M)

and the trajectory is an extremal of the action functional with the Lagrangian

L = K.

Similarly, one may deﬁne an LHS of diﬀuse matter on a manifold M with

boundary. This time, however, the motion takes place on a larger manifold

N without boundary (dim N =dimM). The construction still holds if we

take H

(M,N) as the conﬁguration space. One may also consider an LHS of

diﬀuse matter with an external force.

Note that by Proposition 3.67 the curve (g(t), ˙g(t)) in T D

(M)isaninte-

gral curve of the geodesic spray

Z (see (5.4)), i.e.,

(g(t), ˙g(t)) = Z(g(t), ˙g(t)). (16.2)

Proposition 16.2 For every X ∈ T

(M) there exists a unique solution

g(t) of (16.1) with initial conditions g(0) = e and

g(0) = X that is well-

deﬁned for t in a suﬃciently small interval [0,ε).

Indeed, (16.1) is equivalent to (16.2) (which has a smooth right-hand side).

Recall that the group D

(M) is an open neighborhood of e in H

(M,M)

and so a solution of (16.1)thatstartsate,fort in a suﬃciently small in-

terval [0,ε), belongs to D

(M). A key role in the Euler description of diﬀuse

matter is played by the vector v(t) ∈ T

(M) obtained by right transla-

tion of the velocity ˙g(t) ∈ T

g(t)

(M) to the tangent space at the unit, i.e.,

v(t)=TR

−1

g(t)

˙g(t) (see Section 11.2). Specify some t and consider the right-

invariant vector ﬁeld ¯v(t)onD

(M) constructed by right translations of v(t)

to all points of D

(M). Note that the derivative

∂

∂t

v(t) is a vertical vector in

(e,v(t))

T D

(M)andfrom(16.2) it follows that

∂

∂t

v(t) equals the diﬀerence

between

Z and its component “tangent” to the submanifold ¯v(t)atthegiven

t with respect to the decomposition T

(e,v(t))

T D

(M)=

(e,v(t))

⊕T

(e,v(t))

¯v(t).

In fact, if v(t)isanH

-vector ﬁeld on M,¯v(t) is only continuous (see Section

5.1), but if it is H

s+k

for some integer k>0, then ¯v(t)isC

-smooth and so

it really has the tangent space at (e, v(t)).

We introduce the notation ¯g(t + Δt)=R

−1

g(t)

g(t + Δt). From the construc-

tion of the geodesic spray and of ¯v(t) it follows that

Z(e, v(t)) = lim

Δt→+0

¯v(t, ¯g(t + Δt)) − v(t)

Δt

−

K lim

Δt→+0

¯v(t, ¯g(t + Δt)) − v(t)

Δt

where the ﬁrst summand on the right-hand side is tangent to ¯v(t). Then

∂

∂t

v(t), as the diﬀerence between

Z and the latter summand takes the form

16.1 The Lagrangian Formalism of the Hydrodynamics of an Ideal Barotropic Fluid 389

∂

∂t

v(t)=−

K lim

Δt→+0

¯v(t, ¯g(t + Δt)) − v(t)

Δt

Since

K lim

Δt→+0

¯v(t,¯g(t+Δt))−v(t)

Δt

= ∇

v(t)

v(t), taking into account that the vec-

tor v(t) ∈ T

(M) is a vector ﬁeld v(t, m)onM and that the connector

is deﬁned via the connector K of the Levi-Civit´a connection on M by formula

(5.2), we obtain that v(t) satisﬁes the equation

∂

∂t

v + ∇

v = 0 (16.3)

on M. This is the so-called Hopf equation describing the motion of diﬀuse

matter in the framework of Euler’s approach. Sometimes it is also called the

Burgers equation without viscous term. Note that on the ﬂat torus (16.4)

takes the form

∂

∂t

v +(v ·∇)v =0. (16.4)

It should be pointed out that, unlike (16.1), equations (16.3)and(16.4) lose

derivatives, as is typical in Euler’s approach.

The motion of diﬀuse matter is not of much interest for hydrodynamics

and we shall use it only as a starting point for our further analysis.

