Nonlinear Partial Differential Equations: The Abel Symposium 2010

360 Pages·2012·3.11 MB·English

by  Luigi Ambrosio (auth.)

ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
Forfurthervolumes:
www.springer.com/series/7462

Helge Holden (cid:2) Kenneth H. Karlsen
Editors
Nonlinear Partial
Differential
Equations
The Abel Symposium 2010
Editors
HelgeHolden KennethH.Karlsen
Dept.ofMathematics CentreofMathematicsforApplications
NorwegianUniversityofScienceand UniversityofOslo
Technology Oslo
Trondheim Norway
Norway
ISBN978-3-642-25360-7 e-ISBN978-3-642-25361-4
DOI10.1007/978-3-642-25361-4
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2012930729
MathematicsSubjectClassification(2010): 35-XX,34-XX,76-XX,65-XX,81-XX,70-XX
©Springer-VerlagBerlinHeidelberg2012
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
areliabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot
imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective
lawsandregulationsandthereforefreeforgeneraluse.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface to the Series
The Niels Henrik Abel Memorial Fund was established by the Norwegian gov-
ernment on January 1, 2002. The main objective is to honor the great Norwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstand-
ing scientific work in the field of mathematics. The prize shall contribute towards
raising the status of mathematics in society and stimulate the interest for science
amongschoolchildrenandstudents.InkeepingwiththisobjectivetheBoardofthe
Abel Fund has decided to finance an annual Abel Symposium. The topic may be
selectedbroadlyintheareaofpureandappliedmathematics.TheSymposiashould
beatthehighestinternationallevel,andservetobuildbridgesbetweenthenational
and international research communities. The Norwegian Mathematical Society is
responsiblefortheevents.Ithasalsobeendecidedthatthecontributionsfromthese
Symposia should be presented in a series of proceedings,and Springer Verlag has
enthusiastically agreed to publish the series. The board of the Niels Henrik Abel
Memorial Fund is confident that the series will be a valuable contribution to the
mathematicalliterature.
HelgeHolden
ChairmanoftheboardoftheNielsHenrikAbelMemorialFund
v

