Table Of ContentFrontiers 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
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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
