Table Of Content

Frontiers in Mathematics Rainer Picard · Des McGhee Sascha Trostorff · Marcus Waurick A Primer for a Secret Shortcut to PDEs of Mathematical Physics Frontiers in Mathematics AdvisoryBoard LeonidBunimovich(GeorgiaInstituteofTechnology,Atlanta) WilliamY.C.Chen(NankaiUniversity,Tianjin) BenoîtPerthame(SorbonneUniversité,Paris) LaurentSaloff-Coste(CornellUniversity,Ithaca) IgorShparlinski(TheUniversityofNewSouthWales,Sydney) WolfgangSprößig(TUBergakademieFreiberg) CédricVillani(InstitutHenriPoincaré,Paris) Moreinformationaboutthisseriesathttp://www.springer.com/series/5388 Rainer Picard • Des McGhee (cid:129) Sascha Trostorff (cid:129) Marcus Waurick A Primer for a Secret Shortcut to PDEs of Mathematical Physics RainerPicard DesMcGhee Institutfu¨rAnalysis DepartmentofMathematicsandStatistics TUDresden UniversityofStrathclyde Dresden,Germany Glasgow,UK SaschaTrostorff MarcusWaurick MathematischesSeminar DepartmentofMathematicsandStatistics Christian-Albrechts-Universita¨tzuKiel UniversityofStrathclyde Kiel,Germany Glasgow,UK ISSN1660-8046 ISSN1660-8054 (electronic) FrontiersinMathematics ISBN978-3-030-47332-7 ISBN978-3-030-47333-4 (eBook) https://doi.org/10.1007/978-3-030-47333-4 MathematicsSubjectClassification:35F,35-01 ©SpringerNatureSwitzerlandAG2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownor hereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbookare believedtobetrueandaccurateatthedateofpublication. Neitherthepublishernortheauthorsortheeditors giveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissions thatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmaps andinstitutionalaffiliations. ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.com,bytheregisteredcompany SpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Introduction A typicalentry pointinto the field of (linear)partialdifferentialequationsis to consider generalpolynomialsP (∂)in∂ :=(∂ ,...,∂ )with(complexorreal)matrixcoefficients. 0 n Here∂ denotesthepartialderivativewithrespecttothevariableinthepositionlabelled k with1 k ∈ {0,...,n}, n ∈ N. Evenif we discusssolutionsonlyin thewholeEuclidean spaceRn+1 thesolutiontheoryforanequationoftheform P (∂)u=f involving a general partial differential operator P (∂) is quite involved and one quickly restrictsattentiontoveryspecificpolynomials.Indeed,theequationsrelevanttoapplica- tionsarenotthatvaried.Onecommonlyinvestigatesthreesubclasses,looselylabelledas elliptic,parabolic,andhyperbolic,topresentspecificsolutionmethodsforeachofthem. However,whenviewedfromtherightperspectivethereisasinglesubclasscontaining these three types (and manymore), which can be characterizedconvenientlyand solved with one and the same method. To explain the correspondingrigorousframeworkis the objectiveofthistext. Thetheorywewillpresentinthisbookisrootedin[57],withsomefirstgeneralizations tobefoundin[55,59].Weshallreferalsoto[63,79,83,87,88]forgeneralizationstowards nonlinear or non-autonomous setups. The interested reader will find a more detailed surveyin[68,75].Inthepresentbook,however,weshallpresentthecoreyetsurprisingly elementarysolutiontheoryforwhatwewillcallevolutionaryequations. Thestructureofthisclassofpartialdifferentialexpressionscanbeformallydescribed by two matrices2 M ,M ∈ R(N+1)×(N+1), N ∈ N. The partial differential operator 0 1 1Notethatweusuallyprefertostartournumberingwith0.Inparticular,Ndenotesthesetofnon- negativeintegers. 