BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 44, Number 4, October 2007, Pages 581–602
S 0273-0979(07)01181-0
Article electronically published on June 18, 2007

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

DEMETRIOS CHRISTODOULOU

This article is in celebration of the 300th anniversary of the birth of one of the

greatest mathematicians and physicists in history, Leonhard Euler. The article
is directly concerned with Euler’s work in ﬂuid mechanics, although his work in
the calculus of variations and in partial diﬀerential equations in general have been
instrumental in the developments to be outlined here.

Euler did have predecessors in the ﬁeld of ﬂuid mechanics, who had conceived

some of the basic concepts. His immediate predecessor in this regard was his friend
D. Bernoulli, whose 1738 work [Be] is likely to have had a great inﬂuence on him.
However it was Euler who ﬁrst formulated the general equations describing the mo-
tion of a perfect ﬂuid. The general compressible Euler equations ﬁrst appeared in
published form in [Eu2], the second of three Euler articles on ﬂuid mechanics which
appeared in the same 1757 volume of the

M´

emoires de l’Academie des Sciences de

Berlin

. The third of these articles, [Eu3], is a continuation of the second, while the

ﬁrst, [Eu1], establishes the general validity of the basic concepts and formulates
the equations in the static case. However, it seems that the article [Eu4], which
formulates the equations of motion in the incompressible case and which was pub-
lished only in 1761, was actually the ﬁrst to be composed, as at least a preliminary
version of it was presented to the Berlin Academy in 1752.

Thus Euler’s ﬂuid equations were among the ﬁrst partial diﬀerential equations to

be written down, preceded, it seems, only by D’Alembert’s 1749 formulation [DA]
of the one-dimensional wave equation describing the motion of a vibrating string
in the linear approximation.

Euler was not content to conﬁne himself to the formulation of the basic laws of

ﬂuid mechanics, but he proceeded to investigate and explain on the basis of these
laws some of the basic observed phenomena. Thus in [Eu5] he made the ﬁrst, albeit
incomplete, study of convection, a phenomenon which depends on compressibility
as well as on temperature variation in a gravitational potential. In [Eu7] he stud-
ied incompressible ﬂows in pipes in the linear approximation, while in [Eu8] he
studied compressible ﬂows in the linear approximation, treating the generation and
propagation of sound waves.

The contrast to D’Alembert’s equation however could not be greater, for we are

still, after the lapse of two and a half centuries, far from having achieved an adequate
understanding of the observed phenomena which are supposed to lie within the
domain of validity of Euler’s ﬂuid equations.

The phenomena displayed in the interior of a ﬂuid fall into two broad classes: the

phenomena of sound, the linear theory of which is acoustics, and the phenomena

Received by the editors May 15, 2007.
2000

Mathematics Subject Classiﬁcation.

Primary 76L05, 76-03; Secondary 01A50, 01A55,

01A60, 35L65, 35L67, 76N15.

2007 American Mathematical Society

581

582

DEMETRIOS CHRISTODOULOU

of vortex motion. The sound phenomena depend on the compressibility of a ﬂuid,
while the vortex phenomena occur even in a regime where ﬂuid may be considered
to be incompressible. The formation and evolution of shocks belongs to the class of
sound phenomena but lies in the non-linear regime, beyond the range covered by
linear acoustics. The phenomena of vortex motion include the chaotic form called
turbulence, the understanding of which is one of the great challenges of science.

I shall presently review the history of the study of the phenomena of sound in

ﬂuids since the original formulation by Euler of the laws governing these phenomena
in the works cited above. The review shall concentrate on the non-linear phenomena
of the formation and evolution of shocks. A comprehensive, up-to-date introduction
to the mathematical theory of vortex phenomena is provided by the book [M-B].

Now, at the time when the equations of ﬂuid mechanics were ﬁrst formulated,

thermodynamics was in its infancy; however it was already clear that the local state
of a ﬂuid as seen by a comoving observer is determined by two thermodynamic vari-
ables, say the pressure and the temperature. Of these, only the pressure entered
the equations of motion, while the equations involve also the density of the ﬂuid.
The density was already known to be a function of the pressure and the temper-
ature for a given type of ﬂuid. However in the absence of an additional equation,
the system of equations at the time of Euler, which consisted of the momentum
equations together with the equation of continuity, was underdetermined except in
the incompressible limit. The additional equation was supplied by Laplace in 1816
[La] in the form of what was later to be called the adiabatic condition and allowed
him to make the ﬁrst correct calculation of the speed of sound.

The ﬁrst work on the formation of shocks was done by Riemann in 1858 [Ri].

Riemann considered the case of isentropic ﬂow with plane symmetry, where the
equations of ﬂuid mechanics reduce to a system of conservation laws for two un-
knowns and with two independent variables, a single space coordinate and time.
He introduced for such systems the so-called Riemann invariants, and with the help
of these showed that solutions which arise from smooth initial conditions develop
inﬁnite gradients in ﬁnite time. Riemann also realized that the solutions can be
continued further as discontinuous solutions, but here there was a problem. Up to
this time the energy equation was considered to be simply a consequence of the laws
of motion, not a fundamental law in its own right. On the other hand, the adiabatic
condition was considered by Riemann to be part of the main framework. Now as
long as the solutions remain smooth it does not matter which of the two equations
we take to be the fundamental law, for each is a consequence of the other, modulo
the remaining laws. However this is no longer the case once discontinuities develop,
so one must make a choice as to which of the two equations to regard as fundamental
and therefore remains valid thereafter. Here Riemann made the wrong choice, for
only during the previous decade, in 1847, had the ﬁrst law of thermodynamics been
formulated by Helmholtz [He], based in part on the experimental work of Joule on
the mechanical equivalence of heat, and the general validity of the energy principle
had thereby been shown. In 1865 the concept of entropy was introduced into theo-
retical physics by Clausius [Cl2], and the adiabatic condition was understood to be
the requirement that the entropy of each ﬂuid element remains constant during its
evolution. The second law of thermodynamics, involving the increase of entropy in
irreversible processes, had ﬁrst been formulated in 1850 by Clausius [Cl1] without
explicit reference to the entropy concept. After these developments the right choice

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

583

in Riemann’s dilemma became clear. The energy equation must at all times be
kept as a fundamental law, but the entropy of a ﬂuid element must jump upward
when the element crosses a hypersurface of discontinuity. The formulation of the
correct jump conditions that must be satisﬁed by the thermodynamic variables and
the ﬂuid velocity across a hypersurface of discontinuity was began by Rankine in
1870 [Ra] and completed by Hugoniot in 1889 [Hu].

With Einstein’s discovery of the special theory of relativity in 1905 [Ei] and its

ﬁnal formulation by Minkowski in 1908 [Mi] through the introduction of the concept
of spacetime with its geometry, the domain of geometry being thereby extended to
include time, a unity was revealed in physical concepts which had been hidden up
to this point. In particular, the concepts of energy density, momentum density
or energy ﬂux, and stress where uniﬁed into the concept of the energy-momentum-
stress tensor, and energy and momentum were likewise uniﬁed into a single concept,
the energy-momentum vector. Thus, when the Euler equations where extended to
become compatible with special relativity, it was obvious from the start that it made
no sense to consider the momentum equations without considering also the energy
equation, for these two where parts of a single tensorial law, the energy-momentum
conservation law. This law plus the particle conservation law (the equation of
continuity of the non-relativistic theory) constitute the laws of motion of a perfect
ﬂuid in the relativistic theory. The adiabatic condition is then a consequence for
smooth solutions.

