Table Of ContentLecture Notes in Mathematics 2062
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
FondazioneC.I.M.E.,Firenze
C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International
MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,
Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year
foryear,tothisday.
Many mathematicians from all over the world have been involved in a way oranother in
C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre
maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes
from pure and applied mathematics are offered byapplication to mathematicians from all
countries. ASessionis generally basedonthree orfourmaincourses given byspecialists
ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural
locationinItaly.
TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins,
development, and perspectives of some very active branch of mathematical research. The
topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere
ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational
collaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
Universita`diFirenze Universita`diFirenze
ViaS.Marta,3 vialeG.B.Morgagni67/A
50139Florence 50134Florence
Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
-INdAM(IstitutoNazionalediAltaMatematica)
-MIUR(Ministerodell’Universita’edellaRicerca)
Luigi Ambrosio Alberto Bressan
(cid:2)
Dirk Helbing Axel Klar Enrique Zuazua
(cid:2) (cid:2)
Modelling and Optimisation
of Flows on Networks
Cetraro, Italy 2009
Editors:
Benedetto Piccoli
Michel Rascle
123
LuigiAmbrosio AxelKlar
ScuolaNormaleSuperiore TechnischeUniversita¨t
DepartmentofMathematics Kaiserslautern
Pisa,Italy FachbereichMathematik
Kaiserslautern,Germany
AlbertoBressan
PennStateUniversity EnriqueZuazua
StateCollege BCAM BasqueCenterfor
DepartmentofMathematics AppliedMathema
UniversityPark Depto.Matematicas
PA,USA Bilbao,Spain
DirkHelbing
ETHZu¨rich
SwissFederalInstituteofTechnology
Zurich,Switzerland
ISBN978-3-642-32159-7 ISBN978-3-642-32160-3(eBook)
DOI10.1007/978-3-642-32160-3
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012949580
MathematicsSubjectClassification(2010):35R02,35L65,90B20
Thecontributionsco-authoredorauthoredbyHelbinghavepreviouslybeenpublishedin
TheEuropeanPhysicalJournalB69(4),539–548,DOI:10.1140/epjb/e2009-00192-5(2009),
(cid:2)c EDPSciences,SocietaItalianadiFisica,Springer-Verlag2009,reproductionwithkindpermissionof
TheEuropeanPhysicalJournal(EPJ)
TheEuropeanPhysicalJournalB69(4),549–562,DOI:10.1140/epjb/e2009-00182-7(2009),
(cid:2)c EDPSciences,SocietaItalianadiFisica,Springer-Verlag2009,reproductionwithkindpermissionof
TheEuropeanPhysicalJournal(EPJ)
TheEuropeanPhysicalJournalB69(4),583–598,DOI:10.1140/epjb/e2009-00140-5(2009),
(cid:2)c EDPSciences,SocietaItalianadiFisica,Springer-Verlag2009,reproductionwithkindpermissionof
TheEuropeanPhysicalJournal(EPJ)
Networks and Heterogeneous Media 2(2), 193–210 (2007), (cid:2)c American Institute of Mathematical
Sciences,ReproductionwithkindpermissionofNetworksandHeterogeneousMedia(NHM).
TheEuropeanPhysicalJournalB70(2),257–274,DOI:10.1140/epjb/e2009-00213-5 (2009),(cid:2)c EDP
Sciences, Societa Italiana di Fisica, Springer-Verlag 2009, reproduction with kindpermission ofThe
EuropeanPhysicalJournal(EPJ)
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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•
Preface
ThepresentvolumecollectsnotesfromlecturesdeliveredfortheCIMEcourseon
Modellingandoptimisationofflowsonnetworks,heldinCetrarointhesummerof
2009.
In recent years modelling of flows on networks has been the subject of many
investigationsleadingtoanincreasingnumberofresearchpapers.Moreover,awide
set of possible applications, such as vehicular traffic, blood flow, supply chains
andothers,hasdirectedtheattentionofmathematicianstowardsresearchdomains
usuallypopulatedbyengineers,physicistsorresearcherswithotherexpertise.
