Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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xðtÞ¼aðp
; x
Þt þ bðp
; x
Þþy
þ
ðtÞ
aðp
; x
Þ 6¼ 0; y
þ
ðtÞ!0;
_
y
þ
ðtÞ!0
as t !þ1
½4
furthermore, the set D of all ( p
, x
) 2 R
2d
, p
6¼ 0,
for which [4] holds for fixed v, is an open subset of
R
2d
and Mes(R
2d
nD) = 0.
We say that a, b arising in [4] (and defined on D)are
the scattering data for eqn [1]. In addition, the scattering
data a, b at fixed energy E > 0meansa, b on {(p
, x
) 2
Djp
2
=2 = E}. Roughly speaking, for a particle moving
according to [1], the functions a, b relate the free motion
at time t !1withthefreemotionattimet !þ1.
Note that
aðp
; x
þ t
0
p
Þ¼aðp
; x
Þ
bðp
; x
þ t
0
p
Þ¼bðp
; x
Þþt
0
aðp
; x
Þ
ðp
; x
Þ2D; t
0
2 R
½5
Formula [5] imply that a, b on D are uniquely
determined by a, b on {(p
, x
) 2Djp
x
= 0},
where p
x
is the scalar product of p
and x
.
If v(x) 0, then a(p
, x
) = p
, b(p
, x
) =
x
,(p
, x
) 2 R
d
, p
6¼ 0. Therefore, it is convenient
to use for a, b the following representati on:
aðp
; x
Þ¼p
þ a
sc
ðp
; x
Þ
bðp
; x
Þ¼x
þ b
sc
ðp
; x
Þ; ðp
; x
Þ2D ½6
where the subscript sc is an abbreviation of the word
‘‘scattering.’’
The direct scattering problem for eqn [1], under
the assumptions [2], consists in the following: given
v, find a, b.
The inverse-scattering problem for eqn [1], under
the assumptions [2], consists in the following: given
a, b (or some partial information about a, b), find v.
In the present article, we discuss, mainly, the
aforementioned inverse-scattering problem.
Abel’s Result of 1826
Consider the Newton equation [1] in dimension
d = 1 for x 2] 1, x
1
], x
1
> 0, where
v 2 C
2
ð 1; x
1
; RÞ
vðxÞ¼0 for x < 0
dvðxÞ
dx
> 0 for 0 < x < x
1
½7
Under the assumptions [7], for any p
> 0, where
E = p
2
=2 < v(x
1
), eqn [1] has a unique solution
x 2 C
2
(R,] 1, x
1
]) such that
xðtÞ¼p
t for t 0 ½8
in addition,
xðtÞ¼p
t þ bðp
Þ as t !þ1 ½9
Let
TðEÞ¼
bð
ffiffiffiffiffiffi
2E
p
Þ
ffiffiffiffiffiffi
2E
p
; 0 < E < vðx
1
Þ;
ffiffiffiffiffiffi
2E
p
> 0 ½10
(T(E) is the time during which a particle starting
at x = 0 with the impulse p
=
ffiffiffiffiffiffi
2E
p
returns to
x = 0).
Let x( v), v 2 [0, v(x
1
)], be the inverse function to
v(x), x 2 [0, x
1
]. Then (under the assumptions [7]),
TðEÞ¼
ffiffiffi
2
p
Z
E
0
ðE vÞ
1=2
dxðvÞ
dv
dv
0 < E < vðx
1
Þ½11
xðvÞ¼
1
ffiffiffi
2
p
Z
v
0
ðv EÞ
1=2
TðEÞdE
0 < v < vðx
1
Þ½12
Actually, the formulas [11], [12] re lating the travel
time T and the potential v are the results from
Abel (1826) (see also Keller (1976) for a discus-
sion of this result). Formula [11] is a result on
direct scattering, whereas [12] is a result on
inverse scattering. In addition, if T(E), 0 < E <
v(x
1
), is given, then [11] is the Abel integral
equation for x(v), 0 < v < v(x
1
), a nd [12] solves
this equation.
Concerning further results on inverse scattering
for the one-dimensional Newton equation, see Keller
(1976) and Astaburuaga et al. (1991). Note that for
the one-dimensional case the scattering data a, b do
not in general determine v uniquely.
The Abel integral equation and the Abel
formula solving this equation were used also, in
particular, by Firsov (1953) and Keller et al.
(1956), where inverse scattering was considered
for the three-dimensional Newton equation at
fixed energy for the case of spherically symmetric
monotonous decreasing potential in jxj.
Note also that the Abel method for solving the
integral equation [11] was used by Radon (1917)
for finding the inversion formula for the Radon
transformation. In the next section, we reduce the
inverse-scattering problem for the Newton equa-
tion [1] in di mension d 2, under the assumptions
[2], to the inversion problem for the X-ray
transformation (i.e., the Radon transformation
along straight lines).
