Fuchsian reduction, applications to geometry, cosmology and mathematical physics

Progress in Nonlinear Differential Equations
and Their Applications
Volume 71
Editor
Haim Brezis
Universit´e Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
Antonio Ambrosetti, Scuola Internationale Superiore di Studi Avanzati, Trieste
A. Bahri, Rutgers University, New Brunswick
Felix Browder, Rutgers University, New Brunswick
Luis Caffarelli, The University of Texas, Austin
Lawrence C. Evans, University of California, Berkeley
Mariano Giaquinta, University of Pisa
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
Sergiu Klainerman, Princeton University
Robert Kohn, New York University
P. L. Lions, University of Paris IX
Jean Mawhin, Universit´e Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath

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Satyanad Kichenassamy

Fuchsian Reduction
Applications to Geometry, Cosmology,
and Mathematical Physics

Birkhăauser
Boston ã Basel • Berlin

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Satyanad Kichenassamy
Universite´ de Reims Champagne-Ardenne
Moulin de la Housse, B.P. 1039
F-51687 Reims Cedex 2
France


Mathematics Subject Classification (2000): 35-02, 35A20, 35B65, 35J25, 35J70, 35L45, 35L70, 35L80,
35Q05, 35Q51, 35Q75, 53A30, 80A25, 78A60, 83C75, 83F05, 85A15
Library of Congress Control Number: 2007932088
ISBN-13: 978-0-8176-4352-2

e-ISBN-13: 978-0-8176-4637-0

Printed on acid-free paper.
c 2007 Birkhăauser Boston
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhăauser Boston, c/o Springer Science+Business Media LLC, 233
Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or

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9 8 7 6 5 4 3 2 1
(INT/MP)

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To my parents

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Preface

The nineteenth century saw the systematic study of new “special functions”,
such as the hypergeometric, Legendre and elliptic functions, that were relevant
in number theory and geometry, and at the same time useful in applications.
To understand the properties of these functions, it became important to study
their behavior near their singularities in the complex plane. For linear equations, two cases were distinguished: the Fuchsian case, in which all formal
solutions converge, and the non-Fuchsian case. Linear systems of the form
z

du

+ A(z)u = 0,
dz

with A holomorphic around the origin, form the prototype of the Fuchsian
class. The study of expansions for this class of equations forms the familiar
“Fuchs–Frobenius theory,” developed at the end of the nineteenth century
by Weierstrass’s school. The classification of singularity types of solutions of
nonlinear equations was incomplete, and the Painlev´e–Gambier classification,
for second-order scalar equations of special form, left no hope of finding general
abstract results.
The twentieth century saw, under the pressure of specific problems, the
development of corresponding results for partial differential equations (PDEs):
The Euler–Poisson–Darboux equation
utt +

λ
ut − Δu = 0
t

and its elliptic counterpart arise in axisymmetric potential theory and in the
method of spherical means; it also comes up in special reductions of Einstein’s
equations. In particular, one realized that equations with different values of λ
could be related to each other by transformations u → tm u. Elliptic problems
in corner domains and problems with double characteristics also led to further generalizations. This development was considered as fairly mature in the
1980s; it was realized that some problems required complicated expansions
with logarithms and variable powers, beyond the scope of existing results, but

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viii

Preface

it was assumed that this behavior was nongeneric. Nonlinear problems were
practically ignored.
The word “Fuchsian” had come to stand for “equations for which all formal power series solutions are convergent.” Of course, Fuchsian ODEs have
solutions involving logarithms, but by Frobenius’s trick, logarithms could be
viewed as limiting cases of powers, and were therefore not thought of as
generic.
However, in the 1980s difficulties arose when it became necessary to solve
Fuchsian problems arising from other parts of mathematics, or other fields.
The convergence of the “ambient metric” realizing the embedding of a Riemannian manifold in a Lorentz space with a homothety could not be proved in
even dimensions. When, in the wake of the Hawking–Penrose singularity theorems, it became necessary to look for singular solutions of Einstein’s equations,
existing results covered only very special cases, although the field equations
appeared similar to the Euler–Poisson–Darboux equation. Numerical studies
of such space-times led to spiky behavior: were these spikes artefacts? indications of chaotic behavior?
Other problems seemed unrelated to Fuchsian PDEs. For the blowup problem for nonlinear wave equations, again in the eighties, Hă
ormander, John, and
their coworkers computed asymptotic estimates of the blowup time—which is
not a Lorentz invariant. For elliptic problems Δu = f (u) with monotone nonlinearities, solutions with infinite data dominate all solutions, and come up in
several contexts; the boundary behavior of such solutions in bounded C 2+α
domains is not a consequence of weighted Schauder estimates. Outside mathematics, we may mention laser collapse and the weak detonation problem.
In astrophysics, stellar models raise similar difficulties; equations are singular at the center, and one would like to have an expansion of solutions near
the singularity to start numerical integration. Also, the theory of solitons has
provided, from 1982 on, a plethora of formal series solutions for completely integrable PDEs, of which one would like to know whether they represent actual
solutions. Do these series have any relevance to nearly integrable problems?
The method of Fuchsian reduction, or reduction for short, has provided
answers to the above questions. The upshot of reduction is a representation
of the solution u of a nonlinear PDE in the typical form