The aim of the CIME school was to gather summer courses which could give
a wide view of modelling, analysis, numerics and control for dynamic flows on
networks. Encompassing all application domains (including irrigation channels,
datanetworks,airtrafficmanagementandothers)wasimpossible,thuswefocused
on mathematical approaches, which are feasible for a number of applications,
and a restricted set of specific applications, in particular vehicular traffic and
supplychains.Theattemptoffindingacommonground,fordifferentmathematical
techniquestotreatflowsonnetworks,wasalreadysuccessfulinanumberofcases
both at the levelof research projects (such as the Italian nationalINDAM project
2005)andeditorialinitiatives(thefoundationin2006ofanewappliedmathjournal
entitledNetworksandHeterogeneousMedia).
TheschooltookplaceinCetraro,Italy,onJune15–192009.Thecoursesubjects
werethefollowing:
1. Introductiontoconservationlaws:AlbertoBressan(PennState)
2. Optimaltransportation:LuigiAmbrosio(SNS,Pisa)
3. Pedestrianmotionsandvehiculartraffic:DirkHelbing(ETH)
4. Controlandstabilizationofwaveson1-Dnetworks:EnriqueZuazua(BCAM)
5. Modelling and optimization of scalar flows on networks: Axel Klar
(Kaiserlautern)
6. Fluid dynamic and kinetic models for supply chains: Christian Ringhofer
(ArizonaState)
vii
viii Preface
RationaleBehindtheChoiceof Courses for CIME School
Taking into account the above-mentioned scientific background, courses for the
CIME school were chosen in order to give a wide view over main mathematical
techniquesandtheirapplicationsinspecificcontexts.
1. AnalysisandcontroloflinearPDEsonnetworks
2. AnalysisofnonlinearPDEsonnetworks
3. Optimizationtechniquesforcomplexnetworks
4. NumericalmethodsforPDEsonnetworks
TocoverthefirsttopicandlastoneforthelinearPDEaspect,wedecidedtofocus
onwaveequationsonnetworksofone-dimensionalstructuresand,inparticular,on
the use of spectral methods. Therefore, the choice was made to contact Enrique
Zuazua,DirectoroftheBasqueCenterforAppliedMathematicsandaworldleader
on the subject. Prof. Zuazua also authoreda volume on the subject(SMAI series,
Springer-Verlag,2006).
In many applications it is natural to use conservation laws to model flows on
networks, thus for the second course we contacted Alberto Bressan of PennState
University, who was one of the major contributors of the theory of systems
of conservation laws in last 20 years and author of a well-known monograph
(CambridgeUniversityPress,2000).
The fourth topic for the nonlinear aspect was covered in courses dealing also
with applications, and illustrated below, of Klar and Ringhofer. Finally, for the
third topic, we individuated optimal transportation as one of the most suited
mathematical framework, and thus decided to contact Luigi Ambrosio of Scuola
NormaleSuperioreofPisa,whoauthoredvariousrecentfundamentalpapersinthe
subject and a monograph on the related topics of gradient flows in metric spaces
(Birkauser,2008).
For what concerns applications related to our main theme, Dirk Helbing of
ETH of Zurich accepted to deliver a course covering both pedestrian dynamics
and vehicular traffic. Helbing was one of the pioneers in providing advanced
mathematicalmodellingforpedestrianswithcelebratedpapersinNature.
Then we focused on supply chain dynamics and thus contacted Christian
RinghoferofArizonaStateUniversity,whocoauthoredapioneeringpaperin2006
providing the first model of supply chains using PDEs. The course of Ringhofer
dealtalsowithkineticapproaches.
Finally, Axel Klar of Kaiserlautern TechnicalUniversity provideda course not
onlydealingwithgeneralmodellingandnumericsofconservationlawsonnetworks
butalsotreatingcoupledsystemsofODEsandPDEswithexamplesfromvehicular
traffic,supplychainsandsewersystems.
ThepresentvolumecontainslecturenotesfromthefirstfivecoursesoftheCIME
school.Wewishreadersapleasantandfruitfulreading.