A new basic physical insight on the shock development problem was reached ﬁrst,

it seems, by Landau in 1944 [Ln]. This was the discovery that the condition that
the entropy jump be positive as a hypersurface of discontinuity is traversed from the
past to the future should be equivalent to the condition that the ﬂow is evolutionary,
that is, that conditions in the past determine the ﬂuid state in the future. More
precisely, what was shown by Landau was that the condition of determinism is
equivalent, at the linearized level, to the condition that the tangent hyperplane at
a point on the hypersurface of discontinuity is on one hand contained in the exterior
of the sound cone at this point corresponding to the state before the discontinuity,
while on the other hand intersects the sound cone at the same point corresponding
to the state after the discontinuity. Moreover, this latter condition is equivalent to
the positivity of the entropy jump. This is interesting from a general philosophical
point of view, because it shows that irreversibility can arise, even though the laws
are all time-reversible, once the solution ceases to be regular. To a given state at a
given time there always corresponds a unique state at any given later time. If the
evolution is regular in the associated time interval, then the reverse is also true: to
a given state at a later time there corresponds a unique state at any given earlier
time, the laws being time reversible. This reverse statement is false however if there
is a shock during the time interval in question. Thus determinism in the presence
of hypersurfaces of discontinuity selects a direction of time and the requirement
of determinism coincides, modulo the other laws, with what is dictated by the
second law of thermodynamics, which is in its nature irreversible. This recalls the
interpretation of entropy, ﬁrst discovered by Boltzmann in 1877 [Bo], as a measure
of disorder at the microscopic level. An increase of entropy was thus understood to
be associated to an increase in disorder or to loss of information, and determinism
can be expected only in the time direction in which information is lost, not gained.

584

DEMETRIOS CHRISTODOULOU

An important mathematical development with direct application to the equa-

tions of ﬂuid mechanics in the physical case of three space dimensions was the
introduction by Friedrichs of the concept of a symmetric hyperbolic system in 1954
[F] and his development of the theory of such systems. It is through this theory
that the local existence and domain of dependence property of solutions of the
initial value problem associated to the equations of ﬂuid mechanics were ﬁrst es-
tablished. Another development in connection to this was the general investigation
by Friedrichs and Lax in 1971 [F-L] (see also [Lx1]) of nonlinear ﬁrst order systems
of conservation laws which for smooth solutions have as a consequence an addi-
tional conservation law. This is the case for the system of conservation laws of ﬂuid
mechanics, which consists of the particle and energy-momentum conservation laws,
which for smooth solutions imply the conservation law associated to the entropy
current. It was then shown that if the additional conserved quantity is a convex
function of the original quantities, the original system can be put into symmetric
hyperbolic form. Moreover, for discontinuous solutions satisfying the jump condi-
tions implied by the integral form of the original conservation laws, an inequality
for the generalized entropy was derived. This inequality had been suggested by
Kruˇ

zhkov [Kr].

The problem of shock formation for the equations of ﬂuid mechanics in one

space dimension and, more generally, for systems of conservation laws in one space
dimension was studied by Lax in 1964 [Lx2] and 1973 [Lx3] and by John [J] in
1974. The approach of these works was analytic, the strategy being to deduce an
ordinary diﬀerential inequality for a quantity constructed from the ﬁrst derivatives
of the solution which showed that this quantity must blow up in ﬁnite time under
a certain structural assumption on the system called genuine non-linearity and
suitable conditions on the initial data. The genuine non-linearity assumption is in
particular satisﬁed by the non-relativistic compressible Euler equations in one space
dimension provided that the pressure is a strictly convex function of the speciﬁc
volume. A more geometric approach in the case of systems with two unknowns
was developed by Majda in 1984 [Ma1] based in part on ideas introduced by Keller
and Ting in 1966 [K-T]. In this approach one considers the evolution of the inverse
density of the characteristic curves of each family and shows that under appropriate
conditions this inverse density must somewhere vanish within ﬁnite time. In this
way, not only were the earlier blow-up results reproduced, but, more importantly,
insight was gained into the nature of the breakdown. Moreover Majda’s approach
also covered the case where the genuine non-linearity assumption does not hold,
but we have linear degeneracy instead. He showed that in this case global in time
smooth solutions exist for any smooth initial data.

The problem of the global in time existence of solutions of the equations of ﬂuid

mechanics in one space dimension was treated by Glimm in 1965 [Gl] through an
approximation scheme involving at each step the local solution of an initial value
problem with piecewise constant initial data. The convergence of the approxima-
tion scheme then produced a solution in the class of functions of bounded variation.
Now, by the previously established results on shock formation, a class of functions
in which global existence holds must necessarily include functions with discontinu-
ities, and the class of functions of bounded variation is the simplest class having
this property. Thus, the treatment based on the total variation, the norm in this
function space, in itself an admirable investigation, would have been insuperable if

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

585

the development of the one-dimensional theory were the goal of the scientiﬁc eﬀort
in the ﬁeld of ﬂuid mechanics. However since that goal can only be the mathemat-
ical description of phenomena in real three dimensional space, one had eventually
to face the fact that methods based on the total variation do not generalize to more
than one space dimension. It is in fact clear from the study of the linearized theory,
acoustics, which involves the wave equation, that in more than one space dimension
only methods based on the energy concept are appropriate.

The ﬁrst general result on the formation of shocks in three-dimensional ﬂuids was

obtained by Sideris in 1985 [S]. Sideris considered the compressible Euler equations
in the case of a classical ideal gas with adiabatic index

γ >

1 and with initial data

which coincide with those of a constant state outside a ball. The assumptions of
his theorem on the initial data were that there is an annular region bounded by
the sphere outside which the constant state holds and a concentric sphere in its
interior, such that a certain integral in this annular region of

−

, the departure

of the density

from its value

in the constant state, is positive, while another

integral in the same region of

ρv

, the radial momentum density, is non-negative.

These integrals involve kernels which are functions of the distance from the center.
It is also assumed that everywhere in the annular region the speciﬁc entropy

not less than its value

in the constant state. The conclusion of the theorem

is that the maximal time interval of existence of a smooth solution is ﬁnite. The
chief drawback of this theorem is that it tells us nothing about the nature of the
breakdown. Also the method relies on the strict convexity of the pressure as a
function of the density displayed by the equation of state of an ideal gas and does
not extend to more general equations of state.

Another important work on shocks in three space dimensions was the 1983 work

of Majda [Ma2], [Ma3] on the local in time shock continuation problem. In this
problem we are given initial data in

which is smooth in the closure of each

component of

, where

is a smooth complete surface in

. The data is to

satisfy the condition that there exists a function

such that the jumps of the

data across

satisfy the Rankine-Hugoniot jump conditions as well as the entropy

condition with

in the role of the shock speed. The higher order compatibility

conditions associated to an initial-boundary value problem are also required to be
satisﬁed. We are then required to ﬁnd a time interval [0

, τ

], a smooth hypersurface

in the spacetime slab [0

, τ

]

and a solution of the compressible Euler equations

which is smooth in the closure of each component of [0

, τ

]

and satisﬁes

across

the Rankine-Hugoniot jump conditions as well as the entropy condition, or

equivalently the determinism condition. Majda’s solution of this problem requires
an additional condition on the initial data to ensure the stability of the linearized
problem. The additional condition follows from the other conditions in the case of
a classical ideal gas, but it does not follow for a general equation of state.

In the remainder of this article, I shall summarize my own recent work in this

ﬁeld. All the material which is presented below is expounded in the monograph
[Ch1]. The monograph considers the relativistic Euler equations in three space
dimensions for a perfect ﬂuid with an arbitrary equation of state.

The mechanics of a perfect ﬂuid is described in the framework of the Minkowski

space-time of special relativity by a future-directed unit time-like vectorﬁeld

the ﬂuid

4-velocity

, and two positive functions

and

, the

number of particles

per unit volume

(in the local rest frame of the ﬂuid) and the

entropy per particle

586

DEMETRIOS CHRISTODOULOU

respectively. In terms of a system of rectangular coordinates (

, x

), with

a time coordinate and (

, x

) space coordinates, the metric components

µν

µ, ν

= 0

3, are given by

(1)

−

, g

= 1

, g

µν

= 0 : if

ν.