Inverse Problem in Classical Mechanics 157
Inverse Scattering for the
Multidimensional Newton Equation
Consider
TS
d1
¼fð; xÞ2S
d1
R
d
jx ¼ 0g½13
Consider the X-ray transformation P defined by the
formula
Pf ð; xÞ¼
Z
R
f ðt þ xÞdt; ð; xÞ2TS
d1
½14
where
f 2 CðR
d
; R
m
Þ
f ðxÞ¼Oðjxj
Þ as jxj!1for some >1 ½ 15
Consider the functions a
sc
, b
sc
of [6]
Theorem 1 (Novikov 1999). For the Newton
equation [1], under the assumptions [2], the follow-
ing formulas hold:
PFð; xÞ¼ lim
s!þ1
sa
sc
ðs; xÞ; ð; xÞ2TS
d1
½16
Pvð;xÞ¼ lim
s!þ1
s
2
b
sc
ðs;xÞ; ð; xÞ2TS
d1
½17
in addition,
jPFð; xÞsa
sc
ðs; xÞj
d
3
c
2
2
2þ4
ð 1Þð1 þjxj=
ffiffiffi
2
p
Þ
21
s
3
ðs=
ffiffiffi
2
p
1Þ
4
½18
jPvð; xÞs
2
b
sc
ðs; xÞj
d
3
c
2
2
2þ4
ð 1Þ
2
ð1 þjxj=
ffiffiffi
2
p
Þ
22
s
4
ðs=
ffiffiffi
2
p
1Þ
5
½19
for (, x) 2 TS
d1
, s z(d, c, , jxj), where b
sc
is the
scalar product of and b
sc
, z is the root of the equation
d
2
c2
þ2
ð 1Þð1 þjxj=
ffiffiffi
2
p
Þ
1
z
2
ðz=
ffiffiffi
2
p
1Þ
3
¼ 1
z 2
ffiffiffi
2
p
; þ1½ ½20
c = max (c
1
, c
2
)(and , c
1
, c
2
are the constants of [2]).
Theorem 1 gives a method for finding PF and Pv
from a
sc
and b
sc
at high energies. It has been proved in
Novikov (1999) by means of analysis of the following
nonlinear integral equation for the function y
of [3]:
y
ðtÞ¼A
p
; x
ðy
ÞðtÞ
where
A
p
; x
ðuÞðtÞ¼
Z
t
1
Z
1
Fðp
s þ x
þ uðsÞÞds d
p
6¼ 0
In dimension d 2, Theorem 1 and methods for
the reconstruction of f from Pf (Gelfand et al. 1980,
Natterer 1986, Novikov 1999) give a method for the
reconstruction of F and v from the scattering data a,
b at high energies. Note that for d = 1 Theorem 1
is valid but f cannot be uniquely reconstructed
from Pf.
Theorem 1 is an analog of the Born formula for
the Schro¨ dinger equation at high energies (see, e.g.,
Faddeev (1956), Enss and Weber (1995),and
Novikov (1998) as regards this Born formula and
its variations). On the other hand, Theorem 1 was
preceded by a result of Gerver and Nadirashvili
(1983) on the high-energy asymptotics for the travel
time between boundary points for the Newton
equation in a bounded strictly convex domain with
smooth boundary. There is a considerable similarity
between this result and Theorem 1.
We continue our review on inverse scattering for
the multidimensional Newton equation, and make
the following well-known observation.
Observation 1 Suppose that v(x) > E > 0 for x 2U,
where U is a compact subset of R
d
. Then the scattering
data a, b for energies smaller than or equal to E contain
no information about v(x) for x 2U.
In addition to Theorem 1 and Observation 1, one
has the following conjecture.
Conjecture 1 (Novikov 1999). Suppose that v
satisfies [2], d 2, and the energy E is sufficiently
large, E > E(v). Then the scattering data a, bat
fixed energy E uniquely determine v.
Gerver and Nadirashvili (1983) proved a result
similar to Conjecture 1 for the case of the Newton
equation in a bounded strictly convex domain G
with smooth boundary. Their proof of this result
contains no reconstruction method but does contain
a stability estimate. It is based on the Maupertuis
principle and the results of Muhometov and Roma-
nov (1978), Beylkin (1979), and Bernstein and
Gerver (1980). For the case v 2 C
2
(R
d
, R), supp v
G (where G has the properties mentioned above),
in Novikov (1999) a connection between the
boundary-value data of Gerver and Nadirashvili
(1983) and the scattering data a, b is given and it is
shown that for d 2 the scattering data a, b and the
domain G uniquely determine v at fixed sufficiently
large energy E > E(v, G).
For more information concerning results men-
tioned above, see Novikov (1999) and Gerver and
Nadirashvili (1983). One can see from the review
of this section that very few results on inverse
scattering for the multidimensional Newton equa-
tion are given in the literature, at present. It should
158 Inverse Problem in Classical Mechanics
be remarked that the inverse-scattering theor y in
multidimensions is much more developed for the
Schro¨ dinger equation than for the Newton
equation.
Inverse Scattering for the Schro¨ dinger
Equation in Multidimensions
The inverse-scattering theory for the multidimen-
sional Schro¨ dinger equation has been developed by
many authors (see, e.g., surveys given in Grinevich
(2000) and Novikov (2001)).
Quantum-mechanical analogs of Theorem 1
appear, for example, in Faddeev (1956), Enss
and Weder (1995), Novikov (1998) (see also
references therein). Similarly, the quantum-mechan-
ical analogs of Conjecture 1 have been proved, for
example, in Novikov (1992, 1994) and Grinevich
and Novikov (1995) (see also references therein). On
the other hand, as a rule, classical-mechanical analogs
of results of the works on inverse Schro¨ dinger
scattering in multidimensions are unknown. This
leads to many open problems. For the one-dimen-
sional case some results on finding classical limits of
results on inverse Schro¨ dinger scattering are given in
Lax and Levermore (1983) and Bogdanov (1985).