2Indeed,keepinginmindthatacomplexnumberx+iy canbeunderstoodasa(2×2)-matrixof theform (cid:2) (cid:3) x −y , y x wherex,y ∈R,wemayactuallyassumethatM0andM1haveonlyrealentries. v vi Introduction P (∂)willthenbeassumedtobeoftheform (cid:4) (cid:6) P (∂)=∂ M +M +A (cid:5)∂ , (1) 0 0 1 (cid:4) (cid:6) where A (cid:5)∂ denotesa polynomialin(cid:5)∂ := (∂ ,...,∂ ), that is, borrowingjargonfrom 1 n applied fields, only in the “spatial” variables, if we consider ∂ to be the derivative 0 with respect to “time”. In this terminology,if we focus on “relevant” partial differential equations, we may focuson first-order-in-timesystems. Moreover,in standard cases we have structural features of P (∂) which narrow down the class of differential operators evenfurther.Weassume3 (cid:4) (cid:6) (cid:4) (cid:6) A∗ −(cid:5)∂ =−A (cid:5)∂ and (cid:4) (cid:6) M =M∗and ReM := 1 M +M∗ ≥c >0. (2) 0 0 1 2 1 1 0 Inapplications,thelatterpositivedefinitenessconstraintisrarelysatisfied.However,after asimpleformaltransformation4 weget (cid:4) (cid:6) (cid:4) (cid:4) (cid:6)(cid:6) ∂ M +M(cid:7) +A (cid:5)∂ =exp(−(cid:3)m ) ∂ M +M +A (cid:5)∂ exp((cid:3)m ) 0 0 1 0 0 0 1 0 (cid:4) (cid:6) (cid:4) (cid:6) 3Withthis,A (cid:5)∂ becom(cid:8)esskew(cid:9)-selfadj(cid:4)oin(cid:6)tinL2(cid:10)Rn and—bycanonic(cid:4)al e(cid:6)xtensionto(cid:4)the(cid:6)time- d(cid:10)ependent c(cid:4)ase—(cid:6)in L2 R1+n . If A (cid:5)∂ = α∈NnAα(cid:5)∂α, then A (cid:5)∂ ∗ = A∗ −(cid:5)∂ := α∈NnA∗α −(cid:5)∂ α and this constraint means that the matrix coeffic(cid:10)ients Aα, α = (α1,...,αn), are selfadjoint or skew-selfadjo(cid:4)int(cid:6)depending on the order |α| := nk=1αk being even or odd, (cid:5) respectively.NotethatsinceA ∂ isapolynomial,onlyfinitelymanyofthecoefficientsarenon- vanishing.Inmostcases,themaximalorderisactuallyalsojust1. (cid:8) (cid:9) 4This transformation shifts the rigorous functional analytical discussion from L2 Rn+1 to the (cid:8) (cid:4) (cid:6)(cid:9) moreappropriatesettingintheHilbertspaceH(cid:3),0 R;L2 Rn ,whichisdefinedsuchthat (cid:8) (cid:4) (cid:6)(cid:9) (cid:8) (cid:4) (cid:6)(cid:9) (cid:8) (cid:9) exp(−(cid:3)m0):H(cid:3),0 R;L2 Rn →L2 R;L2 Rn =L2 R1+n ϕ(cid:6)→exp(−(cid:3)m0)ϕ becomes a unitary mapping. Here the multiplication operator exp(−(cid:3)m0) is defined via (exp(−(cid:3)m0)ϕ)(t):=exp(−(cid:3)t)ϕ(t),t ∈R.Wewillbemorepreciseanddetailedlater. Introduction vii where M(cid:7) :=M −(cid:3)M . 1 1 0 Now,theconstraints(2)translateto M =M∗and(cid:3)M +ReM ≥c >0 (3) 0 0 0 1 0 and the latter strict positive definitenessconstraintneedsto holdonly for all sufficiently large(cid:3) ∈ ]0,∞[.Asweshallseelater,theparticularroleoftimeisencodedinthisbias forpositivevaluesofparameter(cid:3). Toimproveontherangeofapplicability,wewillgeneralizetheaboveproblemclassby allowingM andM tobeHilbertspaceoperatorsandAtobeageneralskew-selfadjoint 0 1 operatorsothatoperatorsofthe“space-time”form ∂ M +M +A (4) 0 0 1 can be treated. In the proper setting, ∂0 will be seen to be a continuously(cid:8)inve(cid:9)rtible operator, which, among other things, allows us to consider the operator M ∂−1 := 0 M0 +∂0−1M1, which in a(cid:8)pplica(cid:9)tionoccurswhen describingso-called material laws. We thereforeshallrefertoM ∂−1 aswellastoM andM asmateriallawoperators.This 0 0 1 settingessentiallyyieldsanewnormalformforpartialdifferentialequationsoccurringin numerousapplications. In the following, we shall rigorously develop the solution theory of these abstract equations,which—dueto theirimpliedcausalityproperties—wereferto asevolutionary equations.Weusethetermevolutionaryinasomewhatsubtleattempttodistinguishthem