The conditions on the 4-velocity components

= 0

3, are then:

(2)

µν

−

, u

where we follow the summation convention, according to which repeated upper and
lower indices are summed over their range. The mechanical properties of a perfect
ﬂuid are speciﬁed once we give the

equation of state

, which expresses the

relativistic

mass-energy density

as a function of

and

(3)

(

n, s

)

According to the laws of thermodynamics, the

pressure

and the temperature

are then given by:

(4)

∂ρ

∂n

−

ρ, θ

∂ρ

∂s

The functions

ρ, p, θ

are assumed positive. Moreover, it is assumed that

is an

increasing function of

at constant

and

is an increasing function of

at constant

. In terms of the

volume per particle

(5)

and the

relativistic energy per particle

(6)

ρv,

these relations take the familiar form:

(7)

(

v, s

)

(8)

−

pdv

θds.

We note that the relativistic energy per particle contains the rest mass contribution

being the particle rest mass and

the universal constant represented by

the speed of light in vacuum. Under ordinary circumstances this is in fact the
dominant contribution to

. The corresponding contribution to

nmc

being the rest mass density. In writing down the relativistic equations to follow,
we choose the relation of the units of temporal to spatial lengths so as to set

= 1.

We note moreover that the particle rest mass may be taken to be unity, so that all
quantities per particle are quantities per unit rest mass and

coincides with the

rest mass density.

The function

(9)

√

(

)

or equivalently

(10)

√

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

587

is called

enthalpy per particle

. By virtue of eqs. 3 and 4, or 7, 8,

√

can be

considered to be a function of

and

, and its diﬀerential is given by:

(11)

√

vdp

θds.

We may in fact use

and

instead of

and

as the basic thermodynamic variables.

The

sound speed

is deﬁned by:

(12)

dρ

a fundamental thermodynamic assumption being that the right-hand side of

positive

. Then

is deﬁned to be positive. Another condition on

in the framework

of special relativity is that

η <

1, namely that the sound speed is less than the

speed of light in vacuum.

The

particle current

is the vectorﬁeld

whose components are given by:

(13)

The

energy-momentum-stress

tensor is the symmetric 2-contravariant tensorﬁeld

whose components are:

(14)

µν

= (

)

(

−

)

µν

Here (

−

)

µν

µ, ν

= 0

3, are the components of the reciprocal metric,

(15)

(

−

)

−

(

−

)

= (

−

)

= (

−

)

= 1

(

−

)

µν

= 0 : if

ν.

The

equations of motion

of a perfect ﬂuid are the conservation laws:

∂

= 0

(16)

∂

µν

= 0

(17)

where the symbol

∂

∂x

denotes partial derivative with respect to the rectangular coordinate

One reason for working with the relativistic equations is that there is a sub-

stantial gain in geometric insight because of the spacetime geometry viewpoint of
special relativity. As an example we give the following equation:

(18)

−

θds.

Here

is the vorticity 2-form:

(19)

dβ,

where

is the 1-form deﬁned, relative to an arbitary system of coordinates, by:

(20)

−

√

σu

µν

In (18),

denotes contraction on the left by the vectorﬁeld

. We note here that

the vorticity 2-form is not the exact analogue to the classical notion of vorticity.
What exactly corresponds to the classical notion is the

vorticity vector

(21)

(

−

)

µαβγ

βγ

Here

−

is the reciprocal volume form of the Minkowski metric

or volume form

in the cotangent space at each point. Its components in a rectangular coordinate
system constitute the 4-dimensional fully antisymmetric symbol. The vectorﬁeld

588

DEMETRIOS CHRISTODOULOU

is the obstruction to integrability of the distribution of orthogonal hyperplanes

to the ﬂuid velocity

, the local simultaneous spaces of the ﬂuid.

Equation (18) is equivalent, modulo the particle conservation law (16), to the

energy-momentum conservation laws (17) and is arguably the simplest explicit form
of these equations. The 1-form

plays a fundamental role in the monograph. In

the irrotational isentropic case it is given by

dφ

, where

is a function, which

we call

wave function

. In this case we have

−

(

−

)

µν

∂

φ∂

φ,

(22)

−

∂

√

(

∂

= (

−

)

µν

∂

)

(23)

and the whole content of the equations of motion is contained in the particle current
conservation law (16), which takes the form of a non-linear wave equation:

(24)

∂

(

G∂

) = 0

where

(25)

(

)

(

being in this case a constant) is given by:

(26)

√

Our relativistic treatment has the virtue that, while being more general, it does

not require any special care in extracting information on the non-relativistic limit.
This is due to the fact that the non-relativistic limit is a regular limit, obtained by
letting the speed of light in conventional units tend to inﬁnity while keeping the
sound speed ﬁxed. To allow the results in the non-relativistic limit to be extracted
from our treatment in a straightforward manner, we have chosen to avoid summing
quantities having diﬀerent physical dimensions when such sums would make sense
only when a unit of velocity has been chosen, even though we have followed the
natural choice within the framework of special relativity of setting the speed of light
in vacuum equal to unity in writing down the relativistic equations of motion.

The most important concept on which our treatment is based is that of the

acoustical spacetime manifold

. This consists of the same underlying manifold as the

Minkowski spacetime, but with the

acoustical metric

in the role of the Minkowski

metric

(27)

µν

+ (1

−

)

This is a Lorentzian metric, the null cones of which are the sound cones.

An initial data set for the equations of motion (16), (17) consists of the speciﬁ-

cation of the triplet (

p, s, u

) on a hypersurface Σ in Minkowski spacetime, possibly

with boundary, such that the metric induced on Σ by the acoustical metric (27),
deﬁned along Σ by the initial data, is positive deﬁnite. To any given initial data
set there corresponds a unique

maximal solution

of the equations of motion. The

notion of maximal solution or maximal development of an initial data set is the
following. Given an initial data set the local existence theorem asserts the exis-
tence of a

development

of this set, namely of a domain

in Minkowski spacetime,

whose past boundary is the domain Σ of the initial data, and of a solution deﬁned
in

and taking the given data at the past boundary, such that if we consider any

point

∈ D

and any curve issuing at

with the property that its tangent vector at

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

589

any point

belongs to

−

, the closure of the past component of the open double

cone deﬁned by

, the acoustical metric at

, the curve terminates in the past at

a point of Σ. The local uniqueness theorem asserts that if (

(

, s

, u

)) and

(

, s

, u

)) are two developments of the same initial data set, then (

, s

, u

)

coincides with (

, s

, u

) in

. It follows that the union of all developments

of a given initial data set is itself a development, the unique

maximal development

of the initial data set.

In the monograph [Ch1] we consider regular initial data on a spacelike hyper-

plane Σ

in Minkowski spacetime which outside a sphere coincide with the data

corresponding to a constant state. (Here the notions “spacelike hyperplane” and
“sphere” refer to the Minkowski metric

.) We consider the restriction of the initial

data to the exterior of a concentric sphere in Σ

, and we consider the maximal clas-

sical development of this data. Then, under a suitable restriction on the size of the
departure of the initial data from those of the constant state, we prove certain the-
orems which give a complete description of the maximal classical development. In
particular, the theorems give a detailed description of the geometry of the boundary
of the domain of the maximal classical development and a detailed analysis of the
behavior of the solution at this boundary. A complete picture of shock formation
in three-dimensional ﬂuids is thereby obtained. Also, sharp suﬃcient conditions
on the initial data for the formation of a shock in the evolution are established,
and sharp lower and upper bounds for the temporal extent of the domain of the
maximal solution are derived.