Note that inverse scattering for the two-dimensional
Schro¨ dinger equati on at fixed energy (see Novikov
(1992), Grinevich and Novikov (1995),and
Grinevich (2000) and references therein) has con-
siderable similarity with inverse scattering for the
one-dimensional Schro¨ dinger equation. Therefore,
an interesting open problem consists in extending
the aforementioned study of Lax and Levermore
(1983) and Bogdanov (1985) to the case of inverse
scattering for the two-dimensional Schro¨ dinger
equation at fixed energy . Perhaps, in this way one
can find proper two-di mensional analogs of the Abel
formulas [11] and [12].
Further Reading
Abel NH (1826) Auflo¨ sung einer mechanis chen Aufgabe. J. Reine
Angew. Math. 1: 153–157 (German) (French translation:
Re´solution d’un proble`me de me´canique. In: Sylow L and
Lie S (eds.) Oeuvres comple` tes de Niels Henrik Abel, vol. 1,
pp. 97–101. Grondahl: Christiania (Oslo), (1881)).
Astaburuaga MA, Fernandez C, and Corte´s VH (1991) The
direct and inverse problem in Newtonian scattering. Proceed-
ings of the Royal Society of Edinburgh Section A 118:
119–131.
Bernstein IN and Gerver ML (1980) A condition of distinguish-
ability of metrics by hodographs. Computational Seismology
13: 50–73 (Russian).
Beylkin G (1979) Stability and uniqueness of the solution of the
inverse kinematic problem of seismology in higher dimensions.
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.
(LOMI) 84: 3–6 (Russian) (English translation Journal of
Soviet Mathematics 21: 251–254 (1983)).
Bogdanov IV (1985) Classical limit of the quantum inverse
scattering problem. Teor. Mat. Fiz. 65(1): 35–43 (Russian)
(English translation. Theoretical and Mathematical Physics
65: 992–998 (1985)).
Enss V and Weder R (1995) Inverse potential scattering: A
geometrical approach. In: Feldman J, Froese R, and Rosen L
(eds.) Mathematical Quantum Theory II: Schro¨ dinger Opera-
tors, CRM Proc. Lecture Notes, vol. 8, pp. 151–162.
Providence, RI: American Mathematical Society.
Faddeev LD (1956) Uniqueness of solution of the inverse
scattering problem. Vestnik Leningrad Univ. 11(7): 126–130
(Russian).
Firsov OB (1953) Determination of the force acting between
atoms via differential effective elastic cross section. Zh.
Eksper. Teoret. Fiz. 24: 279–283 (Russian).
Gelfand IM, Gindikin SG, and Graev MI (1980) Integral
geometry in affine and projective spaces. Itogi Nauki i
Tekhniki. Sov. Prob. Mat. 16: 53–226 (Russian) (English
translation. Journal of Soviet Mathematics 18: 39–167
(1980)).
Gerver ML and Nadirashvili NS (1983) Inverse problem of mechanics
at high energies. Computational Seismology 15: 118–125
(Russian).
Grinevich PG (2000) Scattering transform for the two-dimensional
Schro¨ dinger operator with decreasing at infinity potential at
fixed non-zero energy. Usp. Math. Nauk 55(6): 3–70 (Russian)
(English translation. Russian Mathematical Surveys 55: 1015–
1083).
Grinevich PG and Novikov RG (1995) Transparent potentials at
fixed energy in dimension two. Fixed-energy dispersion
relations for the fast decaying potentials. Communications in
Mathematical Physics 174: 409–446.
Keller JB (1976) Inverse problems. The American Mathematical
Monthly 83: 107–118.
Keller JB, Kay I, and Shmoys J (19 56) Determination of the
potential from scattering data. Physical Review 102:
557–559.
Lax PD and Levermore CD (1983) The small dispersion limit of
the Korteweg–de Vries equation. I–III. Communications in
Pure and Applied Mathematics
36: 253–290, 571–593,
809–830.
Muhometov RG and Romanov VG (1978) On the problem of
finding an isotropic Riemannian metric in an n-dimensional
space. Dokl. Akad. Nauk SSSR 243(1): 41–44 (Russian)
(English translation Soviet Mathematics Doklady. 19:
1330–1333).
Natterer F (1986) The Mathematics of Computerized Tomogra-
phy. Stuttgart: Teubner.
Novikov RG (1992) The inverse scattering problem on a fixed
energy level for the two-dimensional Schro¨ dinger operator.
Journal of Functional Analysis 103: 409–463.
Novikov RG (1994) The inverse scattering problem at fixed
energy for the three-dimensional Schro¨ dinger equation with an
exponentially decreasing potential. Communications in Math-
ematical Physics 161: 569–595.
Novikov RG (1998) On inverse scattering for the N-body
Schro¨ dinger equation. Journal of Functional Analysis 159:
492–536.
Novikov RG (1999) Small angle scattering and X-ray transform
in classical mechanics. Arkiv fo¨r Matematik 37: 141–169.
Inverse Problem in Classical Mechanics 159
Novikov RG (2001) Scattering for the Schro¨ dinger equation in
multidimensions. Non-linear
@-equation, characterization
of scattering data and related results. In: Pike ER and
Sabatier P (eds.) Scattering, ch. 6.2.4. New York: Academic
Press.
Radon J (1917) U
¨
ber die Bestimmung von Funktionen durch ihre
Integralwerte la¨ngs gewisser Mannigfaltigkeiten. Ber. Verh.
Sa¨chs. Akad. Wiss. Leipzig, Math.-Nat. K1 69: 262–267.
Reed M and Simon B (1979) Methods of Modern Mathematical
Physics. III. Scattering Theory. New York: Academic Press.