from the classical concept of evolution equations, which are explicit first-order-in-time equations. Althoughthisclasscanbereadilygeneralizedtoincludemorecomplicatedcases,such ∗ asmerelyassumingthatthenumericalrangesofA,A areintheclo(cid:8)sedc(cid:9)omplexrighthalf- planeorallowingformorecomplicatedmateriallawoperatorsM ∂−1 withthepositive 0 (cid:8) (cid:9) definitenessconstraintthatforsomec ∈]0,∞[thenumericalrangeof∂ M ∂−1 −c 0 0 0 0 isintheclosedcomplexrighthalf-plane(forallsufficientlylarge(cid:3) ∈ ]0,∞[)(seeagain e.g.[59,75]),weshallfocushereonthemoreeasilyaccessiblepuredifferentialcase. Eventually, we aim at a solution theory with easy to check assumptions that lead to well-posednessofaratherlargeclassofpartialdifferentialequations.Indeed,wewillsee thatwell-posednessofanevolutionaryequationboilsdowntoprovinganumericalrange constraintforcertainboundedoperatorsonly. In Chap.1, we developthe functionalanalyticalsetting and the basic solutiontheory. Chapter 2 illustrates the theory for a number of model problems from mathematical viii Introduction physics. This concludes the book’s core material. Chapter 3 addresses some of the issuesthatmayarisewhencomparingourapproachwithsomealternative,possiblymore mainstreamideasfordealingwithproblemsofthesametype.Twoappendicescomplement thebook’smaterialbyprovidingadditionalideasforexpandingontheapplicabilityofthe approach,AppendixA,andcollectingsomebackgroundmaterialfromfunctionalanalysis asastudyresource,AppendixB. Contents 1 TheSolutionTheoryforaBasicClassofEvolutionaryEquations............ 1 1.1 TheTimeDerivative........................................................... 1 1.2 AHilbertSpacePerspectiveonOrdinaryDifferentialEquations........... 9 1.3 EvolutionaryEquations ....................................................... 14 2 SomeApplicationstoModelsfromPhysicsandEngineering.................. 31 2.1 AcousticEquationsandRelatedProblems................................... 32 2.2 AReductionMechanismandtheRelativisticSchrödingerEquation....... 38 2.3 LinearElasticity............................................................... 46 2.4 TheGuyer–KrumhanslModelofThermodynamics......................... 60 2.5 TheEquationsofElectrodynamics........................................... 68 2.6 CoupledPhysicalPhenomena ................................................ 88 3 ButWhatAbouttheMainStream?............................................... 103 3.1 WhereistheLaplacian?....................................................... 103 3.2 WhyNotUseSemi-Groups?.................................................. 109 3.3 WhatAboutOtherTypesofEquations?...................................... 114 3.4 WhatAboutOtherBoundaryConditions?................................... 118 3.5 WhyAllThisFunctionalAnalysis?.......................................... 121 A TwoSupplementsfortheToolbox................................................. 123 A.1 MothersandTheirDescendants.............................................. 123 A.2 Abstractgrad-div-Systems ................................................... 126 B RequisitesfromFunctionalAnalysis.............................................. 129 B.1 FundamentalsofHilbertSpaceTheory ...................................... 129 B.2 TheProjectionTheorem ...................................................... 137 B.3 TheRieszRepresentationTheorem .......................................... 141 B.4 LinearOperatorsandTheirAdjoints......................................... 144 ix