The reason why we consider only the maximal development of the restriction of

the initial data to the exterior of a sphere is in order to avoid having to treat the long
time evolution of the portion of the ﬂuid which is initially contained in the interior
of this sphere, for we have no method at present to control the long time behavior of
the pointwise magnitude of the vorticity of a ﬂuid portion, the vorticity satisfying a
transport equation along the ﬂuid ﬂow lines. Our approach to the general problem is
the following. We show that given arbitary regular initial data which coincide with
the data of a constant state outside a sphere, if the size of the initial departure from
the constant state is suitably small, we can control the solution for a time interval of
order 1

/η

, where

is the sound speed in the surrounding constant state. We then

show that at the end of this interval a thick annular region has formed, bounded by
concentric spheres, where the ﬂow is irrotational and isentropic, the constant state
holding outside the outer sphere. We then study the maximal classical development
of the restriction of the data at this time to the exterior of the inner sphere. We
should emphasize here that if we were to restrict ourselves from the beginning to
the irrotational isentropic case, we would have no problem extending the treatment
to the interior region, thereby treating the maximal solution corresponding to the
data on the complete initial hyperplane Σ

. In fact, it is well known that sound

waves decay in time faster in the interior region, and our constructions can readily
be extended to cover this region. It is only our present inability to achieve long
time control of the magnitude of the vorticity along the ﬂow lines of the ﬂuid that
prevents us from treating the interior region in the general case.

The general concept of variation, or variation through solutions, is a basic con-

cept on which the treatment not only of the irrotational isentropic case but also of
the general equations of motion is based. This concept has been discussed in the
general context of Euler-Lagrange equations, that is, systems of partial diﬀerential

590

DEMETRIOS CHRISTODOULOU

equations arising from an action principle, in a previous monograph [Ch2]. To a
variation is associated a linearized Lagrangian, on the basis of which energy currents
are constructed following the ideas of Noether [N]. It is through energy currents and
their associated integral identities that the estimates essential to our approach are
derived. Here the ﬁrst order variations correspond to the one-parameter subgroups
of the Poincar´

e group, the isometry group of Minkowski spacetime, extended by the

one-parameter scaling or dilation group, which leave the surrounding constant state
invariant. The higher order variations correspond to the one-parameter groups of
diﬀeomorphisms generated by a set of vectorﬁelds, the commutation ﬁelds. The
construction of an energy current requires a multiplier vectorﬁeld which at each
point belongs to the closure of the positive component of the inner characteristic
core in the tangent space at that point.

In the irrotational isentropic case the characteristic in the tangent space at a

point consists only of the sound cone at that point, and this requirement becomes
the requirement that the multiplier vectorﬁeld be non-spacelike and future directed
with respect to the acoustical metric (27). We use two multiplier vectorﬁelds in
our analysis of the isentropic irrotational problem. The ﬁrst multiplier ﬁeld is the
vectorﬁeld

(28)

= (

−

)

−

κL

+ 2

Here,

is the inverse density of the hyperplanes Σ

corresponding to the con-

stant values of the time coordinate

, and

is the inverse spatial density of the

wave fronts, both with respect to the acoustical metric. The vectorﬁeld

is the

tangent vectorﬁeld of the bicharacteristic generators, parametrized by

, of a family

of outgoing characteristic hypersurfaces

, the level sets of an acoustical function

. The wave fronts

t,u

are the surfaces of intersection

. The vectorﬁeld

deﬁnes a ﬂow on each of the Σ

, taking each wave front onto another wave front,

the normal, relative to the induced acoustical metric

, ﬂow of the foliation of Σ

by the surfaces

t,u

The second multiplier ﬁeld is the vectorﬁeld

deﬁned by:

(29)

= (

ω/ν

)

Here

is the mean curvature of the wave fronts

t,u

relative to their characteristic

normal

. However

is deﬁned not relative to the acoustical metric

µν

but rather

relative to a conformally related metric ˜

µν

(30)

µν

= Ω

µν

It turns out that there is a choice of conformal factor Ω such that in the isentropic
irrotational case a ﬁrst order variation ˙

of the wave function

satisﬁes the wave

equation relative to the metric ˜

µν

. This choice deﬁnes Ω, and the deﬁnition makes

Ω the ratio of a function of

to the value of this function in the surrounding constant

state; thus Ω is equal to unity in the constant state. It turns out moreover that
Ω is bounded above and below by positive constants. The function

appearing in

(30) is required to satisfy certain conditions, and it is shown that the function

(1+

) does satisfy these requirements. The multiplier ﬁeld

corresponds to the

generator of inverted time tranlsations, which are proper conformal tranformations
of the Minkowski spacetime with its Minkowskian metric

µν

. The latter was ﬁrst

used by Morawetz [Mo] to study the decay of solutions of the initial-boundary value
problem for the classical wave equation outside an obstacle. The vectorﬁeld

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

591

an analogue of the multiplier ﬁeld of Morawetz for the acoustical spacetime which,
as we explained above, is the same underlying manifold but equipped with the
acoustical metric

µν

. To each variation

, of any order, there are energy currents

associated to

and to

and

respectively. These currents deﬁne the energies

[

](

[

](

), and ﬂuxes

[

](

[

](

). For given

and

the energies

are integrals over the exterior of the surface

t,u

in the hyperplane Σ

, while the

ﬂuxes are integrals over the part of the outgoing characteristic hypersurface

between the hyperplanes Σ

and Σ

. It is these energy and ﬂux integrals, together

with a spacetime integral

[

](

t, u

) associated to

, to be discussed below, which

are used to control the solution.

Evidently, the means by which the solution is controlled depend on the choice

of the acoustical function

, the level sets of which are the outgoing characteristic

hypersurfaces

. The function

is determined by its restriction to the initial

hyperplane Σ

. The divergence of the energy currents, which determines the growth

of the energies and ﬂuxes, itself depends on

(

)

, in the case of the energy current

associated to

, and

(

)

, in the case of the energy current associated to

Here for any vectorﬁeld

in spacetime, we denote by

(

)

the Lie derivative of

the conformal acoustical metric ˜

with respect to

. We call

(

)

the deformation

tensor corresponding to

. In the case of higher order variations, the divergences

of the energy currents depend also on the

(

)

, for each of the commutation ﬁelds

to be discussed below.

All these deformation tensors ultimately depend on the acoustical function

or, what is the same, on the geometry of the foliation of spacetime by the outgoing
characteristic hypersurfaces

, the level sets of

. The most important geometric

property of this foliation from the point of view of the study of shock formation
is the density of the packing of its leaves

. One measure of this density is the

inverse spatial density of the wave fronts, that is, the inverse density of the foliation
of each spatial hyperplane Σ

by the surfaces

t,u

. This is the function

which

appears in (28) and is given in arbitrary coordinates on Σ

by:

(31)

−

= (

−

)

∂

u∂

where

is the induced acoustical metric on Σ

. Another measure is the inverse

temporal density of the wave fronts, the function

given in arbitrary coordinates

in spacetime by:

(32)

−

(

−

)

µν

∂

t∂

The two measures are related by:

(33)

ακ

where

is the inverse density, with respect to the acoustical metric, of the foliation

of spacetime by the hyperplanes Σ

. The function

also appears in (28) and is

given in arbitrary coordinates in spacetime by:

(34)

−

(

−

)

µν

∂

t∂

It is expressed directly in terms of the 1-form

. It turns out, moreover, that

it is bounded above and below by positive constants. Consequently

and

are

equivalent measures of the density of the packing of the leaves of the foliation of

592

DEMETRIOS CHRISTODOULOU

spacetime by the

. Shock formation is characterized by the blowup of this density

or equivalently by the vanishing of

The other entity besides

which describes the geometry of the foliation by

the

is the second fundamental form of the

. Since the

are null hypersur-

faces with respect to the acoustical metric

, their tangent hyperplane at a point is

the set of all vectors at that point which are

-orthogonal to the generator

, and

itself belongs to the tangent hyperplane, being

-orthogonal to itself. Thus the

second fundamental form

is intrinsic to

, and in terms of the metric

induced by the acoustical metric on the

t,u

sections of

, it is given by:

(35)

= 2

where

for a covariant

t,u

tensorﬁeld

denotes the restriction of

T S

t,u

The acoustical structure equations are:
The propagation equation for

along the generators of

The Codazzi equation which expresses div

/ χ

, the divergence of

intrinsic to

t,u

in terms of

, the diﬀerential on

t,u

of tr

, and a component of the acoustical

curvature and of

, the second fundamental form of the Σ

relative to

The Gauss equation which expresses the Gauss curvature of (

t,u

, h

) in terms of

and a component of the acoustical curvature and of

An equation which expresses

in terms of the Hessian of the restriction of

t,u

and another component of the acoustical curvature and of

These acoustical structure equations seem at ﬁrst sight to contain terms which

blow up as

tend to zero. The analysis of the acoustical curvature then shows

that the terms which blow up as

tend to zero cancel.