Preface
The topic of the 2010 Abel Symposium was Nonlinear Partial Differential Equa-
tions. The study of differential equations is of fundamental importance in math-
ematics and in almost all of the applications of mathematics in natural sciences,
economics, and engineering. This area of mathematics is currently in the midst of
anunprecedenteddevelopmentworldwide.Differentialequationsareusedtomodel
phenomenaofincreasingcomplexity,andinareasthathavetraditionallybeenout-
sidetherealmofmathematics.Newanalyticaltoolsandmathematicaltheories,cou-
pledwithnewnumericalmethods,aredramaticallyimprovingourunderstandingof
nonlinearmodels.Nonlinearitygivesraisetosolutionshavingsingularities,oscilla-
tions,orconcentrationeffects,whichintherealworldarereflectedintheappearance
of shock waves, turbulence, material defects, etc. These effects frequently require
new techniques,and offer challengingnovel problems for mathematicians.On the
otherhand,newmathematicaldevelopmentsprovide(numerical)solutionsandnew
insightinmanyapplications.ThepurposeoftheseAbelSymposiumproceedingsis
to present a selection of the latest exciting results by world leading researchers in
theareaofnonlinearpartialdifferentialequations.
The Abel Symposium was hosted at the Norwegian Academy of Science and
Letters, Oslo, from September 28 to October 2, 2010. Attendance was by invita-
tion only, and the symposium had a total of 74 participants, out of which 32 were
fromNorwegianuniversities.TheScientificCommitteeconsistedofAlbertoBres-
san (Penn State), Helge Holden (Trondheim), Kenneth H. Karlsen (Oslo), Sergiu
Klainerman(Princeton),andEitanTadmor(Maryland).
Talkswerepresentedby
LuigiAmbrosio(Pisa)
AlbertoBressan(PennState)
LuisA.Caffarelli(Texas)
Gui-QiangChen(Oxford)
CamilloDeLellis(Zürich)
MariaJ.Esteban(Paris)
EduardFeireisl(Prague)
vii
viii Preface
GerhardHuisken(Golm)
CarlosKenig(Chicago)
AlexKiselev(Madison)
SergiuKlainerman(Princeton)
RobertV.Kohn(NewYork)
Pierre-LouisLions(Paris)
AndrewJ.Majda(NewYork)
IgorRodnianski(Princeton)
LaureSaint-Raymond(Paris)
EitanTadmor(Maryland)
JuanLuisVazquez(Madrid)
CédricVillani(Lyon)
These proceedings include most of the lectures presented at the Abel Sympo-
sium, and the editors appreciate the efforts made by the speakers to present their
talksintheseproceedings.
Participantswere:
Andreianov,Boris Karper,Trygve Risebro,NilsHenrik
Berre,Inga Klausen,RunhildAae Rønquist,Einar
Borluk,Handan Kvamsdal,Trond Sande,Hilde
Bouchut,François Kværnø,Anne Selberg,Sigmund
Brenier,Yann Langtangen, Serre,Denis
Carrillo,JoséA.
HansPetter Signahl,Mikael
Celledoni,Elena
Li,Qifan Svanstedt,Nils
Christiansen,SnorreH.
Lie,KnutAndreas Süli,Endre
Colombo,RinaldoM.
Lindqvist,Peter Titi,EdrissS.
Dafermos,Constantine
Lyubarskii,Yurii Trivisa,Konstantina
Engquist,Bjørn
Malinnikova,Eugenia Tveito,Aslak
Evje,Steinar
Marcati,Pierangelo Vasiliev,Alexander
Fuchs,Franz
Markina,Irina Wasylewicz,
Golse,François
Grong,Erlend Munthe-Kaas,Hans Agnieszka
Hanche-Olsen,Harald Nordbotten,Jan Winther,Ragnar
Jakobsen,EspenR. Nørsett,SyvertP. Xin,Zhouping
Kalisch,Henrik Owren,Brynjulf Yaguchi,Takaharu
Kalise,Dante Raynaud,Xavier Zanna,Antonella
WegratefullyacknowledgefinancialsupportfromtheNielsHenrikAbelMemo-
rial Fund, The Centre for Advanced Study at the Norwegian Academy of Science
andLetters,aswellastheCentreofMathematicsforApplicationsattheUniversity
ofOslo.ThekindassistanceofMsMaritF.Strømisgreatlyappreciated.Thanksgo
toMariusThauleforhisassistanceinthepreparationoftheseproceedings.
Trondheim,Norway HelgeHolden
Oslo,Norway KennethH.Karlsen
Contents
ConvergenceofWignerTransformsinaSemiclassicalLimit . . . . . . . 1
LuigiAmbrosio
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 WellPosednessoftheLiouvilleEquationandFlows . . . . . . . . 5
3 StabilityoftheFlows . . . . . . . . . . . . . . . . . . . . . . . . 8
4 ConvergenceofWigner/HusimiTransforms . . . . . . . . . . . . 9
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ContractiveMetricsforNonsmoothEvolutions . . . . . . . . . . . . . . . 13
AlbertoBressan
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 DiscontinuousODEs. . . . . . . . . . . . . . . . . . . . . . . . . 15
3 NonlinearWaveEquations. . . . . . . . . . . . . . . . . . . . . . 18
3.1 AMetricInducedbyOptimalTransportation . . . . . . . . 20
3.2 AMetricInducedbyRelabelingEquivalence . . . . . . . . 23
4 HyperbolicConservationLaws . . . . . . . . . . . . . . . . . . . 26
4.1 AnExplicitFunctional. . . . . . . . . . . . . . . . . . . . 27
4.2 ARiemannTypeDistance . . . . . . . . . . . . . . . . . . 32
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Non-localDiffusions,DriftsandGames . . . . . . . . . . . . . . . . . . . 37
LuisCaffarelli
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 DivergenceStructure. . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Non-divergenceEquations . . . . . . . . . . . . . . . . . . . . . . 42
4 Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 TheCorrespondingABPforIntegralDiffusions[6] . . . . . . . . 47
6 Non-localInfiniteLaplacian . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix