Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

291 Pages·2007·2.52 MB·English

by  Horst Reinhard Beyer (auth.)

Lecture Notes in Mathematics (cid:49)(cid:56)(cid:57)(cid:56)
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Horst Reinhard Beyer
Beyond Partial Differential
Equations
On Linear and Quasi-Linear
Abstract Hyperbolic
Evolution Equations
(cid:65)(cid:66)(cid:67)
Author
HorstReinhardBeyer
CenterforComputation&Technology
LouisianaStateUniversity
(cid:51)(cid:51)(cid:48)JohnstonHall
BatonRouge
LA70803
USA
e-mail:[email protected]
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DedicatedtoGodtheFather
Preface
SemigroupTheoryusesabstractmethodsofOperatorTheorytotreatinitialbound-
aryvalueproblemsforlinearandnonlinearequationsthatdescribetheevolutionofa
system.Duetothegeneralityofitsmethods,theclassofsystemsthatcanbetreated
inthiswayexceedsbyfarthosedescribedbyequationscontainingonlylocaloper-
atorsinducedbypartialderivatives,i.e.,PDEs.Inparticular,thatclassincludesthe
systemsofQuantumTheory.
Another important application of semigroup methods is in field quantization.
Simple examples are given by the cases of free fields inMinkowski spacetime like
Klein-Gordon fields, the Dirac field and the Maxwell field, whose field equations
aregivenbysystemsoflinearPDEs.Thesecondquantizationofsuchafieldreplaces
the field equation by a Schro¨dinger equation whose Hamilton operator is given by
thesecondquantizationofanon-localfunctionofaself-adjointlinearoperator.That
operator generates the semigroup given by the time-development of the solutions
of the field equation corresponding to arbitrary initial data as a function of time.
Moregenerally,inthesecasesthestructuresusedintheformulationofawell-posed
abstract initial value problem for the field equation also provide the mathematical
framework for the quantization of the field. Quantum Theory is an abstract theory,
thereforeitshouldbeexpectedthatonlyanabstractapproachtoclassicalfieldequa-
tions using methods from Operator Theory is capable of providing the appropriate
structuresforquantizationinthelesssimplecasesofnonlinearfields,likethegravi-
tationalfielddescribedbyEinstein’sfieldequations.
Ademonstrationofthestrengthofsemigroupmethodscanbeseeninthefirstrig-
orousproofofwell-posedness(localintime)oftheinitialvalueproblemforquasi-
linear symmetric hyperbolic systems by T. Kato in 1975 in [110]. This result is a
particular application of a theorem on the well-posedness of the initial value prob-
lemforabstractquasi-linearequations1,whichhasbeensuccessfullyappliedalsoto
Einstein’s equation’s [102], the Navier-Stokes equations, the equations of Magne-
tohydrodynamics [109] and more. To my knowledge, there is no other approach to
quasi-linearequationsleadingtoatheoremofsuchgenerality.
1SeeTheorem11.0.7below.
VIII Preface
The semigroup approach goes beyond that of a tool for deciding the well-
posedness of initial boundary value problems. For a autonomous nonlinear equa-
tion the important question of the linearized stability of a particular solution leads
onaspectralproblemfortheoperatorgeneratingthesemigroupgivenbythetime-
development of the solutions of the linearized equation around that solution corre-
spondingtoarbitraryinitialdataasafunctionoftime.2Thesemethodsalsoprovide,
forautonomouslinearequations,arepresentationofthesolutionoftheinitialvalue
problemasanintegralovertheresolventoftheinfinitesimalgeneratoroftheasso-
ciated semigroup.3 The resolvent operators can often be represented in the form of
integraloperatorswithkernelswhicharedefinedintermsofspecialfunctions.This
is not only true in simple cases where the generator is a partial differential opera-
torwithconstantcoefficients,butalsoinanumberofcasesinvolvingnon-constant
coefficients.Inthisway,anintegralrepresentationofthesolutionoftheinitialvalue
problemisachievedforallsuchcases.4Finally,semigroupmethodsprovideaframe-
work which is general enough to include numerical forms of evolution equations,
openingthepossibilityofcomputingtrueerrorestimatestotheexactsolution,rather
thanresidualerrors.
