Table Of ContentProgress in Nonlinear Differential Equations
and Their Applications
Volume71
Editor
HaimBrezis
Universite´ PierreetMarieCurie
Paris
and
RutgersUniversity
NewBrunswick,N.J.
EditorialBoard
AntonioAmbrosetti,ScuolaInternationaleSuperiorediStudiAvanzati,Trieste
A.Bahri,RutgersUniversity,NewBrunswick
FelixBrowder,RutgersUniversity,NewBrunswick
LuisCaffarelli,TheUniversityofTexas,Austin
LawrenceC.Evans,UniversityofCalifornia,Berkeley
MarianoGiaquinta,UniversityofPisa
DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh
SergiuKlainerman,PrincetonUniversity
RobertKohn,NewYorkUniversity
P.L.Lions,UniversityofParisIX
JeanMawhin,Universite´ CatholiquedeLouvain
LouisNirenberg,NewYorkUniversity
LambertusPeletier,UniversityofLeiden
PaulRabinowitz,UniversityofWisconsin,Madison
JohnToland,UniversityofBath
Satyanad Kichenassamy
Fuchsian Reduction
Applications to Geometry, Cosmology,
and Mathematical Physics
Birkha¨user
Boston • Basel • Berlin
SatyanadKichenassamy
Universite´deReimsChampagne-Ardenne
Moulin de la Housse, B.P. 1039
F-51687 Reims Cedex 2
France
[email protected]
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To my parents
Preface
The nineteenth century saw the systematic study of new “special functions”,
suchasthehypergeometric,Legendreandellipticfunctions,thatwererelevant
in number theory and geometry, and at the same time useful in applications.
Tounderstandthepropertiesofthesefunctions,itbecameimportanttostudy
their behavior near their singularities in the complex plane. For linear equa-
tions, two cases were distinguished: the Fuchsian case, in which all formal
solutions converge,and the non-Fuchsian case. Linear systems of the form
du
z +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
nonlinearequationswasincomplete,andthe Painlev´e–Gambierclassification,
forsecond-orderscalarequationsofspecialform,leftnohopeoffindinggeneral
abstract results.
The twentieth century saw, under the pressure of specific problems, the
developmentofcorrespondingresultsforpartialdifferentialequations(PDEs):
The Euler–Poisson–Darbouxequation
λ
u + u −Δu=0
tt t
t
and its elliptic counterpart arise in axisymmetric potential theory and in the
methodofsphericalmeans;italsocomesupinspecialreductionsofEinstein’s
equations.Inparticular,onerealizedthatequationswithdifferentvaluesofλ
couldbe relatedtoeachotherby transformationsu(cid:2)→tmu.Ellipticproblems
in corner domains and problems with double characteristics also led to fur-
ther generalizations.Thisdevelopmentwasconsideredas fairlymatureinthe
1980s; it was realized that some problems required complicated expansions
withlogarithmsandvariablepowers,beyondthescopeofexistingresults,but
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 for-
mal 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 arosewhen it became necessaryto solve
Fuchsian problems arising from other parts of mathematics, or other fields.
The convergence of the “ambient metric” realizing the embedding of a Rie-
mannianmanifoldinaLorentzspacewithahomothetycouldnotbeprovedin
evendimensions.When,inthewakeoftheHawking–Penrosesingularitytheo-
rems,itbecamenecessarytolookforsingularsolutionsofEinstein’sequations,
existing results covered only very special cases, although the field equations
appeared similar to the Euler–Poisson–Darbouxequation. Numerical studies
of suchspace-times led to spiky behavior:were these spikes artefacts?indica-
tions of chaotic behavior?
OtherproblemsseemedunrelatedtoFuchsianPDEs.Fortheblowupprob-
lemfornonlinearwaveequations,againintheeighties,Ho¨rmander,John,and
their coworkerscomputedasymptoticestimatesofthe blowuptime—whichis
notaLorentzinvariant.Forelliptic problemsΔu=f(u)withmonotonenon-
linearities,solutionswith infinite datadominate allsolutions,andcomeupin
several contexts; the boundary behavior of such solutions in bounded C2+α
domains is nota consequence ofweightedSchauder estimates.Outside math-
ematics, we may mention laser collapse and the weak detonation problem.
