ARNOLD'S IDEAS IN FLUID DYNAMICS
AND THEIR FURTHER DEVELOPMENT
A.Shnirelman
Concordia University, Montreal & IAS, Princeton
Edinburgh, October 3, 2011
1. ARNOLD'S PRESENTATION OF BASICS OF THE FLUID
DYNAMICS
Consider a Lagrangian system whose configuration space is a Lie group G
with a unit element e . Suppose the kinetic energy is defined by a rightinvariant Riemannian metric whose restriction on the Lie algebra H=T eG
1
1
is a quadratic form E(u)= 〈 u,u 〉= ( Au,u) where A:H →H * is the
2
2
inertia operator. For any trajectory (geodesic) g(t )∈G the velocity
−1
u=g * ġ(t )∈H . If we define the bilinear operator B:H ×H →H by
〈[u ,v],w 〉=〈 B(w ,u),v 〉 for all u,v,w∈H , then the Euler equation will
have the form
u̇=B(u ,u) .
The dual space H * is foliated into the orbits of coadjoint action of G ; each
orbit O bears a canonic symplectic structure ΩO(ξ,η) , and for any
geodesic g(t ) with the velocity u(t )=R g (t)∗ ġ(t )∈H , the momentum
m(t)=Au(t) on each orbit O is a solution of the Hamiltonian system with
the Hamiltonian E=( A−1 m,m) . Hence, the critical points of the energy on
each orbit O are steady solutions. The surfaces S=A−1O∈H , proimages
of the orbits, are invariant for the Euler equations, and the critical points of
the energy E on S are steady solutions of the Euler equations, which
correspond to geodesics which are one-parameter subgroups of G or their
right shifts.
−1
If the energy E restricted to S has a local
maximum or minimum at u∈S , the steady
solution u appears stable, for the level
surfaces of E encircle u ; however, this
is obvious only in a finite-dimensional case.
For example, in the case of the group SO(3)
the surfaces S are ellipsoids; the level lines of
the energy are shown here:
The critical points A and C correspond to
stable permanent rotations, while C is unstable.
In fact, for the rigid body the metric is not right- but left-invariant. We can
either develop a similar theory for left-invariant metric (which is exactly the
same), or introduce another system.
A “pseudo-rigid” body is an example of a system
on SO(3) with a right-invariant metric. The
system consists of a bundle of massless rods
rigidly connected to each other and rotating around
a fixed point O . A small point mass is moving
along each rod; their angular density ρ(ω) is
fixed in the frame of the rods. Each mass is forced
to remain on a fixed surface S . If S is a sphere
with the center O , then for any function ρ(ω)
our system is the same as a usual solid body with a
fixed point O (the metric on SO(3) is left-invariant) . If, alternatively,
ρ(ω)=const , and S is arbitrary convex surface, then the metric is rightinvariant. For the generic ρ and S the metric is neither right nor left
invariant, and the system is not integrable.
The most interesting case is an ideal incompressible fluid. Let M ⊂R n be a
bounded domain, or a compact Riemannian manifold.
In what follows we consider only the case n=2 .
Consider the motion of ideal incompressible fluid in M . The configuration
space is the group D=SDiff (M ) of volume preserving diffeomorphisms.
The Lie algebra is the space V of divergence-free vector fields u( x) in
−1
M tangent to ∂M . If ξ t is a geodesic on D , then u( x,t )=ξ̇ t∘ξt ( x) is
the Eulerian velocity field.
The kinetic energy
1
1
E=∫ |u( x)|2 dx= ||u||2L .
2
M 2
2
The Euler equations are
∂u
+∇u⋅u+∇ p=0;
∂t
∇⋅u=0;
u n|∂M =0 .
∂u1 ∂u 2
−
and
∂ x2 ∂ x1
−1
satisfies ω( x ,t )=ω(ξ t ( x),0) , i.e. ω is transported by the flow.
The vorticity ω( x ,t)=curl u is defined (for n=2 ) as ω=
Two fields u,v∈V are called isorotated if there exists a difeomorphism
η∈D such that curl u( x)=curl v(η( x)) . The invariant surface S consists
of isorotated fields.
For any field u∈V there exists unique stream function ψ( x) such that
∂ψ
∂ψ
u1=
, u 2=−
∂ x2
∂x 1 ; ψ=const on each component of ∂M , and
ω=Δψ.
If u( x) is a steady solution, then ω=const along any flow line
ψ=const; in this case ∇ ψ and ∇ ω=∇Δψ are proportional.
A field u∈S⊂V is a steady solution if it is a critical point of E on S .
The second variation of E on S at the point u is given by the quadratic
form
δ 2 E(ϕ)=∫ (∇ ϕ)2+
M
∇ψ
(Δϕ)2 dx.
∇Δψ
Solution u is called Arnold stable if this form is either positive or negative
definite.