16.1.2 A barotropic ﬂuid

Let us now turn to Lagrangian hydrodynamical systems of an ideal barotropic

ﬂuid. The major diﬀerence between such an LHS and that described in Sec-

tion 16.1.1 is the presence of a force ﬁeld, the potential of which is called the

internal energy of the ﬂuid. Strictly speaking, an LHS of an ideal barotropic

ﬂuid is not covered by the general construction of Chapter 11 because the

force ﬁeld loses smoothness, i.e., it is an H

s−1

-smooth “vector ﬁeld” on M.If

we considered only C

∞

-diﬀeomorphisms, then the force ﬁeld would be C

∞

smooth. However, then the conﬁguration space D

∞

(M) would be modeled

on a locally convex space, rather than on a Banach space. As a consequence,

the whole analysis would become much more complicated.

Here we brieﬂy outline some deﬁnitions and results on LHSs of a barotropic

ﬂuid on a closed manifold. A more detailed account can be found in [59, 210,

211].

Let M be a compact Riemannian manifold without boundary and let

(M) be the group of H

-diﬀeomorphisms of M with s>n/2+2. De-

note by V

s−1

the space of H

s−1

-smooth volume forms ν on M such that



ν =



μ,

390 16 Hydrodynamics

where μ is the Riemannian volume form. Following Smolentsev [210], consider

the map Ψ

: D

(M) →V

s−1

deﬁned as Ψ

(g)=(g

−1

)

∗

μ,where(g

−1

)

∗

μ is

the pull-back of μ under the map g

−1

. Since any two volume forms diﬀer

by a multiplier (see Section 1.6) which is a scalar function on M ,wehave

(g)=ρ(g)μ,whereρ(g): M → (0, +∞)isanH

s−1

function called the

density of the ﬂuid at g ∈D

(M).

Let U

:(0, ∞) → (0, ∞) be a smooth function. The composition

(ρ): M → (0, ∞) is called the speciﬁc internal energy of the system. Then

the internal energy U: D

(M) → (0, ∞) is deﬁned as

U(g)=



(ρ) ρμ=



(ρ) ν,

where ν = ρμ = Ψ

(g). In a true physical system, the function U

depends

on the properties of the ﬂuid.

Consider also the function p : M → R given by p(ρ)=ρ

d U

dρ

, which

is known as the state equation in mechanics. The function p is called the

pressure of the ﬂuid at g,whereΨ

(g)=ρμ.

Remark 16.3. To explain the terminology, we emphasize that the ﬂuid un-

der consideration is compressible, since we have taken the entire group D

(M)

as the conﬁguration space. In mechanics a compressible ﬂuid is said to be

barotropic if the pressure depends only on the density.

The gradient of U with respect to (·, ·)onD

(M) might not exist because

(·, ·) is just a weak Riemannian metric. However, as the following theorem

shows, the gradient exists in the class of H

s−1

-vector ﬁelds on M.

Theorem 16.4 Let F be the vector ﬁeld on D

(M) deﬁned by the equation

= TR



grad p(ρ)



where Ψ

(g)=ρμ. Then for any Y ∈ T

(M), we have d U(Y )=(Y,F

i.e., F =gradU.

Proof. [219]Letg(t) be the ﬂow of Y on M. Diﬀerentiating the equation

μ = g

∗

(ρ(t)μ), we see that

∗

(ρ(t)μ) = 0 or, equivalently,

∂ρ

∂t

+div (ρV )=0

where V = TR

−1

g(t)

. Therefore, we have

16.1 The Lagrangian Formalism of the Hydrodynamics of an Ideal Barotropic Fluid 391

d U(Y )=

U(g(t))



t=0



(ρ) ρdμ



t=0



∂U

∂ρ

(−div (ρY )) ρdμ +



(ρ)(−div (ρY )) ρdμ





grad



d U

dρ





ρdμ +



grad U

,Yρdμ





grad



ρdμ +



d U

dρ

grad ρ, Y ρdμ



grad p, Y ρdμ −



grad p, Y ρdμ



grad p, Y ρdμ =





grad p



ρdμ



grad p(ρ)



,TR



=(F, Y )

. 2

Deﬁnition 16.5. An LHS of an ideal barotropic ﬂuid without external force

is the mechanical system on D

(M) with kinetic energy K(X)=

(X,X)

given

by (5.1) and potential energy U.