u = s + T mv ,
where s is known in closed form, is singular for T = 0, and may involve a finite
number of arbitrary functions. The function v determines the regular part of
u. This representation has the same advantages as an exact solution, because
one can prove that the remainder T m v is indeed negligible for T small. In
particular, it is available where numerical computation fails; it enables one to
compute which quantities become infinite and at what rate, and to determine
which combinations of the solution and its derivatives remain finite at the
singularity. From it, one can also decide the stability of the singularity under

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Preface

ix

perturbations, and in particular how the singularity locus may be prescribed
or modified.
Reduction consists in transforming a PDE F [u] = 0, by changes of variables and unknowns, into an asymptotically scale-invariant PDE or system of
PDEs
Lv = f [v ]
such that (i) one can introduce appropriate variables (T, x1 , . . . ) such that
T = 0 is the singularity locus; (ii) L is scale-invariant in the T -direction;
(iii) f is “small” as T tends to zero; (iv) bounded solutions v of the reduced
equation determine singular u that are singular for T = 0. The right-hand
side may involve derivatives of v . After reduction to a first-order system, one
is usually led to an equation of the general form
T

d
+ A w = f [T, w],
dT

where the right-hand side vanishes for T = 0. PDEs of this form will be called
“Fuchsian.” The Fuchsian class is itself invariant under reduction under very
general hypotheses on f and A. This justifies the name of the method.
Since v is typically obtained from u by subtracting its singularities and
dividing by a power of T , v will be called the renormalized unknown. Typically,
the reduced Fuchsian equations have nonsmooth coefficients, and logarithmic
terms in particular are the rule rather than the exception. Since the coefficients
and nonlinearities are not required to be analytic, it will even be possible to
reduce certain equations with irregular singularities to Fuchsian form. Even
though L is scale-invariant, s may not have power-like behavior. Also, in many
cases, it is possible to give a geometric interpretation of the terms that make
up s.
The introduction, Chapter 1, outlines the main steps of the method in
algorithmic form.
Part I describes a systematic strategy for achieving reduction. A few general principles that govern the search for a reduced form are given. The list of
examples of equations amenable to reduction presented in this volume is not
meant to be exhaustive. In fact, every new application of reduction so far has
led to a new class of PDEs to which these ideas apply.
Part II develops variants of several existence results for hyperbolic and
elliptic problems in order to solve the reduced Fuchsian problem, since the
transformed problem is generally not amenable to classical results on singular
PDEs.
Part III presents applications. It should be accessible after an upperundergraduate course in analysis, and to nonmathematicians, provided they
take for granted the proofs and the theorems from the other parts. Indeed,
the discussion of ideas and applications has been clearly separated from statements of theorems and proofs, to enable the volume to be read at various
levels.

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x

Preface

Part IV collects general-purpose results, on Schauder theory and the distance function (Chapter 12), and on the Nash–Moser inverse function theorem
(Chapter 13). Together with the computations worked out in the solutions to
the problems, the volume is meant to be self-contained.
Most chapters contain a problem section. The solutions worked out at
the end of the volume may be taken as further prototypes of application of
reduction techniques.
A number of forerunners of reduction may be mentioned.
1. The Briot–Bouquet analysis of singularities of solutions of nonlinear ODEs
of first order, continued by Painlev´e and his school for equations of higher
order. It has remained a part of complex analysis. In fact, the catalogue
of possible singularities in this limited framework is still not complete
in many respects. Most of the equations arising in applications are not
covered by this analysis.
2. The regularization of collisions in the N -body problem. This line of
thought has gradually waned, perhaps because of the smallness of the
radius of convergence of the series in some cases, and again because the
relevance to nonanalytic problems was not pursued systematically.
3. A number of special cases for simple ODEs have been rediscovered several
times; a familiar example is the construction of radial solutions of nonlinear elliptic equations, which leads to Fuchsian ODEs with singularity at
r = 0.
In retrospect, reduction techniques are the natural outgrowth of what is traditionally called the “Weierstrass viewpoint” in complex analysis, as opposed
to the Cauchy and Riemann viewpoints. This viewpoint, from the present perspective, puts expansions at the main focus of interest; all relevant information

is derived from them. For this approach to be relevant beyond complex analysis, it was necessary to understand which aspects of the Weierstrass viewpoint
admit a generalization to nonanalytic problems with nonlinearities—and this
generalization required a mature theory of nonlinear PDEs which was developed relatively recently. The development of reduction techniques in the early
nineties seems to have been stimulated by the convergence of five factors:
1. The emergence of singularities as a legitimate field of study, as opposed
to a pathology that merely indicates the failure of global existence or
regularity.
2. The existence of a mature theory of elliptic and hyperbolic PDEs, which
could be generalized to singular problems.
3. The failure of the search for a weak functional setting that would include
blowup singularities for the simplest nonlinear wave equations.
4. The rediscovery of complex analysis stimulated by the emergence of soliton
theory.
5. The availability of a beginning of a theory of Fuchsian PDEs, as opposed
to ODEs, albeit developed for very different reasons, as we saw.