Examples of Arnold stable flows - parallel flows in a periodic strip (shown
are velocity profiles):
2. DIFFICULTIES OF THE ARNOLD'S APPROACH.
1. The group D is infinite-dimensional; its topology is usually stronger than
topology defined by the Riemannian metric. Hence the existence and
uniqueness of geodesics are not certain.
2. The surfaces S may be nonsmooth, and the partition of H into these
surfaces may be locally nontrivial.
3. It is unclear whether the energy functional E attains a maximum or a
minimum on a given orbit.
3. GROUP D AS A BANACH MANIFOLD
The group D can be endowed with a structure of a Banach manifold
modelled on a Banach functional space X (where X may be a Sobolev
n
space H s , s> +1 , or a Hölder space C k ,α , k≥1,0<α<1 , or some other
2
function space). Correspondingly the group D is denoted DX , and its Lie
algebra by VX . The Riemannian metric defines an L 2 -topology in D
which is weaker than the topology defined by X ; hence the name “weakly
Riemannian metric”.
Theorem. (Lichtenstein, Giunter, ...) (1) For any initial velocity u0∈X ,
where X is one of the above spaces, there exists T>0 and a unique
solution u( x ,t)∈X of the Euler equations with the initial velocity u0
defined for ∣t∣<T ;
(2) If n=2 , T=∞ (i.e. there exists a global in time solution). (Volibner,
Yudovich, Kato, ...)
Hence two basic problems in the MFD:
(1) If dimM ≥3 , what happens with the solution as t grows? Does solution
exist for infinite time, or there occurs some sort of a catastrophe? (This is the
celebrated Singularity Problem.)
(2) If dimM =2 , how does the solution behave as t →∞ ? What is the longtime asymptotics?
I will discuss the second problem; so, from now on dimM =2 , and all
solutions exist for infinite time.
4. MIXING OPERATORS
In 2-d domain M the vorticity ω is transported by the flow; it is distorted,
and effectively mixed. The further is a formalization of this property.
Consider the class K of operators in L 2(M ,R) having the form
Kf (x)=∫ K(x, y) f ( y)dy
M
where the kernel K( x , y) has the following properties:
(i) K( x , y)≥0 (i.e. K is a nennegative measure in M ×M );
(ii)
(iii)
∫M K( x, y )dx≡1 ;
∫M K( x, y )dy≡1 ;
K is a weakly compact convex semigroup of contractive operators in
2
2
L ( M ) . It defines a partial order in L in the following way:
f ≪g if f =Kg for some K∈K . Now, we write u≪v for two vector
fields u,v∈V if curl u≪curl v .
Now suppose u0∈X is a given “initial” vector field. Let us denote
Ωu ={u
0
∣
u≪u0, ∥u∥L =∥u 0∥L
2
2
}
By the Zorn Lemma (which is applicable in this case) there exists a minimal
element v∈Ωu ; all such elements are called minimal flows.
0
Theorem. Any minimal flow v is a steady-state solution of the Euler
equations. If ψ( x) is a stream function of v , and ω=Δψ is vorticity,
then ω=F( ψ) for some monotone function F . Thus, any minimal flow is
Arnold stable (in fact, these two classes of flows practically coincide).
Minimal flow is called energy excessive if F '≤0 ; it is called
energy deficient if F '≥0 . For energy excessive (energy deficient) flow
every mixing of its vorticity results in decreasing (increasing) its kinetic
energy. The third class contains a single flow such that ω=const ; no mixing
of vorticity can change it.
Thus, Arnold stable flows are, in a sense, the most “degenerate” ones, such
that any further mixing of vorticity changes the energy.
5. THE LONG-TIME BEHAVIOUR OF THE FLOW; THE MIXING
OF VORTICITY
The vorticity is transported by the flow; it is distorted and effectively mixed.
There is no visible obstacle to this mixing but the energy conservation. Hence
the first hypothesis: the mixing of vorticity by its own flow proceeds until
any further mixing changes the energy, i.e. any (or almost any) flow tends to
a minimal flow as t →∞ . However, numerical simulations show that this
conjecture is WRONG.
6. THE EVIDENCE OF IRREVERSIBILITY: LIAPUNOV FUNCTION
Anyway, the flow behaviour looks irreversible, while the Euler equations are
time reversible (by the change of variables t →−t , u→−u , p→ p ). This
situation is similar to the classical statistical mechanics, and its understanding
meets similar difficulties. However, for the fluid there exist Liapunov
functions which may help to quantify the irriversibility.
Consider a dynamical system in a phase space X defined by equation
dx
= f ( x) . A function L( x) defined and continuous in the whole space
dt
dL( x(t ))
≥0 along any trajectory
X is called a Liapunov function if
dt
x(t ) .
Not every system has a Liapunov function; for example, there is no
Liapunov function for a harmonic oscillator (because all the trajectories are
closed). In the contrary, for a particle moving freely in R n there is a
dL ˙2
= x ).
Liapunov function, namely L( x, ẋ)=x⋅ẋ (for
dt
The first Liapunov function for the fluid was found by V. Yudovich (1973).