Inverse Problems in Wave Propagation see Boundary Control Method and Inverse Problems of Wave Propagation
Inviscid Flows
R Robert, Universite
´
Joseph Fourier,
Saint Martin D’He
`
res, France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The equations governing the motion of an ideal
(inviscid) fluid were derived by Euler in 1755. They
were, together with the equation of vibrating strings,
the first partial differential equations introduced in the
field of mathematical physics. While several partial
differential equations, coming from the modeling of
physical phenomena, have had a satisfactory mathe-
matical solution, it is piquant to note that the old Euler
equations remain essentially unsolved. Together with
the Navier–Stokes equations of viscous fluids, the
Euler equations play a central role in the modern
analysis of partial differential equations.
The mathematical difficulties encountered in the
study of Euler equations seem to be deeply linked with
the understanding of turbulence, which remains one of
the great open problems in the field of macroscopic
physics.
The relevance of Euler equations as a model of
fluid flow is rather subtle, and the discussion is far
from closed. On the one hand, Euler equations have
disturbing aspects, which, in their most visible form,
yield paradoxes. On the other hand, the systematic
recourse to some viscosity seems to put a serious
obstacle to a proper understanding of turbulence. In
this article we will try to give some insight into this
issue.
To be rigorous, every fluid has some compressi-
bility, that is to say the density varies with the
pressure. Compressibility gives rise to pressure
waves, which propagate in the fluid with some finite
speed. When the velocity of the fluid particles is
slow relative to the speed of the pressure waves, it is
legitimate to make the approximation that the flow
is incompressible; it is the case for meteorological
flows, for example. Then, there are no more
pressure waves; nevertheless the motion can be
very unstable and intricate (turbulent). Although
very often in physical flows these two features
coexist, following the tradition, we clearly separate
the compressible and incompressible cases.
The Equations of the Perfect Fluid
Until now a rigorous derivation of the fluid
equations from a system of interacting particles
governed by Newton’s laws is not known. Thus,
the mathematical models of fluid motion result
from heuristic considerations.
Let us specify some notations.
The fluid motion is supposed to take place in
some domai n (not necessarily bounded) of the
physical space <
3
.
We shall use the so-called Eulerian description of
the fluid motion: (t, x) denotes the local density of
the fluid at time t and position x, and u(t, x) the
velocity of the fluid particle located at x at time t.
The first equation (conservation equation) expresses
the conservation of mass:
@
@t
þ divðuÞ¼0 ½1
The second equation (momentum equation)
expresses Newton’s law (in the absence of internal
friction):
@u
@t
þðu rÞu
¼rp ½2
where the scalar function p(t, x) is the pressure
inside the fluid, and
ðu rÞu ¼
X
i
u
i
@
i
u
With [1] and [2], we have five scalar unknow n
functions (, u
i
, p) and only four equations. To get a
160 Inviscid Flows
closed set of equations, we need to add a supple-
mentary relationship:
divðuÞ¼0; for the incompressible flows ½3
In the case of compressible flows, eqns [1] and [2] must
be completed by a thermodynamical description of the
fluid, which yields a relationship between , p,the
internal energy, the specific entropy, etc. We will only
consider here the simple case of an isentropic gas
which is modeled by the relationship
p ¼ pðÞ½4
with p() = cp
for a perfect gas (c > 0, >1).
Condition at the Boundary @W of the Domain
In the case of a perfect fluid, we simply have to
write that the velocities of the fluid particles at the
boundary are tangent to the boundary, that is,
u n ¼ 0on@ ½5
where n denotes the unit normal vector to the
boundary (pointing outward).
The Incompressible Perfect Fluid: Main
Properties of Smooth Flows
We shall suppose = 1. Equations [1]–[3] and [5]
then yield the classical Euler syst em:
@u
@t
þðu rÞu ¼rp on
div u ¼ 0; u n ¼ 0on@
½6
The Constants of the Motion
Let us examine the constants of the motion of the
dynamical system defined by [6], that is, the functionals
which are conserved by the motion of the fluid.
First we have the classical constants of motion
associated with the natural symmetries by Noether’s
theorem.
The time translational invariance of the system
implies that the kinetic energy is conserved:
E
c
¼
1
2
Z
u
2
dx
In the case =<
3
, the homogeneity of space implies
the conservation of the impulsion:
Z
u dx
The space isotropy, on the other hand, yields the
conservation of the angular momentum:
Z
x ^ u dx
There is a more hidden constant of the motion,
called helicity, which was discovered in 1961 by
J J Moreau (1961) (see, e.g., Serre (1979)).
Let us define the vorticity of the flow:
! ¼ curl u
then the helicity is
Z
! u dx
Of course, here, we suppose u to be vanishing at
infinity in such a manner t hat the above integrals
make sense.
One may wonder about the existence of other
constants of the motion of the form (first-order
functionals):
Z
Fðx; uðxÞ; ruðxÞÞdx
The answer, due to Serre (1979), is that any
functional of the above form which is conserved by
the flow is a linear function of the energy, the
impulsion, the angular momentum, the helicity plus
a trivial term (i.e., taking the same value for any
field u such that div u = 0).
Beltrami Equation and Kelvin’s Theorem
Another important issue is to know how the vorticity
field evolves in a regular flow. If we apply the operator
curl to the equation [6] in order to eliminate the
pressure term, we get:
@!
@t
þðu rÞ! ð! rÞu ¼ 0 ½7
which is the Beltrami equation.