The most important acoustical structure equation from the point of view of the

formation of shocks is the propagation equation for

along the generators of

(36)

Lµ

µe

where the function

is given by:

(37)

(

)

dσ

(

T σ

)

Here

is the function deﬁned by:

(38)

−

σH

where

is the sound speed. In (36), the function

depends only on the derivatives

of the

, the rectangular components of

, tangential to the

. It is the function

which determines shock formation, when negative, causing

to decrease to zero.

The derivative of

with respect to

at constant

is thus seen to play a central

role in shock theory. This quantity is expressed by:

(39)

dσ

−

√

where

is the positive function:

σv

The sign of (

dH/dσ

)

in the state ahead of a shock determines the sign of the jump in

pressure in crossing the shock to the state behind. The jump in pressure is positive

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

593

if this quantity is negative; the reverse otherwise. The value of (

dH/dσ

)

in the

surrounding constant state is denoted by

. This constant determines the character

of the shocks for small initial departures from the constant state. In particular
when

= 0, no shocks form and the domain of the maximal classical solution is

complete. Consider the function (

dH/dσ

)

as a function of the thermodynamic

variables

and

. Suppose that we have an equation of state such that at some

value

the function (

dH/dσ

)

vanishes everywhere along the adiabat

We show that in this case the irrotational isentropic ﬂuid equations corresponding
to the value

of the entropy are equivalent to the minimal surface equation, the

wave function

deﬁning a minimal graph in a Minkowski spacetime of one more

spatial dimension. (That is, equation (24) reduces in the case in question to the
non-parametric minimal surface equation.)

Thus the minimal surface equation

deﬁnes a dividing line between two diﬀerent types of shock behavior. Now, the
relativistic enthalpy is dominated by the term

, the contribution of the particle

rest mass

to the energy per particle

being again the speed of light in vacuum.

Thus in the non-relativistic limit the second term in parentheses in (39) vanishes
and the expression in parentheses reduces simply to (

v/dp

)

. Whereas the case

where (

v/dp

)

0, the adiabats being convex curves in the

p, v

plane so that

(

dH/dσ

)

0, is the more commonly found in nature, the reverse case does occur in

the gaseous region near the critical point in the liquid to vapor phase transition and
in similar transitions at higher temperatures associated to molecular dissociation
and to ionization (see [Z-R]).

The path I have followed in attacking the problem of shock formation in three-

dimensional ﬂuids illustrates the following approach in regard to quasilinear hyper-
bolic systems of partial diﬀerential equations: that the quantities which are used to
control the solution must be deﬁned using the causal, or characteristic, structure of
spacetime determined by the solution itself, not an artiﬁcial background structure.
The original system of equations must then be considered in conjunction with the
system of equations which this structure obeys, and it is only through the study of
the interaction of the two systems that results are obtained. The work with Klain-
erman [C-K] on the stability of the Minkowski space in the framework of general
relativity was the ﬁrst illustration of this approach. In the present case, however,
the structure, which is here the acoustical structure, degenerates as shocks begin
to form, and the precise way in which this degeneracy occurs must be guessed be-
forehand and established in the course of the argument of the mathematical proof.
The fact that the underlying structure degenerates implies that our estimates are
no longer even locally equivalent to standard energy estimates, which would of
necessity have to fail when shocks appear.

I ﬁrst establish a theorem, the fundamental energy estimate, which applies to a

solution of the homogeneous wave equation in the acoustical spacetime, in particular
to any ﬁrst order variation. The proof of this theorem relies on certain bootstrap
assumptions on the acoustical entities.

The most crucial of these assumptions

concern the behavior of the function

. These assumptions are established later on

the basis of the ﬁnal set of bootstrap assumptions, which consists only of pointwise
estimates for the variations up to certain order. To give an idea of the nature
of these assumptions, one of the assumptions required to obtain the fundamental
energy estimate up to time

is:

(40)

−

(

T µ

)

≤

(

) : for all

∈

, s

]

594

DEMETRIOS CHRISTODOULOU

where

(

) is a function such that:

(41)

(1 +

)

−

[1 + log(1 +

)]

(

)

≤

with

a constant independent of

. Here

is the vectorﬁeld deﬁned above, and we

denote by

and

−

, respectively, the positive and negative parts of an arbitrary

function

. This assumption is then established by a certain proposition with

(

)

the following function:

(42)

(

) =

(1 +

)

√

−

Cδ

(1 +

)

where

= log(1 +

), and

is a small positive constant appearing

in the ﬁnal set of bootstrap assumptions.

The spacetime integral

[

](

t, u

) mentioned above is essentially the integral of

−

(

ω/ν

)(

Lµ

)

−

/ψ

in the spacetime exterior to

and bounded by Σ

and Σ

. Another assumption

states that there is a positive constant

independent of

such that in the region

below Σ

where

µ < η

4 we have:

(43)

Lµ

≤ −

−

(1 +

)

−

[1 + log(1 +

)]

−

In view of this assumption, the integral

[

](

t, u

) gives eﬀective control of the

derivatives of the variations tangential to the wave fronts in the region where shocks
are to form. The same assumption, which is then established by a certain proposi-
tion, also plays an essential role in the study of the singular boundary.

The ﬁnal stage of the proof of the fundamental energy estimate is the analysis of

system of integral inequalities in two variables

and

satisﬁed by the ﬁve quantities

[

](

[

](

[

](

[

](

), and

[

](

t, u

After this, the commutation ﬁelds

, which generate the higher order variations,

are deﬁned. They are ﬁve: the vectorﬁeld

which is tranversal to the

, the

ﬁeld

= (1 +

)

along the generators of the

, and the three rotation ﬁelds

= 1

3 which are tangential to the

t,u

sections. The latter are deﬁned to

be Π

◦

= 1

3, where the

◦

= 1

3 are the generators of spatial rotations

associated to the background Minkowskian structure, while Π is the

-orthogonal

projection to the

t,u

. Expressions for the deformation tensors

(

)

(

)

, and

(

)

= 1

3 are then derived, which show that these depend on the acoustical

entities

and

. The last however depend in addition on the derivatives of the

restrictions to the surfaces

t,u

of the spatial rectangular coordinates

= 1

as well as on the derivatives of the

with respect to

and

, that is, on the

rectangular components

and

of the vectorﬁelds

and

The higher order variations satisfy inhomogeneous wave equations in the acous-

tical spacetime, the source functions depending on the deformation tensors of the
commutation ﬁelds. These source functions give rise to error integrals, that is to
spacetime integrals of contributions to the divergence of the energy currents.