The force ﬁeld in such an LHS is −grad U, so that Newton’s equation takes

the form

˙g(t)=−grad U. (16.5)

Using standard properties of the Levi-Civit´a connection, it is not hard to

show that the total energy E = K + U◦π is constant along a trajectory of

the LHS and that every trajectory is an extremal with ﬁxed end-points of

the action functional with the Lagrangian L = K−U◦π.

Let Φ be a vector ﬁeld on M and

Φ the induced right-invariant vector ﬁeld

on D

(M).

Deﬁnition 16.6. An LHS of an ideal barotropic ﬂuid with the external force

Φ is the mechanical system on D

(M) with K as in Deﬁnition 16.5 and the

total force ﬁeld −grad U +

Φ. Newton’s equation for this system is

˙g(t)=−grad U +

Φ. (16.6)

Let us now show how to pass to Euler’s equation for a barotropic ideal

ﬂuid. Let g(t)beatrajectoryoftheLHS(16.6). Consider the curve u(t)=

−1

g(t)

˙g(t)inT

(M). According to the deﬁnitions given in Section 11.2,

the curve u(t) satisﬁes Euler’s equation, which, in this particular case, co-

incides with Euler’s equation of an ideal barotropic ﬂuid. To see this, we

observe that by the deﬁnition of

Φ and Theorem 16.4, the right-hand side

of (16.6) turns into −

grad p + Φ after being right-translated to T

(M).

Then, applying the same arguments as in Section 16.1.1, we obtain that

392 16 Hydrodynamics

u(t, m)=−∇

u(t,m)

u(t, m) −

grad p + Φ where u(t, m)isu(t)considered

as a vector ﬁeld on M . On the other hand, similarly to the proof of Theo-

rem 16.4, we can deduce that

∂ρ

∂t

+div(ρu) = 0. As a result, we obtain the

system of Euler equations



∂u

∂t

+ ∇

u +

grad p = Φ

∂ρ

∂t

+div(ρu)=0.

(16.7)

Remark 16.7. We ﬁnish this section by giving some general references on

the material presented here. The manifold of C

∞

-diﬀeomorphisms was stud-

iedin[210, 211]. In this case the principle of least action in Maupertuis’

form has been proved and the integrals of motion have been analyzed. The

manifold D

(M) was studied in [59]. Note that, in general, it is harder to

work with D

(M) than with D

∞

(M), yet the existence theorems have been

proved only for D

(M). In [59], an LHS of an ideal barotropic ﬂuid was re-

garded as a system with a strong constraint force given by a potential having

a minimum on the manifold D

(M) of volume preserving diﬀeomorphisms.

The latter group is the conﬁguration space for an ideal incompressible ﬂuid.

It has been shown that a trajectory on D

(M) approaches the submanifold

(M) as the parameters of the system go to inﬁnity, so that the ﬂuid be-

comes incompressible.

16.2 Lagrangian Hydrodynamical Systems of an Ideal

Incompressible Fluid

The LHS of an ideal incompressible ﬂuid is described as the LHS of dif-

fuse matter subjected to a constraint in the sense of Section 11.6.This

constraint is described as follows. Both in the case of a ﬁnite-dimensional

manifold M without boundary and with boundary introduce the notation

β = T

(M) ⊂ T

(M) and translate this subspace to tangent spaces at

all points of D

(M). As a result we obtain the distribution

β, a sub-bundle

of T D

(M) that plays the role of a constraint. This constraint is holonomic,

its integral manifold passing through e is D

(M) and so we shall generally

restrict the constraint equations to T

(M). However, in our considerations

below it is sometimes more convenient to deal with

β as a constraint rather

than restricting ourselves to D

(M).

Note that in this case the covariant derivatives



∇ =

P ◦

∇ and

P ◦

introduced by formulae (5.8), where the operator

P is deﬁned by

formula (5.6), coincide with the reduced covariant derivatives by means of

Section 11.1.

Let F be an H

-smooth vector force ﬁeld on M, i.e., a vector in T

(M).

