Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Approximations
All manipulations so far have been exact and, in
fact, rigorous (or can be made so with little extra
effort). Now we turn to a sequence of approxima-
tions that have been used by physicists to develop a
quantitative understanding of weakly disordered
quantum dots, wires, films, etc. While physically
satisfactory, not all of these approximations are
under full mathematical control. We will briefly
comment on their validity as we go along.
Saddle-Point Manifold
We continue to consider G
(2)
kl
(E þ i", E i") and
focus on E = 0 (the center of the energy band) for
simplicity. By varying the exponent on the right-
hand side of [14] and setting the variation to zero
one obtains, for t
1
= t
2
= 0,
X
j
w
ij
sQ
j
s NQ
1
i
¼ 0
which is called the saddle-point equation.
Let us now assume translational invariance,
w
ij
= f (ji jj). Then, if =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N=
j
w
ij
p
, the saddle-
point equation has i-independent solutions of the
form
Q
i
¼
q
BB
0
0 q
FF
where for q
FF
there are three possibilities: two
isolated points q
FF
= 1 (unit matrix) coexist with
a manifold
q
FF
¼
cos
1
sin
1
e
i
1
sin
1
e
i
1
cos
1
½15
which is two-dimensional; whereas the solution
space for q
BB
consists of a single connected
2-manifold:
q
BB
¼
cosh
0
sinh
0
e
i
0
sinh
0
e
i
0
cosh
0
½16
The solutions q
FF
= 1 are usually discarded in the
physics literature. (The a rgument is that they break
supersymmetry and therefore get suppressed by
fermionic zero modes. For the simpler case of the
one-point function [1] and in three space dimen-
sions, such suppression has recently been proved by
Disertori, Pinson, and Spencer.) Other solutions for
q
BB
are ruled out by the requirement ReQ
i
> 0 for
the Scha¨fer–Wegner domain.
The set of matrices [16] and [15] – the ‘‘saddle-
point manifold’’ – is diffeom orphic to the product of
a 2-hyperboloid H
2
with a 2-sphere S
2
. Moving
along that manifold M := H
2
S
2
leaves the Q-field
integrand [14] unchanged (for z
1
= z
2
= t
1
= t
2
= 0).
One can actually anticipa te the existence of such a
manifold from the symmetries at hand. These are
most transparent in the starting point of the
formalism as given by the characteristic function
he
iK
H
i with
K
H
¼ð
’
1
; H’
1
Þð
’
2
; H’
2
Þþð
1
; H
1
Þ
þð
2
; H
2
Þ
The signs of this quadratic expression are what is
encoded in the signature matrix s = diag(1, 1, 1, 1)
(recall that the first two entries are forced by Im z
1
>
0 and Im z
2
< 0). The Hermitian form K
H
is
invariant under the product of two Lie groups:
U(1, 1) acting on the ’’s, and U(2) acting on the ’s.
This invariance gets transferred by the formalism to
the Q-side; the saddle-point manifold M is in fact an
‘‘orbit’’ of the group action of G := U(1, 1) U(2)
on the Q-field. In the language of physics, the
degrees of freedom of M correspond to the Gold-
stone bosons of a broken symmetry.
K
H
also has some supersymmetries, mixing ’’s
with ’s. At the infinitesimal level, these combine
with the generators of G to give a Lie superalgebra
of symmetries g := u (1, 1j2). One therefore expects
some kind of saddle-point supermanifold, say M ,on
the Q-side.
M can be constructed by extending the above
solution q
0
:= diag(q
BB
, q
FF
) of the dimensionless
saddle-point equation sqs = q
1
to the full 4 4
supermatrix space. Putting q = q
0
þ q
1
with
q
1
¼
0 q
BF
q
FB
0
and linearizing in q
1
, one gets
sq
1
s ¼q
1
0
q
1
q
1
0
½17
The solution space of this linear equation for q
1
has
dimension 4 for all q
0
2 M. Based on it, one expects
four Goldstone fermions to emerge along with the
four Goldstone bosons of M.
For the simple case under consideration, one can
introduce local co ordinates and push the analysis to
nonlinear order, but things get quickly out of hand
(when done in this way) for more challenging,
higher-rank cases. Fortunately, there exists an
alternative, coordinate-independent approach, as
the mathematical object to be constructed is
completely determined by symmetry!
Riemannian Symmetric Superspace
The linear equation [17] associates with every point
x 2 M a four-d imensional vector space of solutions,
158 Supersymmetry Methods in Random Matrix Theory
V
x
. As the point x moves on M the vector spa ces V
x
turn and twist; thus, they form what is called a
vector bundle V over M. (The bundle at hand turns
out to be nontrivial, i.e., there exists no global
choice of coordinates for it.)
A section of V is a smooth mapping : M !V
such that (x) 2 V
x
for all x 2 M. The sections of
V are to be multiplied in the exterior sense, as they
represent anticommuting degrees of freedom;
hence the proper object to consider is the exterior
bundle, ^V.
It is a beautiful fact that there exists a unique
action of the Lie superalgebra g on the sections of
^V by first-order differential operators, or deriva-
tions for short. (Be advised however that this
canonical g -act ion is not well known in physics or
mathematics.)
The manifold M is a symmetric space, that is, a
Riemannian manifold with G-invariant geometry.
Its metric tensor, g, uniquely extends to a second-
rank tensor field (still denoted by g) which maps
pairs of derivations of ^ V to sections of ^V, and is
invariant with respect to the g -action. This collec-
tion of objects – the symmetric space M, the
exterior bundle ^V over it, the action of the Lie
superalgebra g on the sections of ^V, and the
g-invariant second-rank tensor g – form what
the author calls a ‘‘Riemannian symmetric super-
space,’’ M .