In astrophysics, stellar models raise similar difficulties; equations are singu-
lar at the center, and one would like to have an expansion of solutions near
the singularityto startnumericalintegration.Also, the theory ofsolitons has
provided,from1982on,aplethoraofformalseriessolutionsforcompletelyin-
tegrablePDEs,ofwhichonewouldliketoknowwhethertheyrepresentactual
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+Tmv,
wheresisknowninclosedform,issingularforT =0,andmayinvolveafinite
number of arbitrary functions. The function v determines the regular part of
u.This representationhas the sameadvantagesas anexactsolution,because
one can prove that the remainder Tmv is indeed negligible for T small. In
particular,it is availablewherenumericalcomputationfails;itenables one to
compute whichquantitiesbecome infinite andatwhatrate,andto 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
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 vari-
ablesandunknowns,into anasymptoticallyscale-invariantPDEorsystemof
PDEs
Lv =f[v]
such that (i) one can introduce appropriate variables (T,x ,...) such that
1
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 involvederivatives of v. After reduction to a first-ordersystem, one
is usually led to an equation of the general form
(cid:2) (cid:3)
d
T +A w =f[T,w],
dT
wheretheright-handsidevanishesforT =0.PDEsofthisformwillbecalled
“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
dividingbyapowerofT,v willbecalledtherenormalizedunknown.Typically,
the reduced Fuchsianequations havenonsmooth coefficients,and logarithmic
termsinparticulararetheruleratherthantheexception.Sincethecoefficients
and nonlinearities are not required to be analytic, it will even be possible to
reduce certain equations with irregular singularities to Fuchsian form. Even
thoughLisscale-invariant,smaynothavepower-likebehavior.Also,inmany
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.
PartI describes a systematic strategy for achieving reduction. A few gen-
eralprinciplesthatgovernthe searchforareducedformaregiven.The listof
examples of equations amenable to reduction presented in this volume is not
meantto be exhaustive.In fact,everynew applicationofreductionso farhas
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
transformedproblemisgenerallynotamenabletoclassicalresultsonsingular
PDEs.
Part III presents applications. It should be accessible after an upper-
undergraduate course in analysis, and to nonmathematicians, provided they
take for granted the proofs and the theorems from the other parts. Indeed,
thediscussionofideasandapplicationshasbeenclearlyseparatedfromstate-
ments of theorems and proofs, to enable the volume to be read at various
levels.
x Preface
Part IV collects general-purpose results, on Schauder theory and the dis-
tancefunction(Chapter12),andontheNash–Moserinversefunctiontheorem
(Chapter 13).Togetherwith the computations workedout 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. TheBriot–BouquetanalysisofsingularitiesofsolutionsofnonlinearODEs
offirstorder,continuedbyPainlev´eandhisschoolforequationsofhigher
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. AnumberofspecialcasesforsimpleODEshavebeenrediscoveredseveral
times; a familiarexample is the constructionofradialsolutions ofnonlin-
ear elliptic equations, which leads to Fuchsian ODEs with singularity at
r =0.
In retrospect, reduction techniques are the natural outgrowth of what is tra-
ditionally calledthe “Weierstrassviewpoint” in complex analysis,as opposed
totheCauchyandRiemannviewpoints.Thisviewpoint,fromthepresentper-
spective,putsexpansionsatthemainfocusofinterest;allrelevantinformation
isderivedfromthem.Forthisapproachtoberelevantbeyondcomplexanaly-
sis,itwasnecessarytounderstandwhichaspectsoftheWeierstrassviewpoint
admit a generalizationto nonanalytic problems with nonlinearities—andthis
generalization required a mature theory of nonlinear PDEs which was devel-
opedrelativelyrecently.Thedevelopmentofreductiontechniquesintheearly
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 searchfor a weak functional setting that would include
blowup singularities for the simplest nonlinear wave equations.
4. Therediscoveryofcomplexanalysisstimulatedbytheemergenceofsoliton
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.
Preface xi
On a more personalnote, a number of mathematicians have,directly or indi-
rectly,helpedtheauthorintheemergenceofreductiontechniques: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
whomI havemetorread,including theanonymousreferees.H.Brezis,whose
mathematical influence may be felt in severalof 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 Birkha¨user, for their kind help with this
project.
Paris Satyanad Kichenassamy
February 27, 2007