The expressions for the source functions and the associated error integrals show

that the error integrals corresponding to the energies of the

+ 1st order variations

contain the

th order derivatives of the deformation tensors, which in turn contain

the

th order derivatives of

and

+ 1st order derivatives of

. Thus to achieve

closure, we must obtain estimates for the latter in terms of the energies of up to

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

595

the

+ 1st order variations. Now, the propagation equations for

and

give

appropriate expressions for

and

Lµ

. However, if these propagation equations,

which may be thought of as ordinary diﬀerential equations along the generators
of the

, are integrated with respect to

to obtain the acoustical entities

and

themselves and their spatial derivatives are then taken, a loss of one degree of

diﬀerentiability would result and closure would fail. We overcome this diﬃculty in
the case of

by considering the propagation equation for

. We show that by

virtue of a wave equation for

, which follows from the wave equations satisﬁed by

the ﬁrst variations corresponding to the spacetime translations, the principal part
on the right-hand side of this propagation equation can be put into the form

−

of a derivative of a function

−

with respect to

. This function is then brought

to the left-hand side, and we obtain a propagation equation for

+ ˇ

. In this

equation ˆ

, the trace-free part of

enters, but the propagation equation in question

is considered in conjunction with the Codazzi equation, which constitutes an elliptic
system on each

t,u

for ˆ

, given tr

. We thus have an ordinary diﬀerential equation

along the generators of

coupled to an elliptic system on the

t,u

sections. More

precisely, the propagation equation which is considered at the same level as the
Codazzi equation is a propagation equation for the

t,u

1-form

µd

, which is

a consequence of the equation just discussed. To obtain estimates for the angular
derivatives of

of order

we similarly consider a propagation equation for the

t,u

1-form:

(

...i

)

µd

(

...R

) +

(

...R

)

In the case of

the aforementioned diﬃculty is overcome by considering the prop-

agation equation for

/ µ

, where

/ µ

is the Laplacian of the restriction of

to the

t,u

. We show that by virtue of a wave equation for

T σ

, which is a diﬀerential

consequence of the wave equation for

, the principal part on the right-hand side

of this propagation equation can again be put into the form

of a derivative

of a function ˇ

with respect to

. This function is then likewise brought to the

left-hand side, and we obtain a propagation equation for

/ µ

−

. In this equation

, the trace-free part of the Hessian of the restriction of

to the

t,u

enters, but

the propagation equation in question is considered in conjunction with the elliptic
equation on each

t,u

for

, which the speciﬁcation of

/ µ

constitutes. Again we

have an ordinary diﬀerential equation along the generators of

coupled to an

elliptic equation on the

t,u

sections. To obtain estimates of the spatial derivatives

of order

+2 of which

are derivatives with respect to

, we similarly consider

a propagation equation for the function:

(

...i

−

)

m,l

−

µR

−

...R

(

)

/ µ

−

...R

(

)

This allows us to obtain estimates for the top order spatial derivatives of

, of which

at least two are angular derivatives. A remarkable fact is that the missing top order
spatial derivatives do not enter the source functions, hence do not contribute to the
error integrals. In fact it is shown that the only top order spatial derivatives of the
acoustical entities entering the source functions are those in the 1-forms

(

...i

)

and the functions

(

...i

−

)

m,l

−

The paradigm of an ordinary diﬀerential equation along the generators of a char-

acteristic hypersurface coupled to an elliptic system on the sections of the hypersur-
face as the means to control the regularity of the entities describing the geometry of
the characteristic hypersurface and the stacking of such hypersurfaces in a foliation

596

DEMETRIOS CHRISTODOULOU

was ﬁrst encountered in the work [C-K] on the stability of the Minkowski space. It is
interesting to note that this paradigm does not appear in space dimension less than
three. In the case of the work on the stability of the Minkowski space however, in
contrast to the present case, the gain of regularity achieved in this treatment is not
essential for obtaining closure, because there is room of one degree of diﬀerentia-
bility. This is due to the fact that the Einstein equations of general relativity arise
from a Lagrangian which is quadratic in the derivatives of the unknown functions,
in contrast to the equations of ﬂuid mechanics or, more generally, of continuum
mechanics, which in the Lagrangian picture are equations for a mapping of space-
time into the material manifold, each point of which represents a material particle,
the Lagrangian not depending quadratically on the diﬀerential of this mapping (see
[Ch2]). As a consequence, the metric determining the causal structure depends in
continuum mechanics on the derivatives of the unknowns rather than only on the
unknowns themselves.

In the present case, the appearance of the factor of

, which vanishes where

shocks originate in front of

...R

and

−

...R

(

)

/ µ

in the deﬁnitions

(

...i

)

and

(

...i

−

)

m,l

−

above, makes the analysis far more delicate. This

is compounded with the diﬃculty of the slower decay in time, which the addition
of the terms

−

...R

and

−

...R

(

)

forces. The analysis requires a

precise description of the behavior of

itself, given by certain propositions, and a

separate treatment of the condensation regions, where shocks are to form, from the
rarefaction regions, the terms referring not to the ﬂuid density but rather to the
density of the stacking of the wave fronts. To overcome the diﬃculties the following
weight function is introduced:

(44)

m,u

(

) = min

m,u

(

)

m,u

(

) = min

where Σ

is the exterior of

t,u

in Σ

, and the quantities

[

](

[

](

[

](

[

](

), and

[

](

t, u

) corresponding to the highest order variations are

weighted with a power, 2

, of this weight function. The following lemma then plays

a crucial role here as well as in the proof of the main theorem, where everything
comes together. Let

(45)

(

) = max

−

(

Lµ

)

−

a,u

−

m,u

(

)

(

)

Then under certain bootstrap assumptions in the past of Σ

, for any constant

≥

there is a positive constant

independent of

and

such that for all

∈

, s

]

we have:

(46)

a,u

(

)

≤

−

m,u

(

)

Now, estimates for the derivatives of the spatial rectangular coordinates

with

respect to the commutation ﬁelds must also be obtained, the derivatives of the

with respect to the vectorﬁelds ˆ

and

being the spatial rectangular components

and

of these vectorﬁelds. Here ˆ

−

is the vectorﬁeld of unit magnitude

with respect to

corresponding to

. Thus, although the argument depends mainly

on the causal structure of the acoustical spacetime, the underlying Minkowskian
structure, to which the rectangular coordinates belong, has a role to play as well,
and it is the estimates in question which analyze the mutual relationship of the
two structures. The derivation of these estimates occupies a considerable part of

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

597

the work. The required estimates for the deformation tensors of the commutation
ﬁelds in terms of the acoustical entities are then obtained.

After this, the acoustical assumptions on which the previous results depend are

established, using the method of continuity, on the basis of the ﬁnal set of bootstrap
assumptions, which consists only of pointwise estimates for the variations up to
certain order. Then, the estimates for up to the next to the top order angular
derivatives of

and spatial derivatives of

are derived. These, when substituted

in the estimates established earlier, give control of all quantities involved in terms
of estimates for the variations. A fundamental role is played by the propositions
which establish the coercivity hypotheses on which the previous results depend.
These propositions roughly speaking show that for any covariant

t,u

tensorﬁeld

the sum

bounds pointwise

/ϑ

and that if

is any

t,u

-tangential

vectorﬁeld and

any covariant

t,u

tensorﬁeld, then we can bound pointwise

in terms of the

and the

= [

, X

We then analyze the structure of the terms containing the top order spatial

derivatives of the acoustical entities, showing that these can be expressed in terms
of the 1-forms

(

...i

)

and the functions

(

...i

−

)

m,l

−

. These terms are shown

to contribute

borderline error integrals

, the treatment of which is the main source of

diﬃculty in the problem. These borderline integrals are all proportional to the con-
stant

mentioned above, hence are absent in the case

= 0. We should make clear

here that the only variations which are considered up to this point are the varia-
tions arising from the ﬁrst order variations corresponding to the group of spacetime
translations. In particular the ﬁnal bootstrap assumption involves only variations
of this type, and each of the ﬁve quantities

[

]

(

[

]

(

[

]

(

[

]

(

and

[

]

(

t, u

), which together control the solution, is deﬁned to be the sum of

the corresponding quantity

[

](

[

](

[

](

[

](

), and

[

](

t, u

over all variations

of this type, up to order

. To estimate the borderline in-

tegrals, however, we introduce an additional assumption which concerns the ﬁrst
order variations corresponding to the scaling or dilation group and to the rotation
group and the second order variations arising from these by applying the commu-
tation ﬁeld

. This assumption is later established through energy estimates of

order 4 arising from these ﬁrst order variations and derived on the basis of the ﬁnal
bootstrap assumption, just before the recovery of the ﬁnal bootstrap assumption
itself. It turns out that the borderline integrals all contain the factor