Nonlinear Sigma Model
According to the Landau–Ginzburg–Wilson (LGW)
paradigm of the theory of phase transitions, the
large-scale physics of a statistical mechanical system
near criticality is expected to be controlled by an
effective field theory for the long-wavelength excita-
tions of the order parameter of the system.
Wegner is credited for the profound insight that
the LGW paradigm applies to the random matrix
situation at hand, with the role of the order
parameter being taken by the matrix Q. He argued
that transport observables (such as the electrical
conductivity) are governed by slow spatial variations
of the Q-field inside the saddle-point manifold.
Efetov skilfully implemented this insight in a super-
symmetric variant of Wegner’s method.
While the direct construction of the effective
continuum field theory by gradient expansion of
[14] is not an entirely easy task, the outcome of the
calculation is predetermined by symmetry. On
general grounds, the effective field theory has to be
a nonlinear sigma model for the Goldstone bosons
and fermions of M :if{
A
} are local coordinates for
the bundle V with metri c g
AB
() = g(@=@
A
, @=@
B
),
the action functional is
S ¼
Z
d
d
x@
A
g
AB
ðÞ@
B
The coupling param eter has the physic al meaning
of bare (i.e., unrenormalized) conductivity. In the
present model = NW
2
a
2d
, where W is essentially
the width of the band random matrix in units of the
lattice spacing a (the short-distance cutoff of the
continuum field theory). S is the effective a ction in
the limit z
1
= z
2
. For a finite frequency ! = z
1
z
2
,a
symmetry-breaking term of the form i!
R
d
d
xf (),
where = N()
1
a
d
is the local density of states,
has to be added to S.
By perturbative renormalization group analysis, that
is, by integrating out the rapid field fluctuations, one
finds for d = 2that decreases on increasing the cutoff
a. This property is referred to as ‘‘asymptotic freedom’’
in field theory. On its basis one expects exponentially
decaying correlations, and hence localization of all
states, in two dimensions. However, a mathematical
proof of this conjecture is not currently available.
In three dimensions and for a sufficiently large bare
conductivity, the renormalization flow goes toward
the metallic fixed point ( !1), where G-symmetry
is broken spontaneously. A rigorous proof of this
important conjecture (existen ce of disordered metals
in three space dimensions) is not available either.
Zero-Mode Approximation
For a system in a box of linear size L, the cost of
exciting fluctuations in the sigma model field is
estimated as the Thouless energy E
Th
= =L
2
. In the
limit of small frequency, j!jE
Th
, the physical
behavior is dominated by the constant modes
A
(x) =
A
(independent of x). By computing the
integral over these modes, Efetov found the energy-
level correlations in the small-frequency limit to be
those of the GUE.
See also: Random Matrix Theory in Physics; Symmetry
Classes in Random Matrix Theory.
Further Reading
Berezin FA (1987) Introduction to Superanalysis. Dordrecht: Reidel.
Disertori M, Pinson H, and Spencer T (2002) Density of states of
random band matrices. Communication in Mathematical
Physics 232: 83–124.
Efetov KB (1997) Supersymmetry in Disorder and Chaos.
Cambridge: Cambridge University Press.
Fyodorov YV (2002) Negative moments of characteristic poly-
nomials of random matrices: Ingham–Siegel integral as an
Supersymmetry Methods in Random Matrix Theory 159
alternative to Hubbard–Stratonovich transformation. Nuclear
Physics B 621: 643–674.
Mirlin AD (2000) Statistics of energy levels and eigen functions in
disordered systems. Physics Reports 326: 260–382.
Scha¨fer L and Wegner F (1980) Disordered system with n orbitals
per site: Lagrange formulation, hyperbolic symmetry, and
Goldstone modes. Zeitschrift fu¨r Physik B 38: 113–126.
Wegner F (1979) The mobility edge problem: continuous
symmetry and a conjecture. Zeitschrift fu¨r Physics B 35:
207–210.
Zirnbauer MR (1996) Riemannian symmetric superspaces and
their origin in random matrix theory. Journal of Mathematical
Physics 37: 4986–5018.
Symmetric Hyperbolic Systems and Shock Waves
S Kichenassamy, Universite
´
de Reims
Champagne-Ardenne, Reims, France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Many systems of partial differential equations
arising in mathematical physics and differential
geometry are quasilinear: the top-order derivatives
enter only linearly. They may be cast in the form
of first-order systems by introducing, if needed,
derivatives of the unknowns as additional unknowns.
For such systems, the theory of symmetric–hyperbolic
(SH) systems provides a unified framework for
proving the local existence of smooth solutions if
the initial data are smooth. It is also convenient for
constructing numerical schemes, and for studying
shock waves. Despite what the name suggests, the
impact of the theory of SH systems is not limited to
hyperbolic problems, two examples being Tricomi’s
equation, and equations of Cauchy–Kowalewska
type.
Application of the SH framework usually requires a
preliminary reduction to SH form (‘‘symmetrization’’).
After comparing briefly the theory of SH systems
with other functional-analytic approaches, we col-
lect basic definitions and notation. We then present
two general rules, for symmetrizing conservation
laws and strictly hyperbolic equations, respectively.
We next turn to special features possessed by linear
SH systems, and give a general procedure to prove
existence, which covers both linear and nonlinear
systems. We then summarize those results on shock
waves, and on blow-up singularities, which are
related to SH structure. Examples and applications
are collected in the last section.