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Preface

xi

On a more personal note, a number of mathematicians have, directly or indirectly, helped the author in the emergence of reduction techniques: D. Aronson,
C. Bardos, L. Boutet de Monvel, P. Garrett, P. D. Lax, W. Littman,
L. Nirenberg, P. J. Olver, W. Strauss, D. H. Sattinger, A. Tannenbaum,
E. Zeidler. In fact, my indebtedness extends to many other mathematicians
whom I have met or read, including the anonymous referees. H. Brezis, whose
mathematical influence may be felt in several of my works, deserves a special
place. I am also grateful to him for welcoming this volume in this series, and

to A. Kostant and A. Paranjpye at Birkhă
auser, for their kind help with this
project.

Paris
February 27, 2007

Satyanad Kichenassamy

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Singularity locus as parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The main steps of reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 A few definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 An algorithm in eight steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Simple examples of reduced Fuchsian equations . . . . . . . . . . . . . 5
1.6 Reduction and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Part I Fuchsian Reduction
2

Formal Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 The Operator D and its first properties . . . . . . . . . . . . . . . . . . .

2.2 The space A and its variants . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Formal series with variable exponents . . . . . . . . . . . . . . . . . . . . .
2.4 Relation of A to the invariant theory of binary forms . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
24
27
35
39
42

3

General Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Reduction of a single equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Introduction of several time variables and second reduction . .
3.3 Semilinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Structure of the formal series with several time variables . . . . .
3.5 Resonances, instability, and group invariance . . . . . . . . . . . . . . .
3.6 Stability and parameter dependence . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45
45
51
52
54
58
64

65

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xiv

Contents

Part II Theory of Fuchsian Partial Differential Equations
4

Convergent Series Solutions of Fuchsian Initial-Value
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Theory of linear Fuchsian ODEs . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Initial-value problem for Fuchsian PDEs with analytic data . .
4.3 Generalized Fuchsian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
69
71
75
82
83

5

Fuchsian Initial-Value Problems in Sobolev Spaces . . . . . . . . . 85

5.1 Singular systems of ODEs in weighted spaces . . . . . . . . . . . . . . 86
5.2 A generalized Fuchsian ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Fuchsian PDEs: abstract results . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Optimal regularity for Fuchsian PDEs . . . . . . . . . . . . . . . . . . . . . 97
5.5 Reduction to a symmetric system . . . . . . . . . . . . . . . . . . . . . . . . . 101
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6

Solution of Fuchsian Elliptic Boundary-Value Problems . . . . 105
6.1 Basic Lp results for equations with degenerate characteristic
form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 Schauder regularity for Fuchsian problems . . . . . . . . . . . . . . . . . 109
6.3 Solution of a model Fuchsian operator . . . . . . . . . . . . . . . . . . . . . 113
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Part III Applications
7

Applications in Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1 Notions on stellar modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Polytropic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Point-source model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8

Applications in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.1 The big-bang singularity and AVD behavior . . . . . . . . . . . . . . . 129
8.2 Gowdy space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.3 Space-times with twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9

Applications in Differential Geometry . . . . . . . . . . . . . . . . . . . . . 143
9.1 Fefferman–Graham metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.2 First Fuchsian reduction and construction
of formal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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xv

9.3

Second Fuchsian reduction and convergence
of formal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.4 Propagation of constraint equations . . . . . . . . . . . . . . . . . . . . . . . 151
9.5 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.6 Conformal changes of metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.7 Loewner–Nirenberg metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10 Applications to Nonlinear Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.1 From blowup time to blowup pattern . . . . . . . . . . . . . . . . . . . . . . 163
10.2 Semilinear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.3 Nonlinear optics and lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

10.4 Weak detonations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.5 Soliton theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.6 The Liouville equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.7 Nirenberg’s example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11 Boundary Blowup for Nonlinear Elliptic Equations . . . . . . . . 217
11.1 A renormalized energy for boundary blowup . . . . . . . . . . . . . . . 218
11.2 Hardy–Trudinger inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
11.3 Variational characterization of solutions
with boundary blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
11.4 Construction of the partition of unity . . . . . . . . . . . . . . . . . . . . . 225
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Part IV Background Results
12 Distance Function and Hă
older Spaces . . . . . . . . . . . . . . . . . . . . . . 231
12.1 The distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.2 Hă
older spaces on C 2+ domains . . . . . . . . . . . . . . . . . . . . . . . . . . 233
12.3 Interior estimates for the Laplacian . . . . . . . . . . . . . . . . . . . . . . . 239
12.4 Perturbation of coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
13 Nash–Moser Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . 247
13.1 Nash–Moser theorem without smoothing . . . . . . . . . . . . . . . . . . 247
13.2 Nash–Moser theorem with smoothing . . . . . . . . . . . . . . . . . . . . . 249
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