To exploit the Beltrami equation, we need the
Lagrangian flow ’(t, x), associated with the field u,
which is defined by the differential equation:
@’
@t
ðt; xÞ¼uðt;’ðt; xÞÞ;’ð0; xÞ¼x
Then we can state the following proposition.
Proposition During the smooth motion of an
incompressible perfect fluid, we have:
!ðt;’ðt; xÞÞ ¼ D’ðt; xÞ½!ð0; xÞ; for all t; x
where D’(t, x) denotes the derivative at the point x
(t fixed) of the mapping x !’(t, x).
The first consequence of this result is to point
out the class of irrotational flows, for which
Inviscid Flows 161
!(t, x) = 0. Indeed, if t he vorticity vanishes initi-
ally, it follows from the proposition that it will
vanish for ever.
Another consequence is the behavior of vortex
lines. By definition, a vortex line is any integral
curve of the vorticity field. More precisely, a
vorticity line at time t , C(s) is defined by the
differential equation
dC
ds
ðsÞ¼!ðt; Cð sÞÞ
Now we can check that vortex lines are merely
transported by the flow: if C(s) is a vortex line at
time t = 0, ’(t, C(s)) is a vortex line at time t.
We end this section with the famous Kelvin’s
circulation theorem (1869) (see, e.g., Marchioro and
Pulvirenti (1994)).
Theorem Let L be a closed (oriented) contour drawn
inside the fluid. We suppose that L is transported by
the flow; ’
t
(L) denotes the contour at time t. Then the
circulation of the velocity field u(t, x) along ’
t
(L) is
independent of t.
Stationary Solutions: D’Alembert’s Paradox
Let us focus now on the flow around a bounded
body
, whose complement
c
will be supposed to
be simply connected.
A stationary solution u(x), p(x) satisfies:
ðu rÞu ¼rp
div u ¼ 0; u n ¼ 0on@
But since (u r)u = r(
1
2
u
2
) þ (curl u) ^ u, any
stationary field u(x) satisfying curl u = 0, div u = 0,
u n = 0on@, defines a stationary solution with
associated pressure p =
1
2
u
2
.
We also need to specify a condition at infinity
for the field u. We impose that the velocity is equal
(at infinity) to some constant value U. Since
c
is
simply con nected, the condition curl u = 0 implies
that the flow is potential, that is, there is a scalar
function F(x) such that u = U þrF.
Thus, the determination of an irrotational flow
around an obstacle amounts to solving the following
exterior Neuman problem.
Find F satisfying:
F ¼ 0in
c
@F
@n
¼U n on @
rF ¼ 0 at infinity
This problem is well known and has a unique
solution, which satisfies, at infin ity:
FðxÞ¼Oð 1=jxj
2
ÞrFðxÞ¼Oð1=jxj
3
Þ
Then a classical calculation (integration by parts) gives
the resulting force exerted by the flow on the body:
R ¼
Z
@
pn d ¼
Z
@
1
2
u
2
n d ¼ 0
This property of inviscid potential flows was first
noticed by Jean Le Rond d’Alembert (1717–1783).
Furthermore, d’Alembert performed a series of
experiments to measure the drag on a sphere in a
flowing fluid and he expected that the force would
go to zero as the viscosity of the fluid ap proached
zero. But this was not the case: the drag seemed
to converge toward a nonzero value. Hence, this
property was called d’Alembert’s paradox.
Of course, d’Alembert’s paradox tells us that some-
thing is going wrong: this model of flow around a body
is not physically relevant. But it is not obvious to
identify precisely what is going wrong.
Physics tells us that in a flow around a flying
airplane, the viscous term (as measured by an
dimensionless number called Reynolds number) is
very small. The main effect of the viscosity is then
to alter the limit condition at the boundary of the
body. The relevant boundary condition is no longer
u n = 0, but the purely viscous condition u = 0,
or more realistically a condition of friction type
(turbulent boundary condition).
A common approach is to disqual ify the perfect-
fluid model in arguing that this modification of the
boundary condition has important consequences on
the flow near the body (giving rise to a turbulent
boundary layer, for example).
It seems to us that such a disqualification of the
perfect-fluid model discards prematurely interesting
issues. Indeed, we must notice first that the
stationary solution on which d’Alembert’s reasoning
is based is highly unstable and not acceptable
physically. Thus, a realistic solution would necessa-
rily be either nonstationary or with some vorticity.
On this basis, we can imagine other scenarios to
explain the existence of a resulting force exerted
on the body. For example, we may imagine a
stationary solution with a discontinuous velocity
field (i.e., with a vortex sheet ). The process
conducive to such a stationary solution is called
Prandtl’s scenario (Batchelor 1967). The mathema-
tical proof that Prandtl’s scenario does exist is a
difficult (open) issue, which seems closely related to
the (probable) nonuniqueness of weak solutions of
the Cauchy problem.
162 Inviscid Flows
The Cauchy Problem for the
Incompressible Perfect Fluid
The Case W R
3
In the Cauchy problem, given an initial velocity field
u
0
(x), we want to determine the corresponding
solution u(t, x)of[6] at each time t.
The first significant result on the Cauch y problem
for three-dimensional Euler equations was given by
Kato (1975).
Theorem For u
0
in the Sobolev space H
s
(<
3
), for
s > 5=2, there is T > 0 and a unique classical solution
(of the Cauchy problem) u(t, x) on [0, T] <
3
. u
depends continuously on t in the space H
s
.
By a classical solution we mean that the field
u(t , x) is derivable in terms of the variables t, x and
satisfies the equations in the usual sense.