The advantages of SH theory are: a standardized
procedure for constructing solutions; the availability
of standard numerical schemes; a natural way to
prove that the speed of propagation of support is
finite. On the other hand, the symmetrization
process is sometimes ad hoc, and does not respect
the phy sical or geometric nature of the unknowns;
to obviate this defect to some extent, we remark that
symmetrizers may be viewed as introducing a new
Riemannian metric on the space of unknow ns. The
search for a comprehensive criterion for identifying
equations and boundary conditions compatible with
SH structure is still the object of current research.
The most important fields of application of the
theory today are general relativity and fluid
dynamics, including magnetohydrodynamics.
Context of SH Theory in Modern Terms
The basic reason why the theory works may be
summarized as follows for the modern reader; the
history of the subject is, however, more involved.
Let H be a real Hilbert space. Consider a linear
initial-value problem du=dt þ Au = 0; u(0) = u
0
2 H,
where A is unbounded, with domain D(A). By
Stone’s theorem, one can solve it in a generalized
sense, if the unbounded operator A satisfies A þ
A
= 0. This condition contains two ingredients: a
symmetry condition on A, and a maximality condi-
tion on D(A), which incorporate boundary condi-
tions (von Neumann, Friedrichs). Semigroup theory
(Hille and Yosida, Phillips, and many others)
handles more general operators A: it is possible to
solve this problem in the form u(t ) = S(t)u
0
for t > 0,
where {S(t)}
t0
is a continuous contraction semi-
group, if and only if (Au, u) 0, and equation x þ
Ax = y has a solution for every y in H (this is a
maximality condition on D(A)). One then says that
A is maximal monotone. For such operators, A þ
A
0. SH systems are systems Qu
t
þ Au = F,
satisfying two algebraic conditions ensuring for-
mally that A þ A
is bounded, and that Q is
symmetric and positive definite. This algebraic
structure enable s one to solve the problem directly,
without explicit reference to semigroup theory.
Precise definitions are given next.
We assume throughout that all coefficients,
nonlinearities, and data are smooth unless otherwise
specified.
160 Symmetric Hyperbolic Systems and Shock Waves
Definitions
Consider a quasilinear system
M
A
B
ðx; uÞ@
u
B
¼ N
A
ðx; uÞ½1
where u = (u
A
)
A = 1,..., m
, x = (x
)
= 0, ..., n
, and @
=
@=@x
. The compone nts of u may be real or
complex. We follow the summation convention on
repeated indices in different positions; x
0
= t may be
thought of as the evolution variable; we write
x = (t, x), with x = (x
1
, ..., x
n
). Indices A, B, ... run
from 1 to m, indices j, k, ... from 1 to n, and Greek
indices from 0 to n. The complex conjugate of u
A
is
written
u
A
.
Equation [1] is symmetrizable if there are func-
tions
AB
(x, u) such that
M
AB
:¼
AC
M
C
B
satisfies the condition M
AB
=
M
BA
for every .
It is symmetric if it is symmetrizable with
AB
=
AB
.
It is symmetric-hyperbo lic with respect to k
if it
is symmetric and if k
M
AB
is positive definite:
k
M
AB
A
B
> 0 for = (
A
) 6¼ 0.
Thus, a symmetrizer (
AB
) gives rise to a
Riemannian metric (k
AC
M
C
B
) on the space of
unknowns, independent of any Riemannian struc-
ture on x-space. The system is SH with respect to x
0
if k
=
0
.
The simplest class of SH systems is provided by
real semilinear systems of the form
A
0
ðxÞ@
t
u þ A
i
ðxÞ@
i
u ¼ Nðx; uÞ½2
where the A
are real symmetric matrices, A
0
is
symmetric and positive definite, and k
=
0
. Writ-
ing A
0
= P
2
, with P symmetric and positive definite,
one finds that v = Pu solves a SH system with A
0
= I
(identity matrix).
Conservation laws (with ‘‘reaction’’ or ‘‘source’’
term N
A
) are usually defined as quasilinear systems
of the form
@
f
A
ðx; uÞ¼N
A
ðx; uÞ½3
They are common in fluid dynamics and combus -
tion. They are limiting cases of nonlinear diffusion
equations of the typical form
@
f
A
ðx; uÞ¼N
A
ðx; u Þþ"@
j
ðB
Ajk
B
@
k
u
B
Þ½4
The determination of the form of the coefficients
B
Ajk
B
is a nontrivial modeling issue; they may reflect
varied physical processes such as heat conduction,
viscosity, or bulk viscosity. They may depend on x,
u, and the derivatives of u. The simplest case is
B
Ajk
B
= D
jk
A
B
with (D
jk
) diagonal. Some authors
require the symmetry condition
AC
B
Cjk
B
¼
BC
B
Ckj
A
½5
Equations in which f
A
= u
A
0
are called reaction–
diffusion equations; they arise in physical a nd
biological problems in which chemical reactions
and diffusion phenomena are combined, and in
population dynamics.
A conservation law is symmetric if and only if
@f
A
=@u
B
is symmetric in A and B, which means
that there are, locally, functions g
(x, u) such that
f
A
= @g
=@u
A
.
A more fundamental derivation of conservation
laws would take us beyond the scope of this survey.
Symmetrization
Two general procedures for symmetrization are
available: one for conservation laws, the other for
semilinear strictly hyperbolic problems.
Conservation Laws with a Convex Entropy
Consider, for simplicity, a conservation law of
the f orm
@
t
u
A
þ @
j
f
Aj
ðuÞ¼0 ½6
We, therefore, assume that the f
A
= f
A
(u) and
f
A0
(u) = u
A
. We show that the following three
statements are equivalent locally: (1) there is a
strictly convex function U(u) such that
AB
=
@
2
U=@u
A
@u
B
is a symmetrizer; (2) eqn [6] imp lies a
scalar relation of the form @
U
= 0, with U
0
strictly
convex; and (3) there is a change of unknowns
v
A
= v
A
(u) such that the system satisfied by v = (v
A
)
is SH and (@v
A
=@u
B
) is positive definite.