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1
Introduction

This introduction defines Fuchsian reduction, or reduction for short, illustrates it with a number of simple examples and outlines its main successes.
The technical aspects of the theory are developed in the subsequent chapters. The upshot of reduction is a parameterized representation of solutions
of nonlinear differential equations, in which singularity locus may be one of
the parameters. We first show, on a very simple example, the advantages of
such a representation. We then describe the main steps of the reduction process in general terms, and show in concrete situations how this reduction is
achieved. We close this introduction with a survey of the impact of reduction
on applications.

1.1 Singularity locus as parameter
Consider the ODE

du
= u2 .
dt
The solution taking the value a for t = t0 is
u(a, t0 , t) =

a
.
1 − a(t − t0 )

The set of solutions may be parameterized by two parameters (a, t0 ). The
2
2
procedure is quite similar
√ to solving the algebraic equation s + t = 1 in
2

the form s = s(t) = ± 1 − t , in which t plays the role of a parameter, or
local coordinate. But unlike the representation s = s(t), the representation
u = u(a, t0 , t) is redundant: only one parameter suffices to describe the general
solution. Indeed, let b = a/(1 + at0 ); we obtain
u(a, t0 , t) =

b
.
1 − bt

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2

1 Introduction

The parameter b gives the position of the (only) singular point of the solution,
namely 1/b.
Quite generally, finding the general solution of a differential equation
amounts to finding a set of parameters that label all solutions close to a given
one. The process is comparable to finding local coordinates on a manifold.
Taking singularity locus as one of the parameters, one obtains a parameterization without redundancy, unlike the parameterization by t0 and the Cauchy
data at time t0 .

1.2 The main steps of reduction
A complete application of the reduction technique to a specific problem
F [u] = 0
follows four steps, detailed below. The square brackets indicate that F may
depend on u and its derivatives, as well as on independent variables.

•
•
•
•

Leading-order analysis.
First reduction and formal solutions.
Second reduction and characterization of solutions.
Invertibility and stability of solutions.

Let us briefly describe how these steps would be carried out for a typical class
of problems: those for which the leading term is a power. Many other types
of leading behavior arise in applications, including logarithms and variable
powers. They will be discussed in due time.
The objective of leading-order analysis is to find a function T and a pair
(u 0 , ν) such that F [u 0 T ν ] vanishes to leading order. The hope is to find
solutions such that
(1.1)
u ∼ u 0T ν .
The objective of the first reduction is to construct a formal solution of the
typical form
∞

u =T

j

ν

ujp T j (ln T )p .

(1.2)

j=0 p=0

It is achieved by introducing a renormalized unknown v , of which a typical
definition has the form
u = T ν (u 0 + T ε v ).
∂
. If
Change variables so that T is the first independent variable. Let D = T ∂T
ε is small enough, v solves a system of the form

(D + A + ε)v = T σ f [T, v].
One then chooses ε such that σ > 0. Such is the typical form of a Fuchsian
first-order system for us. If it is possible to transform a problem into this form
by a change of variables and unknowns, we say that it admits of reduction.

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1.2 The main steps of reduction

3

Chap. 3 gives general situations in which this reduction is possible, and further
special cases are treated in the applications. General results from Chap. 2 give
formal series solutions, and identify the terms containing arbitrary functions
or parameters. The set of arbitrary functions, together with the equation of
the singular set, form the singularity data. In some problems, the singular set

is prescribed at the outset, and is not a free parameter; the singularity data
consist then only of the arbitrary functions or parameters in the expansion.
Remark 1.1. In some cases, it is convenient to reduce first to a higher-order
equation or system, of the form
P (D + ε)v = T σ f [T, v ].
The resonances are then defined as the roots of P .
The objective of the second reduction is to prove that the singularity
data determine a unique solution of the equation F [u] = 0. Introduce a new
renormalized unknown w that satisfies
(D + A + m)w = T τ g[T, w]
with τ > 0. It is typically defined by a relation of the form
v = ϕ + T μ w,
where ϕ is known in closed form, and what contains all the arbitrary elements
in the formal series solution. If μ is large enough, it turns out that m also is.
One then chooses μ such that A+m has no eigenvalue with negative real part.
One then appeals to one of the general results of Chaps. 4, 5, or 6 to conclude
that the equation for w has a unique solution that remains bounded as T →
0+. It may be necessary to take some of the variables that enter the expansion,
such as t0 = T , t1 = T ln T , as new independent variables; this is essential
for the convergence proof, and provides automatically a uniformization of
solutions; see Chap. 4.
We now turn to the fourth step of the reduction process. Denoting by SD
the singularity data, we have now constructed a mapping Φ : SD → u. If, on
the other hand, we have another way of parameterizing solutions, we need to
compare these two parameterizations. For instance, if we are dealing with a
hyperbolic problem, we have a parameterization of solutions by Cauchy data,
symbolically represented by a mapping Ψ : CD → u. The objective of the
“invertibility” step is to determine a map CD → u → SD. This requires
inverting Φ; hence the terminology. At this stage, we know how singularity
data vary: perturbation of Cauchy data merely displaces the singular set or

