Geometric analysis of hyperbolic differential equations : an introduction

130 Pages·2010·1.29 MB·English

by  S Alinhac

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LondonMathematicalSocietyLectureNoteSeries:374
Geometric Analysis of Hyperbolic
Differential Equations:
An Introduction
S. ALINHAC
Universite´ Paris-Sud,Orsay
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,
Sa˜oPaulo,Delhi,Dubai,Tokyo
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521128223
(cid:1)C S.Alinhac2010
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2010
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Alinhac,S.(Serge)
Geometricanalysisofhyperbolicdifferentialequations:anintroduction/S.Alinhac.
p. cm.–(LondonMathematicalSocietylecturenoteseries;374)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-12822-3(pbk.)
1.Nonlinearwaveequations. 2.Differentialequations,Hyperbolic. 3.Quantumtheory.
4.Geometry,Differential. I.Title. II.Series.
QA927.A3886 2010
515(cid:2).3535–dc22 2010001099
ISBN978-0-521-12822-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Preface pagevii
1 Introduction 1
2 Metricsandframes 8
2.1 Metrics,duality 8
2.2 Opticalfunctions 12
2.3 Nullframes 12
3 Computingwithframes 17
3.1 Metricconnexion 17
3.2 Submanifolds 20
3.3 Hessianandd’Alembertian 21
3.4 Framecoefficients 24
4 Energyinequalitiesandframes 29
4.1 Theenergy–momentumtensor 29
4.2 Deformationtensor 31
4.3 Energyinequalityformalism 33
4.4 Energy 34
4.5 Interiortermsandpositivefields 35
4.6 Maxwellequations 41
4.6.1 Duality 41
4.6.2 Energyformalism 43
5 Thegoodcomponents 45
5.1 Theproblem 45
5.2 Animportantremark 46
5.3 Ghostweightsandimprovedstandardenergyinequalities 47
5.4 Conformalinequalities 53
v
vi Contents
6 Pointwiseestimatesandcommutations 57
6.1 Pointwisedecayandconformalinequalities 58
6.2 Commutingfieldsinthescalarcase 59
6.3 ModifiedLorentzfields 61
6.4 CommutingfieldsforMaxwellequations 63
7 Framesandcurvature 65
7.1 Thecurvaturetensor 65
7.2 Opticalfunctionsandcurvature 67
7.3 Transportequations 69
7.4 Ellipticsystems 71
7.5 Mixedtransport–ellipticsystems 75
8 Nonlinearequations,aprioriestimatesandinduction 77
8.1 AsimpleODEexample 77
8.2 Localexistencetheory 80
8.3 Blowupcriteria 81
8.4 InductionontimeforPDEs 84
9 Applicationstosomequasilinearhyperbolicproblems 88
9.1 Quasilinearwaveequationssatisfyingthenullcondition 89
9.2 Quasilinearwaveequations 96
9.3 Lowregularitywell-posednessforquasilinearwaveequations 99
9.4 StabilityofMinkowskispacetime(firstversion) 102
9.5 L2conjectureonthecurvature 106
9.6 StabilityofMinkowskispacetime(secondversion) 108
9.7 Theformationofblackholes 113
References 114
Index 117
Preface
Thefieldofnonlinearhyperbolicequationsorsystemshasseenatremendous
development since the beginning of the 1980s. We are concentrating here on
multidimensional situations, and on quasilinear equations or systems, that is,
when the coefficients of the principal part depend on the unknown function
itself. The pioneering works by F. John, D. Christodoulou, L. Ho¨rmander,
S. Klainerman, A. Majda and many others have been devoted mainly to the
questionsofblowup,lifespan,shocks,globalexistence,etc.Someoverviewof
theclassicalresultscanbefoundinthebooksofMajda[42]andHo¨rmander
[24].Ontheotherhand,ChristodoulouandKlainerman[18]provedinaround
1990thestabilityofMinkowskispace,astrikingmathematicalresultaboutthe
CauchyproblemfortheEinsteinequations.Afterthat,manyworkshavedealt
withdiagonalsystemsofquasilinearwaveequations,sincethisiswhatEinstein