Camden,NJ BenedettoPiccoli
Nice,France MichelRascle
Contents
AUser’sGuidetoOptimalTransport ......................................... 1
LuigiAmbrosioandNicolaGigli
1 Introduction........................................................ 1
2 TheOptimalTransportProblem .................................. 3
2.1 MongeandKantorovichFormulationsofthe
OptimalTransportProblem................................. 3
2.2 NecessaryandSufficientOptimalityConditions .......... 7
2.3 TheDualProblem........................................... 13
2.4 ExistenceofOptimalMaps................................. 16
2.5 BibliographicalNotes....................................... 26
3 TheWassersteinDistanceW ..................................... 28
2
3.1 X PolishSpace .............................................. 29
3.2 X GeodesicSpace........................................... 37
3.3 X RiemannianManifold.................................... 47
3.4 BibliographicalNotes....................................... 58
4 GradientFlows..................................................... 59
4.1 HilbertianTheoryofGradientFlows....................... 59
4.2 TheTheoryofGradientFlowsinaMetricSetting ........ 61
4.3 ApplicationstotheWassersteinCase ...................... 81
4.4 BibliographicalNotes....................................... 92
5 GeometricandFunctionalInequalities........................... 93
5.1 Brunn–MinkowskiInequality............................... 94
5.2 IsoperimetricInequality..................................... 94
5.3 SobolevInequality........................................... 95
5.4 BibliographicalNotes....................................... 96
6 VariantsoftheWassersteinDistance ............................. 97
6.1 BranchedOptimalTransportation.......................... 97
6.2 DifferentActionFunctional................................. 99
6.3 AnExtensiontoMeasureswithUnequalMass............ 100
6.4 BibliographicalNotes....................................... 102
ix
x Contents
7 MoreontheStructureof.P .M/;W /.......................... 103
2 2
7.1 “Duality”BetweentheWassersteinandthe
ArnoldManifolds............................................ 103
7.2 OntheNotionofTangentSpace............................ 106
7.3 SecondOrderCalculus...................................... 107
7.4 BibliographicalNotes....................................... 130
8 RicciCurvatureBounds........................................... 131
8.1 ConvergenceofMetricMeasureSpaces ................... 134
8.2 WeakRicciCurvatureBounds:Definitionand
Properties .................................................... 137
8.3 BibliographicalNotes....................................... 150
References......................................................................... 152
HyperbolicConservationLaws:AnIllustratedTutorial .................... 157
AlbertoBressan
1 ConservationLaws ................................................ 158
1.1 TheScalarConservationLaw............................... 158
1.2 StrictlyHyperbolicSystems ................................ 160
1.3 LinearSystems .............................................. 161
1.4 NonlinearEffects............................................ 163
1.5 LossofRegularity........................................... 164
1.6 WaveInteractions............................................ 166
2 WeakSolutions .................................................... 167
2.1 Rankine–HugoniotConditions.............................. 168
2.2 ConstructionofShockCurves.............................. 172
2.3 AdmissibilityConditions.................................... 173
3 TheRiemannProblem............................................. 179
3.1 SomeExamples.............................................. 179
3.2 AClassofHyperbolicSystems............................. 182
3.3 ElementaryWaves........................................... 184
3.4 GeneralSolutionoftheRiemannProblem................. 187
3.5 TheRiemannProblemforthep-System ................... 190
3.6 ErrorandInteractionEstimates............................. 194
4 GlobalSolutionstotheCauchyProblem......................... 196
4.1 FrontTrackingApproximations............................ 197
4.2 BoundsontheTotalVariation .............................. 200
4.3 ConvergencetoaLimitSolution ........................... 203
5 TheGlimmScheme ............................................... 205
6 ContinuousDependenceontheInitialData...................... 210
6.1 UniqueSolutionstotheScalarConservationLaw......... 211
6.2 LinearHyperbolicSystems ................................. 212
6.3 NonlinearSystems........................................... 213
7 UniquenessofSolutions........................................... 216
7.1 AnErrorEstimateforFrontTrackingApproximations... 217
7.2 CharacterizationofSemigroupTrajectories................ 218
7.3 UniquenessTheorems....................................... 221