T ψ

, where

= 0

3 are the ﬁrst variations corresponding to spacetime translations

and the additional assumption is used to obtain an estimate for sup

−

T ψ

in terms of sup

−

Lµ

, which involves on the right the factor

−

. Upon

substituting this estimate in the borderline integrals, the factors involving

cancel,

and the integrals are estimated using the inequality (46). The above is an outline
of the main steps in the estimation of the borderline integrals associated to the
vectorﬁeld

. The estimation of the borderline integrals associated to the vector-

ﬁeld

is however still more delicate. In this case we ﬁrst perform an integration

by parts on the outgoing characteristic hypersurfaces

, obtaining hypersurface

integrals over Σ

and Σ

and another spacetime volume integral. In this integra-

tion by parts the terms, including those of lower order, must be carefully chosen to
obtain appropriate estimates, because here the long time behavior, as well as the
behavior as

tends to zero, is critical. Another integration by parts, this time on

the surfaces

t,u

, is then performed to reduce these integrals to a form which can

598

DEMETRIOS CHRISTODOULOU

be estimated. The estimates of the hypersurface integrals over Σ

are the most

delicate (the hypersurface integrals over Σ

only involve the initial data) and re-

quire separate treatment of the condensation and rarefaction regions, in which the
properties of the function

, established by the previous propositions, all come into

play.

In proceeding to derive the energy estimates of top order,

+ 2, the power

of the weight

m,u

(

) is chosen suitably large to allow us to transfer the terms

contributed by the borderline integrals to the left-hand side of the inequalities
resulting from the integral identities associated to the multiplier ﬁelds

and

. The argument then proceeds along the lines of that of the fundamental energy

estimate, but is more complex because here we are dealing with weighted quantities.
Once the top order energy estimates are established, we revisit the lower order
energy estimates using at each order the energy estimates of the next order in
estimating the error integrals contributed by the highest spatial derivatives of the
acoustical entities at that order. We then establish a descent scheme which yields,
after ﬁnitely many steps, estimates for the ﬁve quantities

[

]

(

[

]

(

[

]

(

[

]

(

), and

[

]

(

t, u

), for

+ 1

−

[

], where [

] is the integral part of

, in

which weights no longer appear.

It is these unweighted estimates which are used to close the bootstrap argument

by recovering the ﬁnal bootstrap assumption. This is accomplished by the method
of continuity through the use of the isoperimetric inequality on the wave fronts

t,u

and leads to the main theorem. This theorem shows that there is another diﬀer-
ential structure, that deﬁned by the acoustical coordinates

t, u, ϑ

, the

const.

coordinate lines corresponding to the bicharacteristic generators of each

, such

that relative to this structure the maximal classical solution extends smoothly to the
boundary of its domain. This boundary contains however a singular part where the
function

vanishes; hence, in these coordinates, the acoustical metric

degener-

ates. With respect to the standard diﬀerential structure induced by the rectangular
coordinates

in Minkowski spacetime, the solution is continuous but not diﬀeren-

tiable on the singular part of the boundary, the derivative ˆ

∂

blowing up as

we approach the singular boundary. Thus, with respect to the standard diﬀerential
structure, the acoustical metric

is everywhere in the closure of the domain of the

maximal solution non-degenerate and continuous, but it is not diﬀerentiable on the
singular part of the boundary of this domain, while with respect to the diﬀerential
structure induced by the acoustical coordinates

is everywhere smooth, but it is

degenerate on the singular part of the boundary.

After the proof of the main theorem, we establish a general theorem which

gives sharp suﬃcient conditions on the initial data for the formation of a shock
in the evolution. The proof of this theorem is through the propositions describing
the properties of the function

and is based on the study of the evolution with

respect to

of the mean value on the sections

t,u

of each outgoing characteristic

hypersurface

of the quantity

(47)

= (1

−

)

−

(

−

)

where

and

are respectively the volume per particle and pressure in the sur-

rounding constant state. Here

and

are the functions:

(48)

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

599

and

the 1-form:

(49)

= ˙

corresponding to any ﬁrst order variation ( ˙

) of a general solution (

p, s, u

) of

the equations of motion. We consider in particular the variation corresponding to
time translations. The proof of the theorem uses the estimate provided by the
spacetime integral

(

t, u

) associated to this variation. Certain crucial integrations

by parts on the

t,u

sections as well as on

itself are performed, in which the

structure of

as a characteristic hypersurface comes into play. The theorem also

gives a sharp upper bound on the time interval required for the onset of shock
formation.

The last part of the work is concerned with the structure of the boundary of the

domain of the maximal classical solution and the behavior of the solution at this
boundary. The boundary of the domain of the maximal solution consists of a regular
part and a singular part. Each component of the regular part

is an incoming

characteristic hypersurface with a singular past boundary. The singular part of
the boundary of the domain of the maximal solution is the locus of points where
the inverse density of the wave fronts vanishes. It is the union

∂

−

, where

each component of

∂

−

is a smooth embedded surface in Minkowski spacetime,

the tangent plane to which at each point is contained in the exterior of the sound
cone at that point. On the other hand each component of

is a smooth embedded

hypersurface in Minkowski spacetime, the tangent hyperplane to which at each
point is contained in the exterior of the sound cone at that point, with the exception
of a single generator of the sound cone, which lies on the hyperplane itself. The
past boundary of a component of

is the corresponding component of

∂

−

. The

latter is at the same time the past boundary of a component of

In the monograph [Ch1] we ﬁrst establish a proposition which describes the

singular part of the boundary of the domain of the maximal classical solution from
the point of view of the acoustical spacetime. This singular part has the intrinsic
geometry of a regular null hypersurface in a regular spacetime and, like the latter, is
ruled by invariant curves of vanishing arc length. On the other hand, the extrinsic
geometry of the singular boundary is that of a spacelike hypersurface which becomes
null at its past boundary. The main result of the last part of the work is the
trichotomy theorem.

This theorem shows that at each point

of the singular

boundary, the past sound cone in the cotangent space at

degenerates into two

hyperplanes intersecting in a 2-dimensional plane. We thus have a trichotomy of
the bicharacteristics, or null geodesics of the acoustical metric, ending at

, into

the set of outgoing null geodesics ending at

, which corresponds to one of the

hyperplanes; the set of incoming null geodesics ending at

, which corresponds to

the other hyperplane; and the set of the remaining null geodesics ending at

, which

corresponds to the 2-dimensional plane. The intersection of the past characteristic
conoid of

(past null geodesic cone of the acoustical metric

) with any Σ

in the

past of

similarly splits into three parts, the parts corresponding to the outgoing

and to the incoming sets of null geodesics ending at

being embedded discs with

a common boundary, an embedded circle, which corresponds to the set of the
remaining null geodesics ending at

All outgoing null geodesics ending at

have

the same tangent vector at

. This vector is then an invariant characteristic vector

associated to the singular point

. This striking result is in fact the reason why the

considerable freedom in the choice of the acoustical function does not matter in the

600

DEMETRIOS CHRISTODOULOU

end, for, considering the transformation from one acoustical function to another, we
show that the foliations corresponding to diﬀerent families of outgoing characteristic
hypersurfaces have equivalent geometric properties and degenerate in precisely the
same way on the same singular boundary.

The monograph [Ch1] then proceeds to give a detailed description of the bound-

ary of the domain of the maximal classical solution from the point of view of
Minkowski spacetime. Now, the maximal classical solution is the physical solution
of the problem up to

∂

−

, but not up to

. In the last part of the monograph

[Ch1] the problem of the physical continuation of the solution is set up as the

shock

development problem

. This problem is associated to each component of

∂

−

, and

its solution requires the construction of a hypersurface of discontinuity

, lying in

the past of the corresponding component of

but having the same past boundary

as the latter, namely the given component of

∂

−

. It then follows that the tan-

gent hyperplanes to

and

coincide along

∂

−

. The maximal classical solution

provides the right boundary conditions at

∂

−

, as well as a barrier at

. The

actual treatment of the shock development problem and the subsequent shock in-
teractions shall be the subject of a follow up monograph. The monograph [Ch1]
concludes with a derivation of a formula for the jump in vorticity across

, which

shows that while the ﬂow is irrotational ahead of the shock, it acquires vorticity
immediately behind, the vorticity vector being tangential to the shock front and
associated to the gradient along the shock front of the entropy jump.

