Methods for Partial Differential Equations: Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models

473 Pages·2018·4.826 MB·English

by  Marcelo R. Ebert,Michael Reissig (auth.)

Marcelo R. Ebert
Michael Reissig
Methods for
Partial Differential
Equations
Qualitative Properties of Solutions,
Phase Space Analysis,
Semilinear Models

Marcelo R. Ebert • Michael Reissig
Methods for Partial
Differential Equations
Qualitative Properties of Solutions,
Phase Space Analysis, Semilinear Models
MarceloR.Ebert MichaelReissig
UniversityofSaQoPaulo TUBergakademieFreiberg
DepartmentofComputingandMathematics InstituteofAppliedAnalysis
RibeirãoPreto,SãoPaulo,Brazil Freiberg,Germany
ISBN978-3-319-66455-2 ISBN978-3-319-66456-9 (eBook)
https://doi.org/10.1007/978-3-319-66456-9
LibraryofCongressControlNumber:2017963017
MathematicsSubjectClassification(2010):35-01,35-02,35A01,35A02,35A10,35B30,35B33,35B40,
35B44,35E20,35F20,35F35,35J05,35J10,35J99,35K05,35K99,35L05,35L45,35L71,35L99
©SpringerInternationalPublishingAG2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Dedicatedtoour wives Helena andSteffi
Preface
It is the aim of this book to motivate young mathematicians at the postgraduate
leveltostartstudyingdifferenttopicsofthetheoryofpartialdifferentialequations.
The book is addressed to Master and PhD students with interest in this theory.
AfterattendingintroductorycoursesonPDE’s,usuallyattheendofundergraduate
studies,thebackgroundsofthosestudentsmayvarywidely.Sometimesthetheory
onPDE’sisreducedtoFourier’smethodandintegraltransformationsonly.
ThemainpurposeofthisbookistoamplifyanadvancedcourseonPDE’sduring
graduatestudies. The book providesan overviewon differenttopics of the theory
of partial differentialequations. As it is announcedin the title, the explanationof
qualitativepropertiesofsolutionsoflinearmodels,theintroductiontophasespace
analysis on the one hand and modern methods how to treat semilinear models on
theotherhand,formthecoreofthebook.Theauthorshavechosenthesesubtitles
because a deeper knowledge of these topics is an important base from which to
applyordevelopthetheoryofPDE’s.
Usually, introductory courses on PDE’s contain only very few properties of
solutions to basic PDE’s. We will illustrate qualitative properties of solutions of
modelequationsshowingfundamentaldifferencesbetweenpropertiesofsolutions
toLaplace,heat,waveorSchrödingerequations.Adeepknowledgeofsuchprop-
ertieshelpsinunderstandingtheapplicabilityofdifferentmethodsortechniquesto
treatmorecomplexmodels.
Phase space analysis is widely applied in a lot of branches of the theory of
PDE’s. Only a very careful description of Fourier transform of functions or even
distributions together with decomposition techniques of the phase space allows
for attacking a lot of problems from the theory of partial differential equations.
WithouthavingtoolsfromWKBanalysis,thetheoryofpseudo-differentialorpara-
differentialoperators,alotofmodelscannotbetreatedinanoptimalway.Forthis
reasontheauthorsincludedchaptersonbasicsofphasespaceanalysis.
The movefromlinearmodelsto nonlinearonesis a big challenge.Evenif one
addsthemostsimplenonlinearitiesjujpor˙jujp(cid:2)1uasanonlinearright-handsidein
alinearmodelitcausesalotofnewdifficulties.Sometimessuchatermdetermines
newtrendsinthetheory.Theauthorsshowbysemilinearheat,waveorSchrödinger
vii
viii Preface
models the big influence of these nonlinearities on methods, trends and expected
results.Adeepknowledgeofhowtotreatsemilinearmodelssimplifiesthemoveto
understandquasi-ornonlinearmodelsaswell.
TheauthorshaveyearsofexperiencegivingcoursesonPDE’satundergraduate
andgraduatelevelandsupervisingPhDstudents.Thebookcontainstheircombined
teaching and supervising expertise. The courses were held at various universities
throughouttheworld.
In 2009, the first author gave a course about the Cauchy problem for partial
differentialoperatorswithinthe27thBrazilianMathematicsColloquiumatInstituto
Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro. Upon the
invitation of Prof. Sandra Lucente, the second author stayed at the University
of Bari in Italy during November 2013, and gave a series of lectures on partial
differential equations within the “Project Messaggeri della Conoscenza 2012,
ID424”, supported by the government of Italy. The audience was composed of
Master and PhD students not only from University of Bari. This comprehensive
course was only one of a series of nine courses on “Basics of Partial Differential
Equations” the second author gave during the period 2007–2016at the Shanghai
Jiao-Tong University, Kazakh National University Al Faraby Alma-Aty in Kaza-
khstan,theEurasianNationalUniversityL.N.GumilevaAstanain Kazakhstan,the
Hanoi National University of Education and the Hanoi University of Science and
Technology in Vietnam. The second author since 2002 gives the course “Partial
DifferentialEquations1/2” atthe Facultyfor MathematicsandComputerScience
of Technical University Bergakademie Freiberg, and here he supervised 15 PhD
students,severalofthemfromabroad.ThefirstauthorspenttheperiodJuly2014–
July 2015, his sabbatical year, in Freiberg. During this time he gave advanced
seminarson“PartialDifferentialEquations”forMasterstudents.Fromallactivities
arosethejointideatowritethisbook.
Itwillbeourgreatpleasureshouldthisbookstimulateyoungmathematiciansto
becomefamiliarwiththebeautifultheoryofpartialdifferentialequations.