In fluid dynamics, U
0
may sometimes be related
to specific entropy, and U
j
to entropy flux. For this
reason, if (2) holds, one says that U
0
is an entropy
for eqn [6], an d that (U
0
, U
j
) is an entropy pair. A
system may have several entropies in this sense; this
fact is sometimes useful in studying convergence
properties of approximate solutions of eqn [6].
Let us now prove the equivalence of these
properties.
Assume first (3): there are new unknowns
v
A
= v
A
(u) and functions g
(v) such that
f
A
= @g
=@v
A
. One finds that if eqn [6] holds,
@
U
¼ 0 wher e U
¼ v
A
@g
@v
A
g
½7
Furthermore, we have f
A0
= u
A
; therefore, eqn [7]
gives: U
0
= v
A
u
A
g
0
, so that U
0
is the Legendre
Symmetric Hyperbolic Systems and Shock Waves 161
transform (familiar from mechanics) of g
0
. It follows
that v
A
= @U
0
=@u
A
. Finally, (@v
A
=@u
B
) = (@
2
U
0
=
@u
A
@u
B
) is positive definite, and U
0
is strictly
convex.
We have proved that (3) implies (2). Next, assum e
(2): the entropy equality U
t
þ @
j
U
j
= 0 holds identi-
cally – and not just for the solution at hand. Using
[6], we find
0 ¼
@U
@u
A
@
t
u
A
þ
@U
j
@u
B
@
j
u
B
¼
@U
@u
A
@f
Aj
@u
B
þ
@U
j
@u
B
@
j
u
B
Assumption (2), therefore, means that U is strictly
convex and satisfies
@U
@u
A
@f
Aj
@u
B
¼
@U
j
@u
B
½8
Now, letting v
A
= @U=@u
A
and g
j
(v) = v
A
f
Aj
U
j
,
we find
@g
j
@v
A
¼ f
Aj
þ
@U
@u
A
@f
Aj
@u
C
@U
j
@u
C
@u
C
@v
A
¼ f
Aj
½9
Let
AB
= @
2
U=@u
A
@u
B
. Since U is strictly convex,
(
AB
) is positive definite, and so is its inverse. We
have now proved (3). Note that u
A
= @g
0
=@v
A
,
where g
0
(v) = u
A
v
A
U(u) is the Legendre trans-
form of U.
Next, using eqn [9], and the relations
AB
=
@v
A
=@u
B
= @v
B
=@u
A
, we find
0 ¼
AB
½@
t
u
B
þ @
j
f
Bj
¼
AB
@
t
u
B
þ
@v
B
@u
A
@
2
g
j
@v
B
@u
C
@
j
u
C
¼
AB
@
t
u
B
þ
@
2
g
j
@u
A
@u
C
@
j
u
C
which is SH; there fore,
AB
is a symmetrizer for eqn
[6], and (1) is proved. Thus, (2) implies (1) and (3).
Finally, if (1) holds,
AC
@f
Cj
=@u
B
is symmetric in
A and B. It follows that
@
@u
C
@U
@u
A
@f
Aj
@u
B
¼
AC
@f
Aj
@u
B
þ
@U
@u
A
@
2
f
Aj
@u
B
@u
C
is symmetric in B and C, so that there are, locally,
functions U
j
such that eqn [8] holds. Therefore,
(U, U
j
) is an entropy pair, and we see that (1)
implies (2).
This completes the proof of the equivalence of (1),
(2), and (3).
Strictly Hyperbolic Equations
Consider the scalar equation Pf = g(t, x), where P is
the linear operator
P ¼ @
N
t
X
N 1
j¼0
p
N j
ðt; xÞ@
j
t
of order N.Let=(1 )
1=2
, where is the Laplace
operator on the space variables. Then u = (u
A
), where
u
A
= @
A1
t
NA
f for A = 1, ..., N, solves a first-order
pseudodifferential system of the form
u
t
Lu ¼ G
If P is strictly hyperbolic, the principal symbol
a
1
(t, x, )ofL has a diagonal form with real
eigenvalues
j
(t, x, ), and there are projectors
p
j
(t, x, )(p
2
j
= p
j
) which commute with a
1
,suchthat
1 =
P
j
p
j
,anda
1
=
P
j
j
p
j
.Letr
0
=
P
j
p
j
p
j
,and
r
0
(D) the corresponding operator. Equation
r
0
ðDÞ@
t
u r
0
ðDÞLu ¼ r
0
ðDÞG
is formally SH in the following sense: r
0
is positive
definite and r
0
a
1
is Hermitian.
Linear Problems
Consider a linear system
Lu :¼ Qðt; xÞ@
t
u þ A
j
ðt; xÞ@
j
u þ Bðt; xÞu
¼ f ðt ; xÞ½10
We assume that Q and the A
j
are real and
symmetric, Q c with c positive, and all coeffi-
cients and their first-order derivatives are bounded.
Energy Identity
Multiplying the equation by u
T
(transpose of u), one
derives the ‘‘energy identity’’
@
t
ðu
T
QuÞþ@
j
ðu
T
A
j
uÞþu
T
Cu ¼ 2u
T
f ðt; xÞ½11
where C = 2B @
t
Q @
j
A
j
. C is not necessarily
positive. However, v := u exp (t) satisfies a linear
SH system for which C is positive definite if is
large enough.
Propagation of Support
A basic property of wave-like equations is finite
speed of propagation of support: if the right-hand
side vanish es, and if the solution at time 0 is
localized in the ball of radius r, then the solution
at time t is localized in the ball of radius r þ ct for a
suitable constant c.