Here H
S
(<
3
) denotes the Sobolev space of the
fields u, which are square integrable and with spatial
derivatives of order s (in the case where s is an
integer) also square integrable.
Remark These results have been generalized to some
extent during the last few decades, but the following
issues are still open:
1. Do singularities occur at a finite time for such
regular solutions?
2. For a less regular initial datum, do weak solutions
exist (in the sense of distributions)?
The Case W R
2
This case is better understood, the first mathematical
results trace back to Lichtenstein (1925) and Wolibner
(1933); they take a plain form with the famous theorem
of Youdovitch 1963 (see, e.g., Chemin (1995)).
In two dimensions, the vorticity ! = curl u identifies
with a scalar function, and the Beltrami equation
becomes
@!
@t
þ divð ! uÞ¼0 ½8a
curl u ¼ ! ½8b
div u ¼ 0; u n ¼ 0on@ ½8c
This formulation, which appears as a transport
equation [8a] for !, coupled with the elliptic system
[8b]–[8c], which determines u from !, is particularly
convenient.
The constants of motion associated with the usual
symmetries, of course, persist; notice, however, that
the helicity degenerates since, in two dimensions,
! u = 0. But now from [8a] we see that ! is merely
convected by the incompressible velocity field u.We
deduce that, for any continuous function f, the
functional
Z
f ð!ðt; xÞÞdx
is a constant of motion.
Thus, a specific feature of the two-dimensional case
is to introduce an infinite set of constants of motion.
By a skilful exploitation of this fact, Youdovitch
succeeded in proving the following result.
Theorem For a given !
0
in the space L
1
(), there
is a unique weak solution !(t, x) of [8], such that
!(t, x) is in L
1
() for all t, and ! depends
continuously on t in the space L
p
,1 p < 1.
L
p
denotes, in a standard way, the Lebesgue space
of the functions f such that jf j
p
is integrable over
and L
1
(), the space of measurable bounded
functions on .
Thus, if we limit ourselves to initial data with
bounded scalar vorticity, the Cauchy problem for
the two-dimensional incompressible perfect fluid is
satisfactorily solved. The situation is much more
intricate if we consider a less regular initial datum
(e.g., if !
0
is a measure supported by a curve (vortex
sheet)).
Arnol’d’s Work on Two-Dimensional Inviscid Flows
Youdovitch’s theorem implies that the incompressible
Euler equations, with !
0
in L
1
(), is a satisfactory
model of two-dimensional flows – an important issue
to study further the properties of this model.
A famous result due to Arnol’d (see Arnol’d and
Khesin (1998) and Marchioro and Pulvirenti (1994))
deals with the nonlinear stability of the stationary
solutions.
Let us determine the smooth stationary solutions
of the two-dimensional Euler equations in a bounded
domain of the plane. We have to solve:
ðu rÞ! ¼ 0 ½9a
curl u ¼ ! ½9b
div u ¼ 0; u n ¼ 0on@ ½9c
Since we have div u = 0, we may introduce the stream
function of u, , which is given by the Dirichlet’s
problem:
¼ !; ¼ 0on@
so that u = curl .
The system [9] becomes:
r ^r! ¼ 0; ¼ !; ¼ 0on@
Inviscid Flows 163
Let us focus on solutions which are characterized by a
relationship ! = f ( ), where f is a smooth function.
Such solutions are given by the resolution of the
following nonlinear elliptic problem:
¼ f ð Þ; ¼ 0on@ ½10
This problem has always at least a solution, for
example, if f is a bounded function of .
Let
be a solution of [10],and!
= f (
)
the corresponding vorticity function. We shall say that
the stationary solution !
is stable in the L
2
-norm if:
For all ">0, there is a >0, such that for all initial
datum !
0
in L
1
() satisfying
Z
ð!
!
0
Þ
2
dx ; we have :
Z
ð!
!ðtÞÞ
2
dx "; for all t
where !(t) denotes the solution of the Cauchy problem
associated with the initial datum !
0
by Youdovitch’s
theorem.
Now we can state the following result.
Theorem (Arnol’d) Let ! be a stationary solution
given by [10]. We assume that one of the following
assumptions holds:
(C1) There are positive constants c
1
, c
2
, such that
c
1
f
0
c
2
(C2) There are positive constants c
1
, c
2
, with c
2
<
1
(first eigenvalue of the Dirichlet problem on the
domain ) such that:
c
1
f
0
c
2
Then ! is stable in the L
2
-norm.
Remarks
(i) This result was the first nonlinear stability result
for stationary flows.
(ii) The proof makes use of the conservation of the
functionals of the vorticity field.
Another significant contribution of Arnol’d to
hydrodynamics was to reveal the geometrical aspect
of the instability of the perfect-fluid motion. We give
a brief insight into this issue.
Let us come back to the Lagrangian description
of motion. We want to determine the function
’(t, x). Each mapping ’
t
(x) = ’(t, x) is, for t fixed,
a diffeomorphism of
preserving the Lebesgue
measure and the orientation (equivalently stated, it
is an element of SDiff(
)).
In other words, a fluid motion is a curve t ! ’
t
drawn on the ‘‘manifold’’ M = SDiff(
) (the config-
uration space of the system).
At time t, the relationship
@’
@t
ðt; xÞ¼uðt;’ðt; xÞÞ
states that the velocity field u(t, ’
t
(x)) belongs to the
space tangent to M at ’
t
. The tangent space at ’
to M is the space of vector fields v(’(x)), wher e v(x)
is an incompressible vector field on
satisfying
v n = 0on@. This space is naturally endowed
with a norm given by the kinetic energy
1
2
Z
vðxÞ
2
dx
and thus M is endowed with a Riemannian
structure.