Suppose the flow domain M ⊂R 2 is such that
∂M contains a piece of the x -axis. Consider a
fluid particle π which was initially on the
straight piece of ∂M (and hence remains there
all the time). Then
d
2 2
.
( ωω x ω y )|π = ( ω ω x )|π ≥ 0
dt
Thus, L=ωω x ω y |π is a Liapunov function.
Other example of Liapunov function (Shnirelman, 1997) has a microlocal
nature and will not be discussed here. In both examples the Liapunov
function measures the intensity of (weak) singularities of the flow, and shows
their tendency to become sharper in time. It is desirable to find a more
“physical” Liapunov function.
7. GENERALIZED MINIMAL FLOW.
Let u0( x) be an initial velocity field, and u( x,t) the solution of the Euler
equations, u( x,0)=u0( x). Then O(u 0 )=∪t∈R{u(⋅,t)} is called the orbit of
u0 . Let us denote by O(u 0 ) the closure of the orbit in L 2 .
Definition. A flow u(t ) with the initial velocity field u0 is called a
generalized minimal flow (GMF) if for any v∈O(u 0 ) ,
||curlv|| L =||curlu0 ||L .
2
2
8. CONSTRUCTION OF GENERALIZED MINIMAL FLOWS BY
PSEUDOEVOLUTION
Let u0 be any velocity field. If it is not a GMF, we can find a field
u1∈O(u0 ) such that ||curlu1|| L <||curlu0|| L ; otherwise we set u1=u 0 . If
u 1 is still not a GMF, we find u2∈O(u 1) such that ||curlu2|| L <||curlu 1|| L ;
otherwise define u2=u1 . Proceeding this way, we construct a sequence
u0 ,u1 ,...,u k ,... defined for all countable ordinals k (for the limit ordinals
k , we define u k as a limit point of u p for any p s →k ). Let
a k=||curlu k || L . Now we use the following
2
2
2
2
s
2
Lemma. For any nonincreasing sequence {a k } defined for all countable
ordinals k there exists a countable ordinal k 0 such that a k≡a k for all
k≥k 0 .
0
Hence, uk is, by definition, a GMF.
0
If we could replace the usual flow evolution by the above transfinite process
called pseudoevolution, this would mean that the set N of GMFs is an
attractor in the space V of velocity fields.
Conjectures:
(1) The set N is an attractor for the Euler equations in the ordinary sense;
(2) Generalized minimal flows are either stationary or quasiperiodic with at
most countable set of periods (this is hinted at by numerical results);
(3) Strictly speaking, we have to prove that the set N is a proper subset of
V , i.e. there exist flows which are not GMF. (This is analogous to the
Landau damping recently proved for the Vlasov-Poisson equation by Villani.)
9. LOCAL REGULARITY OF PARTITION INTO ISOVORTICAL
SURFACES
This question was addressed recently by V.Sverak and A.Choffrut (2010).
Consider steady flows in an annular domain; for any such flow consider the
distribution function of vorticity
λ(s)=mes{x∈M | ω( x)≤s}
Theorem. (Sverak, Choffrut) Steady solutions close to a fixed Arnold stable
one are in a smooth 1-1 correspondence with distribution functions λ(s) .
The proof is difficult and is based on the Nash-Schwarz implicit function
theorem.
10. THE STRUCTURE OF THE EXPONENTIAL MAP
Consider the exponential map exp:T Id D→D defined as follows: for any
u 0∈T Id D let ξt ∈D be the geodesic such that ξ 0=Id , ξ˙0=u0 . Then
def
expu0 = ξ 1 . This map is smooth in the X -topology (Ebin and Marsden,
1970).
Theorem (Ebin, Misiolek, Preston, 2008). If n=2 , the map exp is a
smooth Fredholm map of index zero.
This result is important because of the following
Theorem (Misiolek, 1993). There exists a geodesic ξ t∈D such that
ξ 0=Id and for some τ>0 , the point ξ τ is conjugate to ξ 0 .
Thus, the singularities of the exp map are “finite-dimensional”.
There is an alternative description of singularities of the exponential map.
Definition. (1)Suppose X and Y are real Banach spaces, and F : X →Y
is a continuous map. Suppose X is foliated into parallel planes
X α , α∈R k , of finite codimension k . A map F is called a ruled map, if
F α=F | X is an affine map F α :X α →Y depending continuously on α .
α
(2) A map F is called Fredholm ruled (FR) if the map F α is a Fredholm
map of index zero for all α∈R k ;
(3) A map F : X →Y is called Fredholm quasiruled (FQR) if it can be
uniformly approximated by ruled maps in any bounded domain Ω⊂X ;
Quasiruled manifolds are defined in an obvious way; they form an interesting
and important category.
Theorem. (Sh. 2006)(1)The group D with the X -topology is a quasiruled
manifold;
(2) The exponential map exp:T Id D→D is Fredholm quasiruled.