This property also holds for SH systems. To see
this, let us consider the set where a solution u
162 Symmetric Hyperbolic Systems and Shock Waves
vanishes: if the initial condition vanishes for jxjR,
we claim that u at some later time vanishes for jxj
R t=a, for a large enough.
Indeed, let us integrate the energy identity on a
truncated cone := {jxja(t
0
t)=t
0
;0 t t
1
}
with t
1
< t
0
. The boundary of consists of three
parts: @=
0
[
1
[ S, where
0
and
1
represent
the portions of the boundary on which t = 0andt
1
,
respectively. The outer normal to S is proportional
to (a, t
0
x
j
=jxj). Let E(s) denote the integral of u
T
Qu
on \ { t = s}. Integrating eqn [11] by parts, we
obtain
Eðt
1
ÞEð0Þþ
Z
S
u
T
u ds
¼
ZZ
ð2u
T
f u
T
CuÞdt dx ½12
where is proportional to aQ þ t
0
P
j
x
j
A
j
=jxj.
Take a so large that is positive definite. The
integral over S is then non-negative. If C is positive
definite and f 0, so that E(0) = 0, we find that
E(t
1
) 0. Since Q is positive definite, this implies
u 0on
1
, as claimed.
A Numerical Scheme
System Lu = f may be discretized, for example, by
the Lax–Friedrichs method: let h be the discretiza-
tion step in space, and k the time step; write
j
u(t, x) = u(t, x
1
, ..., x
j
þ h, ..., x
n
) (translation in
the j direction). One replaces @
j
u by the centered
difference in the j direction: (
j
u
1
j
u)=2h; and the
time derivative by
½uðt þ k; xÞ
1
2n
X
j
ð
j
uðt; xÞþ
1
j
uðt; xÞÞ=k ½13
For consistency of the scheme, we require k=h = >0
to be fixed as k and h tend to zero; stability then
holds if is small.
Nonlinear Problems and Singularities
We give a simple setup for proving the existence of
smooth solutions to SH systems for small times.
Such solutions may develop singularities. We limit
ourselves to two types of singularities, on which SH
structure provides some information: jump disconti-
nuities and blow-up patterns. Caustic formation is
not considered.
Construction of a Smooth Solution
Consider a real SH system (eqn [1]). Recall that a
function of x belongs to the Sobolev space H
s
if its
derivatives of order s or less are square-integrable.
One constr ucts a solution defined for t small, which
is in H
s
, s > n=2 þ 1, as a function of x, by the
following procedure:
(1) Replace spatial derivatives by regularized opera-
tors, which should be bounded in Sobolev
spaces; the regularized equation is an ODE in
H
s
; let u
"
be its solution.
(2) Write the equation satisfied by derivatives of
order s of u
"
, and apply the energy identity to it.
(3) Find a positive T such that the solution is
bounded in H
s
for jtjT, uniformly in "; this
implies a C
1
bound.
(4) Prove the convergence of the approximations
in L
2
.
(5) Prove the continuity in time of the H
s
norm;
conclude that the u
"
tend to a solution in
C(T, T; H
s
).
The result admits a local version, in which
Sobolev spaces are replaced by Kato’s ‘‘uniformly
local’’ spaces. Uniqueness of the solution is proved
along similar lines. We do not attempt to identify
the infimum of the values of s for which the Cauchy
problem is well-posed.
Jump Discontinuities: Shock Waves
A ‘‘shock wave’’ is a weak solution of a system of
conservation laws admitting a jump discontinuity.
By definition, weak solutions satisfy, for any smooth
function
A
(x) with compact support,
ZZ
ff
A
@
A
þ N
A
A
gdt dx ¼ 0
The theory of shock waves is an attempt to
understand solutions of conservation laws which are
limits of solutions of diffusion equations; the hope is
that the influence of second-derivati ve terms is
appreciable only near shocks, and that, for given
initial data, there is a unique weak solution of the
conservation law which may be obtained as such a
limit, if modeling has been done correctly. This
problem may be difficult already for a single shock
(‘‘shock structure’’).
The theory of shock waves follows the one-
dimensional theory closely. We therefore describe
the main facts for a conservation law in one space
dimension (u = u(t, x)):
@
t
u þ @
x
f ðuÞ¼0
If a shock travels at speed c, the weak formulation
of the equations gives the Rankine–Hugoniot rela-
tion c[u] = [f (u)], where square brackets denote
jumps. There may be several weak solutions having
the same initial condition. One restricts solutions by
Symmetric Hyperbolic Systems and Shock Waves 163
making two further requirements: (1) the system
admits an entropy pair (U, F) with a convex entropy
and (2) to be admissible, weak solutions must be
limits of ‘‘viscous approximations’’
@
t
u þ @
x
f ðuÞ¼"@
2
x
u
as " !0. One then finds easily that the entropy
equality (@
t
U þ @
x
F = 0) must be replaced, for such
weak solutions, by the entropy condition: @
t
U þ
@
x
F 0 in the weak sense. This condition admits a
concrete interpretation if the gradient of each
characteristic speed is never orthogonal to the
corresponding right eigenvector (‘‘genuine nonli-
nearity’’); in that case, characteristics must impinge
on the shock (‘‘shock inequalities’’).
For the equations of gas dynamics with polytropic
law (pv
= const.), there is a unique solution with
initial condition u = u
l
for x < 0, u = u
r
for x > 0,
where u
l
and u
r
are constant (‘‘Riemann problem’’)
which satisfies the entropy condition, provided ju
l
u
r
j
is small. More generally, if the equation of state
p = p(v, s) > 0 satisf ies @p =@v < 0 and @
2
p=@v
2
> 0,
the shock inequalities are equivalent to the fact that
the entropy increases after the passage of a shock
with j u
l
u
r
j small.