It is easy to check that the perfect-fluid motions
correspond to the curves ’
t
drawn on M which are
the critical points of the action integral:
1
2
Z
t
2
t
1
dt
Z
@’
@t
ðt; xÞ
2
dx; for all t
1
< t
2
ðwith the constraints ’ðt
1
;:Þ¼’
1
;’ðt
2
;:Þ¼’
2
Þ
That is to say, the perfect-fluid motions are the
geodesics of the Riemannian manifold M.
The main interest of this geometric framework is
to bring back, at least formally, the perfect-fluid
motions to well-know n objects. Indeed, we know
that the Riemannian curvature of a manifold has a
profound impact on the behavior of geodesics on it.
If the Riemannian curvature is positive, then nearby
geodesics oscillate about one another, and if the
curvature is negative, geodesics rapidly diverge from
one another. More precisely, the stability of geode-
sics is expressed in terms of the curvature by means
of Jacobi’s equation [1].If’
t
is a geodesic curve
starting from ’
0
, with velocity field v(t) (whose
norm is supposed equal to 1), if the sectional
curvature of the manifol d in all the 2-planes
containing v(t) is less than c(< 0), a perturbation
of the initial datum will increase at least as exp(ct):
dð’
t
;
~
’
t
Þdð’
0
;
~
’
0
ÞexpðctÞ
where
~
’
0
denotes the perturbed initial datum and d
the geodesic distance on the manifold. Moreover, if
the curvatu re at every point and for all the sections
is less than c,andifM is compact, then the geodesic
flow, that is, the one-parameter group of transfor-
mations (’
0
, v(0)) ! (’
t
, v(t)), is mixing (in the usual
meaning of ergodic theory). A rnol’d succeeded in
calculating t he sectional curvature for flows on the
two-dimensional torus; he showed that the
164 Inviscid Flows
curvature is negative for ‘‘most’’ of the sections. This
gives an enlightening geometrical picture of the
instability of Lagrangian flows.
It was tempting to connect the above considera-
tions on the instability of two-dimensional flows
with the problem of weather forecast. In 1963
EN Lorenz stated that a two-week forecast would be
a theoretical bound for predicting the atmospheric
motion. Lorenz’s asser tion was based on numerical
simulations. He took as model for the large-scale
atmospheric motion the two-dimens ional Euler
equations on the torus, which he truncated to a
small number of Fourier modes (about 20). This
model is highly unstable and displays exponential
sensitivity with respect to the initial datum. How-
ever, the parallel between the behavior of this
system an d the instability of the Lagrangian flow is
misleading. On the one hand, if we again do the
Lorenz computations on Euler equations, taking
into account a large number of Fourier modes, we
note a striking phenomenon: the flow has a tendency
to self-organize into large vortices, called coherent
structures, and simultaneously the exponential
sensitivity, as measured in terms of the energy
norm of the velocity field, disappears. On the other
hand, the problem of predicting the Lagrangian flow
is very different, the Lagrangian flow can be
exponentially unstable, while the corresponding
velocity field quietly converges, in the energy norm,
towards some equilibrium. We must keep in mind
that the meteorologist aims to predict the values of
the velocity field at some future time and not the
trajectories of the fluid particles. In fact, it appears
that Lorenz has ignored phenomena of a statistical
nature which occur when a large number of degrees
of freedom are considered; thus, his theoretical
bound for the prediction of the atmo spheric motion
has no definite basis. More detailed reflections on
this issue can be found in Robert and Rosier (2001).
The Cauchy Problem for the Euler
Equations for Compressible Inviscid
Fluids
As remarked in the introduction, compressible flows
yield pressure waves. The equations of motion being
nonlinear, these waves interact in an intricate
manner giving rise to shocks. This is the main
feature of compressible fluid flows. Compressible
flows are situated in the more general domain of
nonlinear hyperbolic systems, which wer e inten-
sively studied during the last decades. We only give
here an example of the kind of result which can be
obtained.
The following theorem, which states that for a set
of regular initial data, shocks do not occur till some
finite time, is a consequence of a more general result
on hyperbolic systems due to Majda (1984).
We consider =<
3
and the system [1], [2], [4].
Theorem Assume p
0
, u
0
2 H
S
\ L
1
(<
3
), with
s > 5=2 and p
0
(x) > 0. Then there is a finite time
T > 0, depending on the H
s
and L
1
norms of the
initial data, such that the Cauchy problem for [1],
[2], [4] has a unique bounded smooth solution p,
u 2 C
1
([0, T] <
3
), with p(t, x) > 0 for all t, x.
Inviscid Flows and Turbulence
Loosely speaking, turbulence is the intricate motion
of a slightly viscous flow. Going back to the first
half of the last century, there are two main
approaches to turbulence. The first is due to Leray.
The dissipation of energy is a characteristic feature
of three-dimensional turbulence, and Leray thought
that, even if very small, the viscosity of the fluid
plays an important role, so that to understand
turbulence the first step is to study the Navier–
Stokes equations. A radically different approach is
due to Onsager. Onsager (1949) started with the
fundamental remark that the 4/5 law of turbulence,
which relates the dissipation of energy to the
increments of the velocity field, does not involve
viscosity. Furthermore, he observed that the proof of
the conservation of energy for the solutions of Euler
equations uses an integration by parts which
supposes some regularity of the velocity field. He
then imagined that an inviscid dissipation mechan-
ism, due to a lack of regularity of the solutions, was
at work in Euler equations. In modern terminology,
he suggested to model turbulent flows by nonregular
(weak) solutions satisfying the Euler equations in the
sense of distributions. He also conjectured that if a
solution satisfies a Ho¨ lder regularity condition of
order >1=3, then the energy would be conserved.