Thisshouldnotgivetheimpressionthatthesemigroupapproachcouldreplaceall
‘hard’analysisfacts.Instead,itreducessuchapplicationtoabareminimum,which
givestheapproachitsefficiency.5Forinstance,resultsfromharmonicanalysisorthe
theoryofsingularintegraloperatorshaveapplicationsinsocalled‘commutatores-
timates’wherethecommutatorofanintertwiningoperatorwiththeprincipalpartof
apartialdifferentialoperator,usuallytwounboundedoperators,hastobeestimated.
Toachievemostgeneralresults,itisoftennecessarytochooseintertwiningoperators
asnon-localoperators.
These methods are especially attractive if not inevitable for theoretical physi-
cists in view of their comprehension of classical physics and quantum physics. In
addition, their efficiency is of advantage in view of time restrictions in the math-
ematics education of physicists. In spite of their power, efficiency and versatility,
semigroupmethodsaresurprisinglylittleusedintheoreticalphysics.6,7Thisappears
toberelatedtotwomisconceptions.
First, because of the requirements of Special Relativity and General Relativity,
currentproblemsinfundamentaltheoreticalphysicsnecessarilyleadonhyperbolic
2Forinstance,seeChapter5.4.
3Roughlyspeaking,foraprecisestatementsee,e.g.,Chapter4.3.
4See,e.g.,[25].
5Butitisthebeliefoftheauthorthatthenecessityoftheuseof‘hard’toachieveanalyt-
ical facts in the solution of a problem indicates that its structure has not yet been fully
understood.
6Paradoxically, in some sense, one could also hold the opposite view that these methods
havebeenusedforalongtimeintheoreticalphysics,mostlywithoutrealizingthattheyare
rootedininSpectralTheory.
7Inaddition,theauthorisnotawareofasingleintroductiontoPDEbasedsolelyonSemi-
groups/OperatorTheory,althoughthiswouldhavebeenpossibleevenbeforetheappear-
anceofclassicalintroductionslike[20]thatuselessgeneralmethods.
Preface IX
partialdifferentialequationsystems(‘hyperbolicproblems’).Standardtextsonsemi-
groupsoflinearoperatorsmainlyfocusonapplicationstoparabolicpartialdifferen-
tialequationsystems(‘parabolicproblems’).Differentlytothehyperboliccase,this
leads to the consideration of strongly continuous semigroups which are in addition
analytic. Apparently, the consideration of parabolic problems originates from a fo-
cusonengineeringapplications.Engineeringsciencespredominantlyapplyclassical
Newtonian physics where signal propagation speeds are not limited by the speed
of light in vacuum as this is the case in Special Relativity/General Relativity. For
example,theevolutionofacompactlysupportedtemperaturefieldinspaceattime
t “ 0 by the parabolic heat equation leads to a temperature field that has no com-
pact support for every t ą 0. As consequence, in the evolution signal propagation
speeds occur that exceed the speed of light in vacuum, and hence the equation is
incompatible with Special Relativity. The same is also true for Schro¨dinger equa-
tions.8 The evolution of hyperbolic partial differential equations systems preserves
thecompactnessofthesupportofthedataundertimeevolution.Tomyexperience,
the focus on engineering applications has lead to the quite common misconception
among physicists, and to some extent also among mathematicians working in the
field of partial differential equations, that semigroup methods cannot be applied to
hyperbolicproblems.
Second,tomyexperience,anothercommonmisconceptionisthatsemigroupsof
linearoperatorscanonlybeappliedtosystemsoflinearpartialdifferentialequations.
Thismightbeinfluencedbythefactthatmoststandardtextsonsemigroupsoflinear
operators,indeed,focusmainlyonsuchapplications.
Asaconsequence,thecourseshouldleadasrapidlyaspossiblefromautonomous
linearequationstothenonlinear(quasi-linear)equationswhicharenowseeninthe
hyperbolic problems emanating from current physics. Particular stress is on wave
equationsandHermitianhyperbolicsystems.Thelastcovertheequationsdescribing
interacting fields in physics and therefore the major part of nonlinear equations
occurring in fundamental physics. Throughout the course applications to prob-
lemsfromcurrentrelativistic(‘hyperbolic’)physicsareprovided,whichdisplaythe
potentialofthemethodsinthesolutionofcurrentproblemsinphysics.Theseinclude
problems from black hole physics, the formulation of outgoing boundary condi-
tionsforwaveequationsandthetreatmentofadditionalconstraints.Thelasttwoare
important current problems in the numerical evolution of Einstein’s field equations
for the gravitational field. Some of the examples contain new unpublished results
oftheauthor.Tomyknowledge,themajorpartofthematerialinthesecondpartof
thenotes,includingnon-autonomousandquasi-linearHermitianhyperbolicsystems,
hasappearedonlyinsideresearchpapers.