changes the arbitrary parameters in the expansion, or both.
Thus, the main technical point is the reduction to Fuchsian form and its
exploitation. For this reason, we now give a few very simple illustrations of
the process leading to Fuchsian form.

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4

1 Introduction

1.3 A few definitions
Let us define some terminology that will be used throughout the volume. Let
T be one of the independent variables, and let
D=T

∂
.
∂T

A system is said to be Fuchsian if it has the form
(D + A)u = F [T, u],

(1.3)

where F vanishes with T , and A is linear. The eigenvalues of −A for T = 0
are called resonances, or (Fuchs) indices; they determine the exponents λ
such that the equation (D + A)u = 0 may be expected to have a solution that
behaves like T λ for T small, real, and positive. Similarly, an equation will be

called Fuchsian if it has the form
P (D)u = F [T, u],
where P is a polynomial, possibly with coefficients depending on variables
other than T , and F vanishes with T . The roots of P are called resonances or
Fuchs indices, and they are again associated to solutions with power leading
behavior. Unlike Fuchs–Frobenius theory, the right-hand side and the solution
may not have a continuation to a full complex neighborhood of the origin. The
unknown may have several components, and if so, it is written in boldface;
A is generally a matrix, but could be a differential operator—this makes no
difference in the formal theory. Seemingly more general equations in which
A = A(T, u) may usually be reduced to the standard form (1.3); see Problem
2.7. Note that T 2 ∂/∂T = s∂/∂s if s = exp(−1/T ), so that equations with irregular singularities may be reduced to Fuchsian form by a nonanalytic change
of variables. A treatment by reduction of some equations with irregular singular points is given in Problem 4.3. Similarly, higher-order equations may be
reduced to first-order ones, by introducing a set of derivatives of the unknown
as new unknowns, just as in the case of the Cauchy problem.

1.4 An algorithm in eight steps
Let us further subdivide the four basic steps into separate tasks. This yields
an algorithm in eight steps:
•
•

Step A. Choose the expansion variable T . Change variables so that u =
u(y, T ), where y represents new coordinates.
Step B. List all possible leading terms and choose one. This is generally
achieved by writing
F [u0 T ν ] = φ[u0 , ν]T ρ (1 + o(1))

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1.5 Simple examples of reduced Fuchsian equations

5

and choosing u0 and ν by the conditions
φ[u0 , ν] = 0 and u0 ≡ 0.
•
•
•
•
•
•

(1.4)

Step C. Compute the first reduced equation. If convenient, convert the
equation into a first-order system.
Step D. Choose ε and determine the resonance equation.
Step E. Determine the form of the solution. Determine in particular which
coefficients are arbitrary.
Step F. Compute the second reduced equation.
Step G. Show that formal solutions are associated to actual solutions.
Step H. Determine whether the solutions of step G are stable, by inverting
the mapping from singularity data to solutions.

1.5 Simple examples of reduced Fuchsian equations
We show, on prototype situations, how a Fuchsian equation arises naturally,
and how reduction techniques encompass familiar concepts: the Cauchy problem, stable manifolds, and the Dirichlet problem. We also work out completely
a simple example of analysis of blowup, and outline another, which introduces

the need for logarithmic terms.
1.5.1 The Cauchy problem as a special case of reduction
Consider, to fix ideas, the equation
utt = u2 ,
and the solution of the Cauchy problem with data prescribed for t = a. Let
T = t − a and
u = u0 + T (u1 + v),
where u0 = u(a) and u1 = u (a). We obtain
D(D + 1)v = T (u0 + T u1 + T v).
This is a Fuchsian equation. We have taken ν = 0, ε = 1, and the resonances are 0 and 1. The solution, which contains three parameters (u0 , u1 , a),
is redundant, because the mapping (u0 , u1 , a) → u is many-to-one, as in the
example of Sect. 1.1.