Onsager’s views were revolutionary and forgotten
for a long time. Recent works, such as the proof of
Onsager’s con jecture, the construction of weak
solutions with energy dissipation, and the discovery
of the explicit local form of the energy dissipation
for weak solutions, show a renewed interest in these
views (see, e.g., Constantin and Titi (1994), Eyink
(1994), Robert (2003),andShnirelman (2003)).
See also: Compressible flows: Mathematical Theory;
Dissipative Dynamical Systems of Infinite Dimension;
Hyperbolic Dynamical Systems; Incompressible Euler
Equations: Mathematical Theory; Non-Newtonian Fluids;
Partial Differential Equations: Some Examples; Chaos
and Attractors; Turbulence Theories.
Inviscid Flows 165
Further Reading
Arnold VI and Khesin BA (1998) Topological Methods in
Hydrodynamics. Berlin: Springer.
Batchelor GK (1967) An Introduction to Fluid Dynamics.
Cambridge: Cambridge University Press.
Chemin JY (1995) Fluides parfaits incompressibles. Asterisque
no 230, SMF.
Chen GQ and Wang D (2002) The Cauchy problem for the Euler
equations for compressible fluids. In: Friedlander S and Serre D
(eds.) Handbook of Mathematical Fluid Dynamics, vol. 1.
Amsterdam: Elsevier.
Constantin P, Weinan E, and Titi ES (1994) Onsager’s
conjecture on the energy conservation for solutions of Euler’s
equation. Communications in Mathematical Physics 165:
207–209.
Eyink G (1994) Energy dissipation vithout viscosity in ideal
hydrodynamics. Physica D 78(3–4): 222–240.
Frisch U (1995) Turbulence. Cambridge: Cambridge University
Press.
Kato T (1975) Quasi-Linear Equations of Evolution with Applica-
tions to Partial Differential Equations, Lecture Notes in Math.
vol. 448, pp. 25–70. New York: Springer.
Majda A (1984) Compressible Fluid Flow and Systems of Conser-
vation Laws in Several Space Variables. New York: Springer.
Marchioro C and Pulvirenti M (1994) Mathematical Theory of
Incompressible Non-Viscous Fluids. Berlin: Springer.
Onsager L (1949) Statistical hydrodynamics. Nuovo Cimento
(suppl. 6): 279–291.
Robert R (2003) Statistical hydrodynamics. In: Friedlander S and
Serre D (eds.) Handbook of Mathematical Fluid Dynamics,
vol. 2. Amsterdam: Elsevier.
Robert R and Rosier C (2001) Long-range predictibility of atmo-
spheric flows. Nonlinear Processes in Geophysics 8: 55–67.
Serre D (1979) Les invariants du premier ordre de l’equation
d’Euler en dimension 3. Comptes Rendus de l’Academie des
Sciences Paris, Serie A 289: 267–270.
Shnirelman A (2003) Weak solutions of incompressible Euler
equations. In: Friedlander S and Serre D (eds.) Handbook of
Mathematical Fluid Dynamics, vol. 2. Amsterdam: Elsevier.
Ising Model see Two-Dimensional Ising Model
Isochronous Systems
Francesco Calogero, University of Rome, Rome,
Italy and Institute, Nazionale de Fisica Nucleare,
Rome, Italy
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
This paper reviews recent developments, following
closely (sometimes verbatim) the review paper
Calogero F (2004c) (see the Bibliography below);
for more traditional investigations of isochronous
systems see other entries of this Encyclopedia (and
for the mathematical investigation of isochronous
centers in the plane, related to the 16th Hilbert
problem, see for instance the survey paper referred
to at the end of this entry).
The isochronous systems treated herein are char-
acterized by the property to possess an open domai n
having full dimensionality in their phase space such
that all the motions evolving from a set of initial
data in it are completely periodic with the same
fixed period. The natural measure of this open
domain might, or it might not, be infinite when the
measure of the entire phase space is itself infinite:
for instance, if the entire phase space is the two-
dimensional Euclidian plane, such a domain might
be the exterior, or the interior, of a circle of finite
radius.
It is justified to call such systems superintegrable,
or perhaps partially superintegrable inasmuch as the
property of isochronicity of all their motions holds
only in a subregion of the entire phase space. This
terminology is justified by the observation that,
roughly speaking, all confined motions of a super-
integrable system – in which all but one of the
degrees of freedom are constrained by the existence
of the maximal possible number of constants of
motion – are completely periodic, although not
necessarily all with a fixed period – entailing that
isochronicity entails superintegrability, while the
converse is not the case (see the entry Integrable
systems in this Encyclopedia).
A simple trick – amounting essentially to a
change of independent, and possible as well of
dependent, variables, allows to deform a largely
arbitrary dynamical system so that the d efor med
system obtained from it be isochronous.This
‘‘trick’’, which is now explained, entails therefore
that isochronous systems are not rare.Belowwe
provide several examples; others can be found in
the further reading suggested at the end of this
entry, and/or can be manufactured ad libitum using
the trick.
166 Isochronous Systems