On the numerical side, one should mention:
(1) the wi dely used idea of upstream differencing;
(2) the Lax–Wendroff scheme, the complete analysis
of which requires tools from soliton theory; and
(3) the availability of general results for dissipative
schemes for SH systems.
Recent trends include: (1) admissibility conditions
when genuine nonlinearity does not hold and
(2) other approximations of shock wave problems,
most notably kinetic formulations.
Some of the ideas of shock wave theory have been
applied to Hamilton–Jacobi equations and to
motion by mean curvature, with applications to
front propagation problems and ‘‘computer vision.’’
Stronger Singularities: Blow-Up Patterns
The amplitude of a solution may also grow without
bound. Examples include optical pulse propagation
in Kerr media and singularities in general relativity.
The phenomenon is common when reaction terms
are allowed. As we now explain, this phen omenon is
reducible to SH theory in many cases of interest.
Blow-up singularities are usually not governed by
the characteristic speeds defined by the principal
part, because top-order derivatives are balanced by
lower-order terms. In many applications, a systema-
tic process (Fuchsian reduction) enables one to
identify the correct model near blow-up; as a result,
one can write the solution as the sum of a singular
part, known in closed form, and a regular part. If
the singul arity locus is represented by t = 0, the
regular part solves a renormalized equation of the
typical form
tMu þ Au ¼ t
"
N ½14
where Mu = 0 is SH. Under natural conditions, for
any initi al condition u
0
such that Au
0
= 0, there is a
unique solution of eqn [14] defined for small t.
The upshot is an asymptotic representation of
solutions which renders the same services as an
exact solution, and is valid precisely where numeri-
cal computation breaks down.
Fuchsian reduction enables one in particular to
study (1) the blow-up time; (2) how the singularity
locus varies when Cauchy data, prescribed in the
smooth region, are varied; and (3) expressions which
remain finite at blow-up. It is the only known general
procedure for constructing analytically singular
spacetimes involving arbitrary functions, rather than
arbitrary parameters, and is therefore relevant to the
search for alternatives to the big bang.
Examples and Applications
Wave Equation with Variable Coefficients
Consider the equation
@
tt
u þ 2a
j
ðxÞ@
jt
u a
jk
ðxÞ@
jk
u ¼ f ðt; x; u; ruÞ
with (a
jk
) positive definite. Letting v = (v
0
, ...,
v
nþ1
):= (u, @
j
u, @
t
u), we find the system
@
t
v
0
¼ v
nþ1
@
t
v
k
@
k
v
nþ1
¼ 0
@
t
v
nþ1
þ 2a
k
@
k
v
nþ1
a
jk
@
k
v
j
¼ f
It is symmetrizable, using the quadratic form
AB
v
A
v
B
= v
2
0
þ a
jk
v
j
v
k
þ v
2
nþ1
.
One proves directly that, if v
j
= @
j
v
0
for t = 0, this
relation remains true for all t.
Maxwell’s Equations
Maxwell’s equations may be split into six evolution
equations: @
t
E curl B þ j = 0 and @
t
B þ curl E = 0,
and two ‘‘constraints’’ div E = 0, div B = 0. The
system of evolution equations is already in sym-
metric form; the quadratic form
AB
u
A
u
B
is here
jEj
2
þjBj
2
.
164 Symmetric Hyperbolic Systems and Shock Waves
Compressible Fluids
Consider first the case of a polytropic gas:
@
t
v þðv rÞv þ
1
rp ¼ 0
@
t
þ divðvÞ¼0
½15
with p proportional to
. Taking (p, v)as
unknowns, one readily finds the SH system
1
p
@
t
p þ
1
p
ðv rÞp þ div v ¼ 0 ½16
@
t
v þrp þ ðv rÞv ¼ 0 ½17
Symmetrization for more genera l compressible
fluids with dissipation, including bulk viscosity, so
as to satisfy the additional condition [5] may be
achieved if we take as thermodynamic variables
and T, and assume press ure p and internal energy "
satisfy @p=@ > 0 and @"=@T > 0, by taking as
unknowns (, v, (" þjvj
2
=2)). The specific entropy
s satisf ies d" = Tds pd(1=). If the viscosity and
heat conduction coefficients are positive, one finds
that U = s is a convex entropy (in the sense of SH
theory) on the set where >0, T > 0.
Einstein’s Equations
The computation of solutions of Einstein’s equa tions
over long times, in particular in the study of
coalescence of binary stars, has recently led to
unexplained difficulties in the standard Arnowitt–
Deser–Misner (ADM) formulat ion of the initial-
value problem in general relativity. One way to
tackle these difficulties is to rewrite the field
equations in SH form; we focus on this particular
aspect of recent research.
Recall the problem: find a four-dimensional
metric g
ab
with Lorentzian signa ture, such that
R
ab
1
2
Rg
ab
= T
ab
, with r
a
T
ab
= 0, combined
with an equation of state if necessary. R
ab
is the
Ricci tensor and R = g
ab
R
ab
is the scalar curvature;
they depend on derivatives of the metric up to order 2.
In addition to the metric, T
ab
involves physical
quantities such as fluid 4-velocity or an electro-
magnetic field. The conservation laws of classical
mathematical physics are all contained in the
relation r
a
T
ab
= 0.
Now, the field equations cannot be solved for
@
2
t
g
ab
, and, as a consequence, the Taylor series of g
ab
with respect to time cannot be determined, even
formally, from the values of g
ab
and @
t
g
ab
for t = 0
(i.e., the Cauchy data). Furthermore, these data
must satisfy four constraint equations. If the
constraints are satisfied initially, they ‘‘propagate.’’