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6

1 Introduction

1.5.2 Stable manifolds
Let us seek the (one-dimensional) manifold of solutions of
utt − u = 3u2
that decay as t → +∞. Let T = exp(−t). Since ∂t = −T ∂T = −D, we obtain
(D + 1)(D − 1)u = 6u2 .
Letting u = aT + T 2 v, where a is a given constant, we obtain
(D + 1)(D + 3)v = 3(a + T v)2 .
This is not yet a Fuchsian equation, because the right-hand side is not divisible
by T . We therefore let v = a2 + w, and obtain
(D + 1)(D + 3)w = 3[(a + T v)2 − a2 ] = 3T v(2a2 + T v).

We have achieved a Fuchsian reduction with ν = 1 and ε = 1. The resonances
are 0 and −2. The stable manifold is parameterized by a.
1.5.3 Dirichlet problem
Consider the problem
−Δu + f (u) = 0
on a smooth bounded domain Ω, with boundary condition u = ϕ(x) on ∂Ω.
Let d(x) denote the distance from x to the boundary of Ω; it is smooth in a
neighborhood of the boundary; see Part IV. Let u = ϕ + dz and g(x, z) :=
df (ϕ + dz). Substituting into the equation and multiplying by d, we obtain
−d2 Δz − 2d∇d · ∇z + dg(x, z) = 0.
That z admits an expansion in powers of d is a consequence of Schauder theory.
However, scaled Schauder estimates are not sufficient to handle equations of
the form −d2 Δz − ad∇d · ∇z + bz + dg(x, z) = 0, for general values of a and b.
Now, such operators arise naturally from the asymptotic analysis of geometric
problems leading to boundary blowup. We develop an appropriate regularity
theory in Chap. 6 to obtain an expansion of solutions in such cases.
1.5.4 Blowup for an ODE
Consider the equation
utt − 6u2 − t = 0.

(1.5)

We illustrate with this example a practical method for organizing computations. We are interested in solutions that become singular for t = a.

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1.5 Simple examples of reduced Fuchsian equations

7

Leading-order analysis
Let T = t − a. The equation becomes uT T − 6u2 − T − a = 0. We seek a
possible leading behavior of the form u ∼ u0 T ν with u0 = 0. It is convenient
to set up the table1
utt
−6u2 −T −a
Exponent
ν−2
2ν
1 0
Coefficient ν(ν − 1)u0 −6u20 −1 −a
We now seek the smallest exponent in the “Exponent” line in this table.
If ν < 0, the smallest is ν − 2 or 2ν. If these two exponents are distinct,
φ[u0 , ν] is proportional to a power of u0 , and therefore may vanish only if
u0 = 0, which contradicts (1.4). Therefore, ν − 2 = 2ν, or ν = −2. We then
obtain φ[u0 , ν] = 6(u0 − u20 ); hence u0 = 1.
If ν ≥ 0, we obtain 2ν > ν − 2. We are therefore left with the following
cases:
If 0 ≤ ν < 2, φ[u0 , ν] = ν(ν − 1)u0 . Therefore, ν = 0 or 1, corresponding
to solutions such that u ∼ u0 and u ≡ u0 T respectively. These are special
cases of the Cauchy problem in which the Cauchy data are nonzero.
If ν = 2, the terms utt and −a balance each other, leading to 2u0 = −a,
or u ∼ − 21 aT 2 . This is admissible if a = 0.
If ν > 2, the only possibility for a nontrivial balance is a = 0 and ν −2 = 1.
This leads to u ∼ 16 T 3 .

•
•
•

The last two cases correspond to the Cauchy problem in which both Cauchy
data vanish. Since these solutions may be investigated by standard means, we
do not pursue their study any further.
First reduction
Consider first the case u0 = 1, ν = −2. Upon multiplication by T 2 , equation
(1.5) turns into
D(D − 1)u − 6(T u)2 − T 3 − aT 2 = 0.
Let
u=

1
(1 + T ε v(T )).
T2

Substituting into the equation for u, and multiplying through by T 2−ε , we
obtain
1

For ODEs, it is possible to use a variant of Newton’s diagram, as in Puiseux
theory. For PDEs in which u0 may be determined by a differential equation,
rather than an algebraic equation, it is not convenient to do so. For this reason,
we do not use Newton’s diagram.

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8

1 Introduction

(D + ε − 2)(D + ε − 3)v − 6T −ε[(1 + T ε v)2 − 1] − T 2−ε [T 3 + aT 2 ] = 0.
Since
(D + ε − 2)(D + ε − 3) − 12 = (D + ε + 1)(D + ε − 6),
this may be rearranged into
(D + ε + 1)(D + ε − 6)v = 6T ε v 2 + T 4−ε [T + a].
The resonance polynomial is read off from the previous step: P (X) = (X +
1)(X − 6). The resonances are −1 and 6. Since the right-hand side involves
T ε and T 4−ε , we may take ε in the range [0, 4]. The best is to take ε as large
as possible, namely ε = 4. This gives
(D + 5)(D − 2)v = a + T + 6T 4 v 2 .
a
Replacing v by v − 10
leads to a Fuchsian equation with right-hand side
divisible by T . The general results of Chap. 2 yield the formal series solution

u=

a
1
1
− T 2 − T 3 + T 4 (b + · · · ),
T2
10
6

where b is arbitrary. The first reduction is now complete.
Second reduction
The second reduction is carried out as follows: define w by
u=

1
a
1
− T 2 − T 3 + T 4 w(T ).
T2
10
6

a
− 16 T + T 2 w. One finds, by direct computation, that
In other words, v = − 10
w solves
D(D + 7)w = T g(T, a, w),
w(0) = b,

where
g(T, a, w) = 6T 2 −

1
1
− T + T 2w
10 6

2

.