The orientation of this course towards abstract quasi-linear evolution equations
made it necessary to omit a number of topics that were not directly important for
achieving itsgoals. Onthe other hand, texts on semigroups of linear operators that
considerthosetopicsareavailable[39,47,52,57,90,99,106,120,168,179,224].Also,
8ThiswasthereasonforthedevelopmentofQuantumFieldTheory.
X Preface
thereisatheoryofnonlinearsemigroupsthatlargelyparallelsthatofsemigroupsof
linearoperators[15,19,88,138,146,167].
This course assumes some basic knowledge of Functional Analysis which can
be found, for instance, in the first volume of [179]. Some examples assume more
specializedknowledgeofpropertiesofself-adjointlinearoperatorsinHilbertspaces
whichcanbefound,forinstance,inthesecondvolumeof[179].Inaddition,some
applications assume basic knowledge of Sobolev spaces of the L2-type as is pro-
vided,forinstance,in[217].Otherwisethiscourseisself-contained.Thematerialis
presentedinascompressedaformaspossibleanditsresultsareformulatedinview
towards applications. In general, theorems contain their full set of assumptions, so
that a study of their environment is not necessary for their understanding. For this
reason and to limit the size of this text, shorter definitions appear as part of theo-
rems.Inaddition,theabstracttheoryanditsapplicationsappearinseparatechapters
toallowthereadertoestimatethenecessarytimetoacquirethetheory.
Chapters2–5constitutethenotesofacoursegivenattheDepartmentofMathe-
maticsoftheLouisianaStateUniversityinBatonRougeinSpring2005.Theypro-
vide a basic introduction into the properties of strongly continuous semigroups on
Banach spaces and applications to autonomous linear hyperbolic systems of PDEs
from General Relativity and Astrophysics. The theoretical part is kept to a mini-
mumwithaviewtoapplicationsinthefieldofhyperbolicPDEs.Itisformulatedin
a way which is expected to be natural to readers with a knowledge of the spectral
theoryofself-adjointlinearoperatorsinHilbertspaces.Anexception,tothisrestric-
tiontotheminimuminthesechapters,isthetreatmentoftheintegrationofBanach
space-valued maps which is more detailed than is usual in most other comparable
texts.Thisisduetothefactthatsuchintegrationisabasictoolwhichisusuallynot
coveredinstandardFunctionalAnalysiscourses,atleastnottoanextentneededin
thestudyofsemigroupsofoperators.Otherwise,thetheoreticalmaterialisstandard
andforthisreasononlyfewreferencestoliteraturearegiven.Formorecomprehen-
siveintroductionsintothetheoryofsemigroupsoflinearoperatorsthatgiveamore
exhaustivelistofreferences,thereaderisreferred,forinstance,to[57,90,99,168].
Chapters6–12arethenotesofasubsequentcourseatthesameplaceinSpring
2006.Theyintroduceintothefieldofabstractevolutionequationswithapplications
tonon-autonomouslinearandquasi-linearhyperbolicsystemsofPDEs.Thissecond
part of the notes follows closely the late Tosio Kato’s 1993 paper ‘Abstract evolu-
tionequations,linearandquasilinear,revisited’inProceedingsoftheInternational
Conference in Memory of Professor Kosaku Yosida held at RIMS, Kyoto Univer-
sity,Japan,July29-Aug.2,1991,[114].ThoseresultsaremoregeneralthanKato’s
well-knownolderresults9 [107–109]inthattheydon’tassumespecialpropertiesof
theunderlyingBanachspaces.ProofsofKato’sresultsareaddedalongwithdetailed
examples displayingtheirapplication toproblems incurrent relativisticphysics. In
afewplaces,Lemmatawereaddedtohisoutlinethatappearednecessaryforthose
proofs.
9Thoseresultsassumethereflexivityandinsomeplacesalsotheseparabilityoftheunder-
lyingBanachspaces.
Preface XI
Acknowledgments
IamespeciallyindebtedtoOlivierSarbach,whodiligentlyreviewedthemajorpart
ofthenotesandrecommendedvaluablemodificationsandcorrections.Also,Ithank
FrankNeubranderandStephenShipmanforilluminatingdiscussionsinconnection
with the notes. Finally, I thank all who have worked on the book, especially the
editorialandproductionstaffofSpringer-Verlag.
BatonRouge,September2006 HorstBeyer