About the author

Demetrios Christodoulou is professor of mathematics and physics at the ETH in

Zurich. He is a MacArthur Fellow, winner of the AMS Bˆ

ocher Prize, and member

of the American Academy of Arts and Sciences and the European Academy of
Sciences.

References

[Be]

Bernoulli, D.

Hydrodynamica

, Argentorati (1738).

[Bo]

Boltzmann, L. “ ¨

Uber die Bezeihung zwischen dem zweiten Hauptsatzes der mechanis-

chen W¨

armetheorie und der Wahrscheinlichkeitsrechnung respective den S¨

atzen ¨

uber das

W¨

armegleichgewicht”,

Wien Ber.

, 73 (1877).

[Ch1]

Christodoulou, D.

The Formation of Shocks in 3-Dimensional Fluids

, EMS Monographs

in Mathematics, EMS Publishing House, 2007. MR2284927

[Ch2]

Christodoulou, D.

The Action Principle and Partial Diﬀerential Equations

, Ann. Math.

Stud.

146

, Princeton University Press, 2000. MR1739321 (2003a:58001)

[C-K]

Christodoulou, D., and Klainerman, S.

The Global Nonlinear Stability of the Minkowski

Space

, Princeton Mathematical Series

, Princeton University Press, 1993. MR1316662

(95k:83006)

[Cl1]

Clausius, R. “ ¨

Uber die bewegende Kraft der W¨

arme”,

Annalen der Physik und Chemie

, 368-397, 500-524 (1850).

[Cl2]

Clausius, R. “ ¨

Uber verschiedene f¨

u die Anwendung bequeme Formen der Hauptgleichungen

der mechanischen W¨

armetheorie”,

Annalen der Physik und Chemie

125

, 353-400 (1865).

[DA]

D’Alembert, J.-B. le R. “Recherches sur la courbe que forme une corde tendu¨

e mise en

vibration”,

M´

em. Acad. Sci. Berlin

, 214-219 (1849).

[Ei]

Einstein, A. “Zur Electrodynamic bewegter K¨

orper”,

Annalem der Physik

, 891-921

(1905).

[Eu1]

Euler, L. “Principes generaux de l’etat d’equilibre des ﬂuides”,

M´

emoires de l’Academie

des Sciences de Berlin

, 217-273 (1757).

THE EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

601

[Eu2]

Euler, L. “Principes generaux du mouvement des ﬂuides”,

M´

emoires de l’Academie des

Sciences de Berlin

, 274-315 (1757).

[Eu3]

Euler, L. “Continuation des recherrches sur la theorie du mouvement des ﬂuides”,

M´

emoires de l’Academie des Sciences de Berlin

, 316-361 (1757).

[Eu4]

Euler, L. “Principia motus ﬂuidorum”,

Novi Commentarii Academiae Scientiarum

Petropolitanae

, 271-311 (1761).

[Eu5]

Euler, L. “De motu ﬂuidorum a diverso caloris gradu oriundo”,

Novi Commentarii

Academiae Scientiarum Petropolitanae

, 232-267 (1767).

[Eu6]

Euler, L. “Sectio secunda de principiis motus ﬂuidorum”,

Novi Commentarii Academiae

Scientiarum Petropolitanae

, 270-386 (1770).

[Eu7]

Euler, L. “Sectio tertia de motu ﬂuidorum lineari potissimum aquae”,

Novi Commentarii

Academiae Scientiarum Petropolitanae

, 219-360 (1771).

[Eu8]

Euler, L. “Sectio quarta de motu aeris in tubis”,

Novi Commentarii Academiae Scien-

tiarum Petropolitanae

, 281-425 (1772).

[F]

Friedrichs, K.O. “Symmetric hyperbolic linear diﬀerential equations”,

Comm. Pure &

Appl. Math.

, 345-392 (1954). MR0062932 (16:44c)

[F-L]

Friedrichs, K.O., and Lax, P.D. “Systems of Conservation Equations with a Convex Ex-
tension”,

Proc. Nat. Acad. Sci. USA

, 1686-1688 (1971). MR0285799 (44:3016)

[Gl]

Glimm, J. “Solutions in the large for nonlinear hyperbolic systems of equations”,

Comm.

Pure & Appl. Math.

, 697-715 (1965). MR0194770 (33:2976)

[He]

Helmholtz, H. v.

Uber die Erhaltung der Kraft

, G. Reimer, Berlin, 1847.

[Hu]

Hugoniot, H. “Sur la propagation du mouvement dans les corps et sp´

ecialement dans les

gaz parfaits”,

Journal de l’´

ecole polytechnique

, 1-125 (1889).

[J]

John, F. “Formation of singularities in one-dimensional non-linear wave propagation”,

Comm. Pure & Appl. Math.

, 377-405 (1974). MR0369934 (51:6163)

[Kr]

Kruˇ

zhkov, S. N. “First order quasilinear equations in several independent variables”,

Math.

USSR Sbornik

, No. 2 (1970).

[K-T]

Keller, J.B. and Ting, L. “Periodic vibrations of systems governed by nonlinear partial
diﬀerential equations”,

Comm. Pure & Appl. Math.

169

, 371-420 (1966). MR0205520

(34:5347)

[La]

Laplace, P. S. “Sur la vitesse du son dans l’air et dans l’eau”,

Ann. de Chim. et de Phys.

iii, 238 (1816).

[Ln]

Landau, L.D. (1944). See Landau, L.D., and Lifschitz, E.M.

Fluid Mechanics

, 2nd edition,

Oxford, Pergamon Press, 1987, pages 332-333. MR961259 (89i:00006)

[Lx1]

Lax, P.D. “Shock waves and entropy”, pp. 603-634 in

Contributions to Nonlinear Func-

tional Analysis

, edited by E. Zarantonello, Academic Press, 1971. MR0393870 (52:14677)

[Lx2]

Lax, P.D. “Development of singularities of solutions of non-linear hyperbolic partial dif-
ferential equations”,

J. Math. Phys.

, 611-613 (1964). MR0165243 (29:2532)

[Lx3]

Lax, P.D.

Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock

Waves

, Regional Conf. Series in Appl. Math.

, SIAM, 1973. MR0350216 (50:2709)

[Ma1] Majda, A.

Compressible Fluid Flow and Systems of Conservation Laws in Several Space

Variables

, Appl. Math. Sci.

, Springer-Verlag, 1984. MR748308 (85e:35077)

[Ma2] Majda, A.

The Stability of Multi-Dimensional Shock Fronts - A New Problem for Linear

Hyperbolic Equations

, Mem. Amer. Math. Society

275

, 1983.

[Ma3] Majda, A.

The Existence of Multi-Dimensional Shock Fronts

, Mem. Amer. Math. Soc.

285

, 1983. MR699241 (85f:35139)

[M-B] Majda, A., and Bertozzi, A.

Vorticity and Incompressible Flow

, Cambridge Texts in Ap-

plied Mathematics, Cambridge University Press, 2002. MR1867882 (2003a:76002)

[Mi]

Minkowski, H. “Raum und Zeit”, Address at the 80th Assembly of German Natural Sci-
entists and Physicians, Cologne (1908).

[Mo]

Morawetz, C. “The decay of solutions of the exterior initial-boundary value problem for the
wave equation”,

Comm. Pure & Appl. Math.

, 561-568 (1961). MR0132908 (24:A2744)

[N]

Noether, E. “Invariante Variationsprobleme”,

Nach. Ges. Wiss. G¨

ottingen, Math.-Phys.

Kl.

1918

, 235-257 (1918).

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