But in numerical computation, these constraints are
never exactly satisfied, and the computed solution
may deviate considerably from the exact solution.
Also, numerical computations depend heavily on the
way Einstein’s equations are formulated.
The simplest way to derive a SH system is to
replace R
ab
by R
(h)
ab
= R
ab
1
2
[g
bc
@
a
F
c
þ g
ac
@
b
F
c
],
where F
c
:= g
ab
c
ab
. It turns out that R
(h)
ab
=
1
2
g
cd
@
cd
g
ab
þ H
ab
(g, @g), where the expression of
H
ab
is immaterial. Applying to each component of
the metric the treatment of the first example above
(wave equation with variable coefficients), one
easily derives an SH system of 50 equations for 50
unknowns: the ten independent components of the
metric, and their 40 first-order derivatives. Now, if
the
c
are initially zero (coordinates are ‘‘harmo-
nic’’), they remain so at later times.
Unfortunately, the harmonic coordinate condition
does not seem to be stable in the larg e. More recent
formulations start with one of the standard setups
(ADM formalism, conformal equations, tetrad
formalism, Newm an–Penrose formalism) and pro-
ceed by adding combinations of the constraints to
the equations, multiplied by parameters adjusted so
as to ensure hyperbolicity or symmetric–hyperboli-
city if needed. Anoth er recent idea is to add a new
unknown which monitors the failure of the
constraint equations; one adds to the equations a
new relation of the form @
t
= C , where
C = 0 is equivalent to the constraints, and and
are parameters. One then adds coupling terms to
make the extended system SH. It is expect ed that the
set of constraints acts as an attractor.
Reported co mputations indicate that these meth-
ods have resulted in an improvement of the time
over which numerical computations are valid.
Tricomi’s Equation
Let ’(x, y) solve (y@
2
x
@
2
y
)’ = 0. Letting u =
e
x
(@
x
’, @
y
’), one finds a symmetric system Lu = 0,
with
L ¼
y 0
01
ð@
x
þ Þ
01
10
@
y
If
Z ¼
1 y
11
we find that K = ZL = A
1
@
x
þ A
2
@
y
þ B, where
B
1
2
ð@
x
A
1
þ @
y
A
2
Þ¼
1
2
þ y y
y
is positive definite if y is bounded, of arbitrary sign,
and is small.
Symmetric Hyperbolic Systems and Shock Waves 165
Cauchy–Kowalewska Systems
Consider a complex system
@
t
u ¼ A
j
ðz; t; uÞ
@u
@z
j
þ Bðz; t; uÞ½18
where u = (u
A
), z = (z
1
, ..., z
n
). The coefficients are
analytic in their arguments when z and t are close to
the origin and u is bounded by some constant K.
The Cauchy–Kowalewska theorem ensures that, for
any analytic ini tial condition near the origin, this
system has a unique analytic solution near z = 0,
even without any symmetry assumption on the A
j
.
This result is a consequence of SH theory
(Garabedian).
Indeed, write z
j
= x
j
þ iy
j
, @
z
j = (1=2)(@
x
j i@
y
j ), and
@
z
j
= (1=2)(@
x
j þ i@
y
j ). Recall that analytic functions
of z satisfy the Cauchy–Riemann equations @
z
j
u = 0.
Adding (
A
j
)
T
@
z
j
to [18], and using the definition of
@
z
j and @
z
j
, we find the symmetric system
u
t
¼
1
2
ðA
j
þð
A
j
Þ
T
Þ@
x
j
u
þ
1
2i
ðA
j
ð
A
j
Þ
T
Þ@
y
j u þ B ½19
Solving this system, we find a candidate u for a
solution of eqn [18]. To show that u is analytic if the
data are, we solve a second SH system for
w = w
(j)
:= @
z
j
u. If the data are analytic, w vanishes
initially, and therefore remains zero for all t.
Therefore, u is indeed analytic.
See also: Computational Methods in General Relativity:
The Theory; Einstein Equations: Initial Value
Formulation; Evolution Equations: Linear and Nonlinear;
Magnetohydrodynamics; Partial Differential Equations:
Some Examples; Semilinear Wave Equations; Shock
Wave Refinement of the Friedman–Robertson–Walker
Metric.
Further Reading
Dautray R and Lions J (1988) Mathematical Analysis and
Numerical Methods for Science and Technology, vols. 1–6.
(transl. from French). New York: Springer.
Friedrichs KO (1986) In: Morawetz CS (ed.) Selecta, two volumes:
collection of reprints with comments. Boston: Birkha¨user.
Godlewski E and Raviart P-A (1996) Numerical Approximation
of Hyperbolic Systems of Conservation Laws. Berlin: Springer.
Gustafsson B, Kreiss H-O, and Oliger J (1996) Time-Dependent
Problems and Difference Methods. New York: Wiley-Interscience.
John F (1982) Partial Differential Equations, 4th edn. Berlin:
Springer.
Kato T (1975a) Quasilinear equations of evolution with applica-
tions to partial differential equations. Lecture Notes in
Mathematics 448: 25–70.
Kato T (1975b) The Cauchy problem for quasilinear symmetric
hyperbolic systems. Archive for Rational Mechanics and
Analysis 58: 181–205.
Kawashima S and Shizuta Y (1988) On the normal form of the
symmetric hyperbolic–parabolic systems associated with the
conservation laws. Toˆ hoku Mathematical Journal 40: 449–464.
Kichenassamy S (1996) Nonlinear Wave Equations. New York:
Dekker.
Lax PD (1973) Hyperbolic Conservation Laws and the Mathe-
matical Theory of Shock Waves. Regional Conference Series in
Applied Mathematics. Philadelphia: SIAM.