equations reduce to when written in the so-called harmonic coordinates. The
mainfeatureofthisparticularcaseisthatthe(scalar)principalpartofthesystem
isawaveoperatorassociatedtoauniqueLorentzianmetricontheunderlying
space-time.Thisisinstrongcontrastwiththemorecomplicatedcaseofgeneral
symmetric quasilinear systems: the compressible isentropic Euler equations,
forinstance,canbeviewedasaquasilinearwaveequationcoupledtoavector
field;thesystemofnonlinearelasticityinvolvestwodifferentwaveequations,
etc.
Iconsiderhereonlythecaseofquasilinearwaveequations.Weobservetwo
maindomainsofinterest:thestudyofglobalsmoothsolutions,andthestudyof
lowregularity solutions,bothdomainsbeingconnected.Thestrikingfeatureis
theunityofthetechniquesandideasusedintheworksonthesedomains:The
emphasisisalwaysongooddirectionsandgoodcomponents,thesecomponents
beingcomponentsoftensorsrelativetosomespecialframes,thenullframes.
Hence the observed unity comes from the fact that most concepts, such as
vii
viii Preface
metrics,connexions,curvature,etc.,areborrowedfromLorentziangeometry.
Thisis,ofcourse,relatedtomathematicalworkbyPenroseandcollaboratorsin
thedomainofgeneralrelativity,wherenullframeshavebeenusedextensively
(see,forinstance,PenroseandRindler[43]).
SincetheworkofChristodoulouandKlainermancitedabove,manymath-
ematicalpapersonthesubjectofquasilinearwaveequationsorEinsteinequa-
tionsusethelanguageofLorentziangeometryanddealwithenergy–momentum
tensors, deformation tensors, etc. However, there seem to be some difficul-
ties:RiemanniangeometrybooksdonotincludethespecificLorentziantools
such as null frames; most relativity books do not include a description of
the relevant mathematical techniques. Let us, however, draw attention to the
booksofHawkingandEllis[23]andRendall[44];whichincludesubstantial
mathematics.
IbelievethattheuseofLorentziantoolsinthemathematicalstudyofnon-
linear hyperbolic systems is going to intensify further, even in the aspects
of the field not directly related with general relativity. This is what we call
“geometric analysis of hyperbolic equations.” It is true that there are exam-
ples of nonlinear wave equations which are perturbations of the standard
wave equation by small nonlinear terms, where it is enough to consider only
the geometry of the standard wave equation, that is, the Minkowski metric:
Theseexamplesarestriking,butthepossibilityofthissimplificationseemsto
be related to the fact that one is considering only small solutions; for large
solutions,webelievethatitwillbenecessarytotakeintoaccountthegeometry
ofthelinearizedoperator,thatis,aLorentzianmetricdependingonthesolution
itself.
Thisbookismeantforpeoplewantingtoaccessthemathematicalliterature
onthesubjectofquasilinearwaveequationsorEinsteinequations.Itsgoalis
twofold:
(i) To give to analysts in the field of partial differential equations (PDEs) a
self-containedandelementaryaccesstothenecessarytoolsofLorentzian
geometry,
(ii) Toexplainthefundamentalideasconnectedwiththeuseofnullframes.
This book can be read by students or researchers with an elementary back-
ground in distribution theory and linear PDEs, specifically hyperbolic PDEs.
Noknowledgeofdifferentialgeometryisrequired.Thoughthelargestpartof
thetextisaboutgeometricconcepts,thisbookisnotabookaboutLorentzian
geometry: it introduces the geometric tools required to understand the mod-
ern PDE literature only as and when they are needed. The author not being
ageometer,Ideliberatelychosetogivenaiveandself-containedproofstoall