The general results of Chap. 5 show that this problem for w has a unique local
solution. The second reduction is now complete: given any pair (a, b), there is

a unique solution u(t; a, b) of equation (1.5) of the form
u(t; a, b) =

a
1
1
− (t − a)2 − (t − a)3 + b(t − a)4 + · · · ,
(t − a)2
10
6

where the expansion converges on some disk |t − a| < 2R, where R = R(a, b)
depends smoothly on its arguments. The singularity data are (a, b); solutions
have a double pole for t = a, and b is the coefficient of (t − a)4 in the Laurent
expansion of u.

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1.5 Simple examples of reduced Fuchsian equations

9

Stability of singular behavior
To fix ideas, restrict a and b to a neighborhood of 0 in such a way that the series
converges for |t| < 3R(0, 0)/2. We may then compute u and its derivatives for
t = 0, for a = 0. We obtain
u(0; a, b) =

1

2
+ · · · + ba4 + · · · and ut (0; a, b) = 3 + · · · − 4ba3 + · · · .
2
a
a

It follows that
∂(u, ut )
∂(a, b)

=
t=0

−2/a3 + · · · a4 + · · ·
= 14 + · · · = 0.
−6/a4 + · · · −4a3 + · · ·

The map (a, b) → (u(0), ut (0)) therefore satisfies the assumptions of the inverse function theorem near any (a, b) with a = 0 and b both small. To sum
up, we have proved the following theorem:
Theorem 1.2. Consider a solution u = u(t; a, b) with a = 0. If a and b are
small, and if v is a solution with Cauchy data close to (u(0), ut (0)), then
v = u(t; a
˜, ˜b), with (˜
a, ˜b) close to (a, b).
If a becomes large, it is conceivable that the solution has another singularity
between 0 and a. The appropriate stability statement, which involves setting
up a correspondence between singularity data at the two singularities, is left
to the reader.
1.5.5 Singular solutions of ODEs with logarithms
Let us seek singular solutions of

u = u 2 + et .

(1.6)

It is proved in [14, p. 166] that (1.6) has no solution of the form u = T −2 (u0 +
u1 T + · · · ), T = t − a, and that terms of the form T 4 ln T must be included.
However, this leads to higher and higher powers of ln T if the computation
is pushed further. We cope with this difficulty by expanding the solution in
powers of T and T ln T .
Theorem 1.3. There is a family of solutions of (1.6) such that u ∼ 6/T 2.
This family is a local representation of the general solution: the parameters
describing the asymptotics are smooth functions of the Cauchy data at a nearby
regular point.
Proof. The argument is similar to the one just given, and we merely indicate
the differences. The formal solution now takes the form
u=

A
A
A
6
− T 2 − T 3 + T 4 ln T + T 4 v(T, T ln T )
T 2 10
6
14

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(1.7)

10

1 Introduction

with A = ea . Here, v is a power series in two variables T and T ln T , entirely
determined by its constant term. Since we have the form of the solution at
hand, let us directly write the equation solved by v, which is the second
reduced equation:
D(D + 7)v = T 2 −

AT
A
−
+ T 2w
10
6

2

+A

T
T2
+
+ ···
3!
4!

.

(1.8)

It has the general form
D(D + 7)v = T f (T, T ln T, v),

(1.9)

and Theorem 4.3 gives the existence and uniqueness of a local solution with
v(0) prescribed; it is the sum of a convergent power series in T and T ln T .
Let b = v(0). The singularity data are (a, b). We conclude with the stability
analysis. Since A = ea , we have
12 A
1
1
∂u
= 3 T +A − +
∂a
T 5
10 3
∂u
= T 4 + o(T 4 ).
∂b

T 2 + O(T 3 ),

These relations can be differentiated with respect to t. We can therefore compute the Jacobian of the mapping Φ : (a, b) → (u(t0 ), u (t0 )) if t0 is close
enough to zero. In fact, in that case, one may replace ∂u/∂a and ∂u/∂b by
their equivalents. We can then invert the map Φ and conclude, as before, that
we have achieved a local representation of the general solution.