RibeirãoPreto,SãoPaulo,Brazil MarceloR.Ebert
Freiberg,Germany MichaelReissig
June2017
Acknowledgements
Theideatowritethisbookaroseduringthestayofthefirstauthor(July2014–July
2015)attheInstituteofAppliedAnalysisatTUBergakademieFreiberg.Thestay
of the first author was supported by Fundação de Amparo à Pesquisa do Estado
deSãoPaulo(FAPESP),grant2013/20297-8.Thebookwascompletedwithinthe
DFGprojectRE961/21-1andFAPESPGrant2015/16038-2.
TheauthorsthankVladimirGeorgiev,SandraLucente,WinfriedSickel,Mitsuru
Sugimoto, Hiroyuki Takamura and Karen Yagdjian for fruitful discussions on the
contentofsomepartsofthisbook.Moreover,theauthorsthankformerPhDstudents
AbdelhamidMohammedDjaouti,ChristianJäh,WanderleyNunesdoNascimento,
AlessandroPalmieriandMaximilianReichforreadingsomeofthechapters.
Finally,wethankthestafffromBirkhäuserpublishinghouse,inparticular,Sarah
GoobandDr.ThomasHempfling,forthefruitfulco-operationinpreparingthefinal
versionofthisbook.
ix
Contents
PartI
1 Introduction................................................................. 3
2 PartialDifferentialEquationsinModels ................................ 7
2.1 AGeneralConservationLaw....................................... 7
2.2 TransportorConvection............................................ 8
2.3 Diffusion ............................................................ 9
2.4 StationaryModels................................................... 10
2.5 WavesinAcoustics ................................................. 11
2.6 QuantumMechanics................................................ 13
2.7 Gas-andHydrodynamics........................................... 14
2.8 ConcludingRemarks................................................ 14
3 BasicsforPartialDifferentialEquations ................................ 17
3.1 Classification ofLinearPartialDifferentialEquations
ofKovalevskianType............................................... 17
3.2 Classification ofLinearPartialDifferentialEquations
ofSecondOrder..................................................... 19
3.3 ClassificationofLinearSystemsofPartialDifferential
Equations............................................................ 21
3.4 ClassificationofDomainsandStatementofProblems............ 26
3.4.1 StationaryProcesses..................................... 27
3.4.2 NonstationaryProcesses................................. 28
3.5 ClassificationofSolutions.......................................... 32
4 TheCauchy-KovalevskajaTheorem ..................................... 37
4.1 ClassicalVersion.................................................... 37
4.2 AbstractVersion .................................................... 40
xi
xii Contents
4.3 ConcludingRemarks................................................ 43
4.3.1 Generalizations of the Classical
Cauchy-KovalevskajaTheorem......................... 43
4.3.2 Generalizations of the Abstract
Cauchy-KovalevskajaTheorem......................... 44
4.3.3 Applications of the Abstract
Cauchy-KovalevskajaTheorem......................... 45
5 Holmgren’sUniquenessTheorem ........................................ 49
5.1 ClassicalVersion.................................................... 49
5.2 AbstractVersion .................................................... 52
5.3 ConcludingRemarks................................................ 53
5.3.1 ClassicalHolmgrenTheorem ........................... 53
5.3.2 AbstractHolmgrenTheorem............................ 54
6 MethodofCharacteristics................................................. 57
6.1 QuasilinearPartialDifferentialEquationsofFirstOrder......... 57
6.2 The Notion of Characteristics: Relation to Systems
ofOrdinaryDifferentialEquations................................. 58
6.3 InfluenceoftheInitialCondition................................... 59
6.4 ApplicationoftheInverseFunctionTheorem..................... 61
6.5 Summary............................................................ 64
6.6 Examples............................................................ 65
6.6.1 ContinuityEquation ..................................... 65
6.6.2 AnExampleofaCharacteristicCauchyProblem...... 65
6.7 ConcludingRemarks................................................ 66
7 Burgers’Equation.......................................................... 69
7.1 ClassicalBurgers’Equation........................................ 69
7.2 OtherModelsRelatedtoBurgers’Equation....................... 73
7.3 ConcludingRemarks................................................ 74
PartII
8 LaplaceEquation—PropertiesofSolutions—StartingPoint
ofEllipticTheory........................................................... 79
8.1 PoissonIntegralFormula........................................... 79
8.1.1 HowDoesPotentialTheoryComeintoPlay?.......... 79
8.1.2 Green’sFunctionandPoissonIntegralFormula ....... 82
8.2 PropertiesofHarmonicFunctions ................................. 87
8.2.1 MeanValueProperty .................................... 87
8.2.2 Maximum-MinimumPrinciple.......................... 87
8.2.3 RegularityofHarmonicFunctions...................... 88
8.2.4 Weyl’sLemmaandInteriorRegularity................. 89
8.3 OtherPropertiesofEllipticOperatorsorEllipticEquations...... 91
8.3.1 Hypoellipticity........................................... 91
8.3.2 LocalSolvability......................................... 92