Construct the right-invariant ﬁeld

F on D

(M)(

= F ) and consider the

Newton law on D

(M) of the obtained system with constraint in the form

16.2 Lagrangian Hydrodynamical Systems of an Ideal Incompressible Fluid 393

˙g(t)=

P ◦

F. (16.8)

Remark 16.8.

P ◦

F is a right-invariant vector ﬁeld on D

(M) such that

(

P ◦

F )

= P ◦ F is the divergence-free (and tangent to the boundary if M

is with boundary) component of F in the Hodge decomposition (5.7)((5.10),

respectively). For this reason, without loss of generality, we may consider F as

divergence-free (and tangent to the boundary, respectively), i.e., as belonging

to T

(M).

Since, as mentioned above,

β is integrable, following the general theory

we can restrict the system to the integral manifold D

(M) that yields a

mechanical system without constraint on that integral manifold.

Deﬁnition 16.9. A Lagrangian hydrodynamical system of an ideal incom-

pressible ﬂuid without external forces is a mechanical system with conﬁgura-

tion space D

(M) and kinetic energy K(X)=

(X, X), X ∈ T D

(M)where

(·, ·) is a weak Riemannian metric (5.1), and so Newton’s law takes the form

˙g(t)=0, (16.9)

where

is as deﬁned in (5.8).

Note that

is the covariant derivative of the Levi-Civit´a connection of

the metric (·, ·)onD

(M), as follows from general results of Riemannian

geometry. Thus the trajectories of the LHS (16.9) are the geodesics of that

metric (i.e., in particular, they are extremals with ﬁxed ends for the action

functional with Lagrangian L = K).

Let F ∈ T

(M)and

F be the corresponding right-invariant vector ﬁeld

on D

(M).

Deﬁnition 16.10. A Lagrangian hydrodynamical system of ideal incompress-

ible ﬂuid with external force F is a mechanical system with conﬁguration

space and kinetic energy as in Deﬁnition 16.9, with external force

F and

hence with Newton’s law in the form

˙g(t)=

F. (16.10)

Taking into account Remark 16.8, one can easily see that we do not lose

generality by choosing the external forces from T

(M).

Recall that the geodesics of the Levi-Civit´a connection of the metric (·, ·)

on D

(M) are described by the geodesic spray S (see Theorem 5.9).

Theorem 16.11 For every X ∈ T

(M) there exists a unique solution g(t)

of equation (16.9) with ˙g(0) = X that is well-deﬁned on some suﬃciently

small interval t ∈ [0,ε).

394 16 Hydrodynamics

Indeed, the solutions of (16.9), and only these solutions, are described as

curves πγ(t)whereπ : T D

(M) →D

(M) is the natural projection and

γ(t) is an integral curve of the geodesic spray S on T D

(M). Such a curve

γ(t) locally exists and is unique for every initial condition X ∈ T D

(M)(in

particular, X ∈ T

(M)) since S is smooth.

Remark 16.12. Since S is right-invariant, the choice of initial condition X ∈

(M)) does not restrict generality. Indeed, by a right shift the initial point

of a trajectory can be translated to e.

Theorem 16.13 Let F be a divergence-free (and tangent to the boundary if

M is with boundary) H

-vector ﬁeld on M where l>

+2.Ifl ≥ s>

+1,

for every X ∈ T

(M) the solution g(t) of (16.10) with g(0) = e and

˙g(0) = X, exists for t>0 small enough, and is unique.

Proof. F generates on T D

(M) the second order diﬀerential equation (special

vector ﬁeld) S +(

F )

(where (

F )

is the vertical lift of

F ). Its integral curves

are sent by the mapping π to the solutions of (16.10). The ﬁeld S +

T D

(M)isatleastC

-smooth (see Theorem 5.4) which yields the local

existence and uniqueness of a solution. If s = l>

+ 2, the existence and

uniqueness of a solution on T D

(M) follow from its existence and uniqueness

on T D

s−1

(M) and from the smooth dependence of the ﬂows on TM on the

initial data (cf. Theorem 5.5). 

Note the following hydrodynamical interpretation of Theorem 5.10:the

ﬂow of an ideal incompressible ﬂuid without external forces can realize every

conﬁguration of the ﬂuid that is suﬃciently close to the initial conﬁguration.