Lax PD and Phillips RS (1989) Scattering Theory, (revised
edition). Boston: Academic Press.
Majda A (1984) Compressible Fluid Flow and Systems of
Conservation Laws. Berlin: Springer.
Serre D (1999/2000) Systems of Conservation Laws, I & II.
Cambridge: Cambridge University Press.
Smoller J (1983) Shock Waves and Reaction-Diffusion Equations.
Berlin: Springer.
Taylor M (1991) Pseudodifferential Operators and Nonlinear
P.D.E., Progress in Mathematics, vol. 100. Boston: Birkha¨user.
Symmetries and Conservation Laws
L H Ryder, University of Kent, Canterbury, UK
ª 2006 Elsevier Ltd. All rights reserved.
Introduction: Spacetime Symmetries
Symmetries have played , and continue to play, an
important role in fundamental physics, but the part
they play is today seen as more complicated and
many-sided than it was in the early days of particle
physics, just after the Second World War. The area
in which symmetries have had their most dramatic
consequences is elementary particle physic s, or
high-energy physics, and the majority of this article
is concerned with this subject. The article concludes
with some observations about symmetries and
conservation laws in general relativity.
In the early days, considerations of symmetry
were almost limited to Lorentz transformations: we
begin by reviewing this crucially important topic.
Invariance of the laws of nature under translations
in space and time are actually necessary for the
existence of science itself; if experiments did not
yield the same results today and tomorrow, and in
Paris and Moscow and on the Moon, then in effect
there would be no laws of nature. Almost as strong
166 Symmetries and Conservation Laws
a statement could be made about invariance under
rotations; if space were not isotropic, experimental
results would depend on which direction the
apparatus was aligned in, and again any laws
would be extremely hard to find. Turning to the
question of motion, Newton and Galileo realized
that the laws of dynamics are the same in all inertial
frames in relative motion. In the Newton–Galileo
scheme, the rule for relating the space and time
coordinates of two frames of reference is (for
relative motion along the common x-axis)
x
0
¼ x vt; t
0
¼ t ½1
This principle of relativity was reaffirmed by
Einstein, but with the crucial modification that the
rules for relating coordinates in two frames are
given by Lorentz transformations, so that [1] is
replaced by
x
0
¼ ðx vtÞ; t
0
¼ t
vx
c
2
½2
Time is absolute in [1] but relative in [2]. Einstein
was of course motivated by the fact that Maxwell’s
equations are covariant under Lorentz transforma-
tions, but not under Newton–Galileo ones.
The above consideration s reveal that the laws of
nature should be covariant under ten types of
transformation: three translations in space, one in
time, three parameters (angles) for rotations and
three veloc ities. These transformations together
form a group, the inhomogeneous Lorentz, or
Poincare´ group. It is a nonabelian group whose ten
generators correspond to 4-momentum, angular
momentum, and Lorentz boosts. The seminal work
on the significance of this group in fundamental
physics is that of Wigner in 1939. Assuming that the
states of fundamental quantum systems (particles,
atoms, molecules) form the basis states for repre-
sentations of this group, these entities are described
by two quantities, mass and spin. Spin, moreover,
which was already familiar from earlier investiga-
tions in quantum physics, was described by the
rotation group (SU(2), which is homomorphic to
SO(3)) only for states with timelike momentum. For
photons, for example, with null momentum, spin is
described by the (noncompact) Euclidean group in
the plane, with the consequence that there are only
two polarization states for this massless particle.
Noether’s theorem provides the crucial link
between symmetries and conservation laws, via the
principle of least action. Noether showed that the
invariance of the action under a continuous
symmetry operation implied the existence of a
conserved quantity. The conserved quantities
corresponding to invariance under translation in
space and time are momentum and energy; con-
servation of angular momentum follo ws from
invariance under rotations and invariance under
Lorentz transformations gives rise to conservation
of motion of the center of mass.
Gauge Theories: Electromagnetism
and Yang–Mills Theories
A quantity whose conservation has been well known
for a long time is electric charge. The question may
then be asked: invariance under what symmetry
gives rise to conservation of electric charge? A
classical complex field has the Lagrangian density
L ¼ð@
Þð@
Þm
2
½3
which is invariant under
! expðiQ Þ ½4
being the parameter for the transformation.
Noether’s theorem then yields conservation of Q,
interpreted as electric charge. With a constant, as
above, the Lagrangian possesses a ‘‘global’’ symme-
try. This becomes a ‘‘local’’ symmetry when
becomes space and time dependent, (r, t)or
(x
). In that case, however, the La grangian [3] is
no longer invariant under [4], because of the
derivative terms. To preserve invariance an extra
field A
must be introduced, so that [4] then
becomes
! expðiQ ðx
ÞÞ
A
! A
þ
1
Q
@
½5
and the Lagrangian acquires extra terms, involving
A
. The field A
is called a gauge field and is
identified with the electromagnetic potential. The
transformation [5] is called a gauge transformation,
and since the phase factor exp (iQ ) may be
regarded as a unitary 1 1 matrix, we have here a
theory with U(1) gauge invariance, which describes
electromagnetism and conservation of charge.
The notion of isospin had been introduced by
Heisenberg in 1932. Isospin (then called isotopic
spin) was a vector-like quantity conserved in strong
(nuclear) interactions. Yang and Mills in 1954 made
the pioneering suggestion that isospin conservation
could also be recast as a gauge theory, by enlarging
the U(1) group of electromagnetism to SU(2)
(corresponding to rotations in ‘‘isospin space’’),
and at the same time treating the rotation angles as
functions of spacetime. Then, eqn [4] will change: if
Symmetries and Conservation Laws 167