Even though v exhibits branching because of the logarithm, it is obtained
from a single-valued function of two variables by performing a multivalued
substitution. In other words, this representation is a uniformization of the
solution.
1.5.6 Blowup for a PDE
We now move to the next level of difficulty: a PDE that requires logarithmic
terms in the expansion of solutions. Since the manipulations involved are
typical of those required for all the applications to nonlinear waves, we write
out the computations in detail. In particular, all background definitions from
Riemannian geometry are included, so that the treatment is self-contained.
Let us perform the first reduction for the hyperbolic equation:
u = exp u,
where = ∂tt − Δ is the wave operator in n space variables. This equation is
the n-dimensional Liouville equation. In one space dimension, this equation
is exactly solvable; see Sect. 10.6. It was this exact solution that suggested

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1.5 Simple examples of reduced Fuchsian equations

11

the introduction of Fuchsian reduction in the first place [120]. The objective
is to show that near singularities, the equation is not governed by the wave
operator, but by an operator for which the singular set is characteristic. This
suggests that blowup singularities for nonlinear wave equations are not due
to the focusing of rays for the wave operator, and that the correct results
on the propagation of singularities must be based on this Fuchsian principal
part rather than the wave operator. This statement will be substantiated

in Chap. 10, where the other steps of reduction, including stability, will be
carried out.
Leading-order analysis
Let us first define new independent variables:
X 0 = T = φ(x, t) = t − ψ(x),

X i = xi for 1 ≤ i ≤ n.

(1.10)

Note that ∂i T = −ψi and ∂i g = ∂i g if g = g(X), so that Δg = Δ g in
particular. It is convenient to put coordinate indices as exponents, and to use
primed indices to denote derivatives with respect to the coordinates (X, T ).
Lemma 1.4. In these coordinates, the wave operator takes the form
= γ∂T2 −

∂i2 − 2ψi ∂T i

+ (Δψ)∂T ,

(1.11)

i

where
γ = (1 − |∇ψ|2 ).

(1.12)

Proof. For fixed i, ∂ii can be expressed as follows (we write ∂i for δii ∂i ):

(∂i − ψi ∂T )2 = (∂i − ψi ∂T )∂i − ∂i (ψi ∂T )
= (∂i − ψi ∂T )∂i − (Δψ)∂T − ψi ∂iT
= ∂i2 − ψi ∂T i − (Δψ)∂T − ψi (∂i − ψi ∂T )∂T
= ∂i2 − 2ψi ∂T i + |∇ψ|2 ∂T2 − (Δψ)∂T .
The result follows.
By tabulating possible cases as before, we find that there is no consistent
leading term for which u behaves like a power of T ; therefore, we seek u
with logarithmic behavior, and require exp u ∼ u0 T ν . Substituting into the
equation and balancing the most singular terms leads to u0 = 2 and ν = −2.
The leading term is therefore u ≈ ln(2/T 2).

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12

1 Introduction

First Reduction
We define a first renormalized unknown v(X, T ) by
u = ln(

2
) + v(X, T )T ε .
φ2

We leave it to the reader to compute the first reduced equation and check that
ε = 0 leads to a Fuchsian PDE for v. We obtain the resonance polynomial
P (X) = (X + 1)(X − 2). The indices are therefore −1 and 2.
General results from Chap. 2 imply that no logarithms enter the solution

until the term in T 2 , since the smallest positive index is 2; furthermore, since it
is simple, there is a formal solution in powers of T and T ln T , which is entirely
determined by the coefficient of T 2 . To perform reduction, we need to compute
the first few terms of the expansion. Inserting v = v (0) (X) + T v (1) (X) + · · ·
into the first reduction and setting to zero the coefficients of T −2 and T −1 in
it, we obtain
v (0) = ln γ, v (1) = −γ −1 Δψ.
However, it is not possible to continue the expansion with a term v (2) T 2 :
substitution into the equation shows that v (2) does not contribute any term
of degree 0 to the equation. In fact, v (2) is arbitrary, and we must include a
term in R1 (X)T 2 ln T in the expansion.
Second reduction
Define the second renormalized unknown w by
u = ln

2
+ v (0) + v (1) T + R1 T 2 ln T + T 2 w(X, T ),
T2

(1.13)

where R1 will be determined below.
Lemma 1.5. The second reduction leads to the Fuchsian PDE
γ(T ∂T )(T ∂T + 3)w
+ T (Δψ)(R1 + (T ∂T )w) + 2ψ i δii ∂i (R1 + (T ∂T )w)
−T ln T ΔR1 − T Δw
+ 4ψ i δii ∂i (R1 T ln T + T w) + 2(Δψ)(R1 T ln T + T w) − T Δv (1)
= (1 − |∇ψ|2 ) (v (1) + R1 T ln T + T w)2 − [v (1) ]2
+T (v (1) + R1 T ln T + T w)3
1

×

(1 − σ)2 exp(T σ(v (1) + R1 T ln T + T w)) dσ
0

for w.

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(1.14)