Now we can derive the Euler equation in “space coordinates” (see Sec-

tion 11.2) corresponding to the LHS (16.9)or(16.10). Let g(t) be a solution

of (16.10) and consider u(t)=R

−1

◦ ˙g(t) ∈ T

(M). As in the derivation

of the Euler equation (16.7), using the deﬁnition of

, one can easily

show that u(t)inT

(M) satisﬁes the equation P(

∂u

∂t

+ ∇

u)=F .Since

u ∈ T

(M), we have P

∂u

∂t

∂u

∂t

. Hence, since P(∇

u)=∇

u +gradp (see

(5.7)and(5.10)), we obtain the classical Euler equation of the motion of an

ideal incompressible ﬂuid:

∂u

∂t

+ ∇

u +gradp = F. (16.11)

Remark 16.14. Usually one ﬁnds the coeﬃcient

before grad p in the Euler

equation, where ρ is the ﬂuid density. Since the ﬂuid is incompressible, its

density is constant and we can conveniently adopt a system of units in which

ρ =1.

Remark 16.15. The condition div u = 0 (and the condition that u is tangent

to the boundary if M is with boundary) is also incorporated into the Euler

equation. We emphasize that this condition is equivalent to u ∈ T

(M).

16.3 The Regularity Theorem and a Review of Results on the Existence of Solutions 395

Remark 16.16. When we prove the existence and uniqueness of solutions

in Theorems 16.11 and 16.13 in the framework of Lagrangian formalism,

we use the fact that the ﬁeld S on TD

(M) is smooth and right-invariant.

When we pass to the Euler description, i.e., to the Euler equation (16.11)in

(M), we lose information about S. We emphasize that the Euler equation

is a partial diﬀerential equation that is well-deﬁned only on the everywhere

dense subset T

s+1

(M)ofT

(M). The proof of the existence of solutions

for the Euler equation (16.11), which is independent of Theorems 16.11 and

16.13, is rather complicated.

The Newton Law



˙g(t)=

F (t, g(t), ˙g(t)) (16.12)

with non-right-invariant vector force ﬁeld

F (t, g, Y ), Y ∈ T

) describes

ﬂuid motion under a force that depends on the ﬂuid conﬁguration and on

velocity. Equation (16.12) can obviously be reduced to the ﬁrst order equation

on T D

) with special vector ﬁeld S+

(t, g, Y )

|(g,Y )

. Recall that S is C

∞

smooth. Hence, if

F (t, g, Y ) is smooth enough, (16.12) has a solution g(t)of

the Cauchy problem g(0) = e,˙g(0) = X

∈ T

), well-deﬁned on a

suﬃciently small time interval. Denote by u(t) the curve in T

) (i.e., a

divergence-free vector ﬁeld on T

) obtained by right translations of vectors

˙g(t), i.e., u(t)= ˙g(t) ◦ g

−1

(t)=TR

−1

g(t)

˙g(t). Applying the above arguments

one can easily see that u(t) satisﬁes the Euler equation

∂

∂t

u +(u ·∇)u − gradp = TR

−1

g(t)

F (t, g(t),u(t, g(t))). (16.13)

16.3 The Regularity Theorem and a Review of Results

on the Existence of Solutions

As we have shown, a trajectory of an LHS of an ideal incompressible ﬂuid

exists locally on D

(M) provided that s>n/2 + 1. (See Theorem 16.11 and,

for a system with external force, Theorem 16.13.) Passing from the LHS to

(16.11), we obtain the local existence of solutions of Euler’s equation. The

solutions belong to H

with s>n/2 + 1, and so are smooth in the stan-

dard sense. The existence of solutions on the interval (−∞, ∞)hassofar

been proved only on two-dimensional manifolds. For various two-dimensional

problems, results of this kind were obtained in [157, 230, 236]. In higher

dimensions, proving global existence is an important and still unsolved prob-

lem. This diﬀerence between hydrodynamics on two- and three-dimensional

manifolds has its roots in the fact that the geometric properties of the group

of volume preserving diﬀeomorphisms D

(M) change drastically as we pass