Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis: The Helge Holden Anniversary Volume

506 Pages·2018·14.77 MB·English

by  Fritz Gesztesy

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reflecting to a large degree Helge Holden’s longstanding research interests. o su
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topics, including, in particular, infinite-dimensional analysis, linear and nonlinear e ct
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matrix theory, and chaotic dynamics and sestina poetry. They represent to some extent Yu nn Espen R. Jakobsen
the lectures presented at the conference Non-linear PDEs, Mathematical Physics and r as
Stochastic Analysis, held at NTNU, Trondheim, July 4–7, 2016. ii L ly, Yurii Lyubarskii
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analysis, operator theory, and probability theory. b s
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Editors
ISBN 978-3-03719-186-6
www.ems-ph.org
SCR Gesztesy et al. | Egyptienne F | Pantone 116, 287 | RB 33 mm

EMS Series of Congress Reports
EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field
of pure or applied mathematics. The individual volumes include an introduction into their subject and
review of the contributions in this context. Articles are required to undergo a refereeing process and are
accepted only if they contain a survey or significant results not published elsewhere in the literature.
Previously published:
Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.)
K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.)
Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.)
Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.)
Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.)
Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.)
Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.)
Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.)
Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and
Andrzej Skowron´ski (eds.)
Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.)
Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann,
Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.)
Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume,
Jaroslav Dittrich, Hynek Kovarˇík and Ari Laptev (eds.)
Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Jarosław Buczyn´ski, Mateusz
Michałek and Elisa Postinghel (eds.)
Non-Linear Partial
Differential Equations,
Mathematical Physics,
and Stochastic Analysis
The Helge Holden Anniversary Volume
Fritz Gesztesy
Harald Hanche-Olsen
Espen R. Jakobsen
Yurii Lyubarskii
Nils Henrik Risebro
Kristian Seip
Editors
Editors:
Fritz Gesztesy Harald Hanche-Olsen
Department of Mathematics Espen R. Jakobsen
Baylor University Yurii Lyubarskii
Sid Richardson 305I, One Bear Place #97328 Kristian Seip
Waco, TX 76798-7328 Department of Mathematical Sciences
USA Norwegian University of Science and Technology
Alfred Getz vei 1
Email: [email protected]
7491 Trondheim
Norway
Nils Henrik Risebro
Department of Mathematics Email: h [email protected]
University of Oslo [email protected]
P.O. Box 1036, Blindern [email protected]
N-0316 Oslo [email protected]
Norway
Email: [email protected]
2010 Mathematics Subject Classification: Primary: 15B52, 35J10, 35L65, 35Q41, 35Q51, 35Q53, 37K10,
42B20, 46N20, 46N30, 46T12, 47B36, 47F05, 60H20, 68N30, 76S05; secondary: 33C45, 35A01, 35A02,
35L80, 37D45, 39A12, 47A10, 47N20, 47N30, 60B20.
Key words: Infinite-dimensional analysis, partial differential equations, hyperbolic conservation laws,
stochastic analysis, spectral theory, discrete evolution, completely integrable systems, random matrix
theory, chaotic dynamics.
ISBN 978-3-03719-186-6
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and
the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
© European Mathematical Society 2018
Contact address:
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Typeset by the editors using the authors’ TeX files: Harald Hanche-Olsen, Trondheim, Norway
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∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
Preface
ThisFestschriftisdedicatedtoHelgeHolden
ontheoccasionof his60thbirthday.
HelgeHoldenwasborntoparentsFinnHolden(1928–),atextbookauthor,histo-
rian,andhighschoolteacher,andKirstenneeWolfhagen(1931–1999),alibrarian
andpsychologist,onSeptember28,1956,inOslo,Norway. Helgegrewupwith
twoyoungerbrothers,Steinar,whoisprofessorofeconomicsattheUniversityof
Oslo,andLars,whoisManagingDirectorattheNorwegianComputingCenterin
Oslo. Interestingly,Helgehasajointpaper[58]1withhistwobrothers.
Helgedecidedearlytostudymathematicsandphysics,andafterservingthe
mandatoryoneyearmilitaryservice,heenteredtheUniversityof Osloin1976.
Inthefallof 1980hestartedtoworkonhiscand.real. thesiswiththelateProf.
Raphael Høegh-Krohn (1938–1988) as advisor, working on solvable models in
non-relativisticquantummechanics. Inthefallof 1981hegraduatedwiththe
cand.real. exam,hisgradesbeingsoimpressivethattheresultwasreportedtothe
KingofNorwayintheCouncilofState.
RaphaelHøegh-Krohn–acharismaticandbrilliantmathematicalphysicist
attheUniversityofOslo–broughtHelgeincontactwiththegrouparoundProf.
Sergio Albeverio in Bochum, and the young mathematicians Fabio Martinelli,
Werner Kirsch, and, Fritz Gesztesy. Fritz came to Oslo because he had found
an improvement in an important paper by Raphael and Sergio (marking the
beginningofanintensivecollaborationwithbothofthem),andatthatoccasion
FritzandHelgealsostartedworkingtogether,andhavecontinuedtodosoever
since. Inthosedays,theworkcenteredaroundso-calledpointinteractionmodels
innonrelativisticquantummechanics,andresultedinthebookβ[1],publishedby
Springerin1988,translatedintoRussian,andre-issuedina2ndeditionbythe
AmericanMathematicalSocietywithanappendixbyProf.PavelExnerin2005.
Ithasestablisheditselfasthestandardreferenceinthefield,anditisfrequently
citedinboththephysicsandmathematicsliterature.
Intheearly1980s,Norwaywasgoingthroughrapidanddramaticdevelopment.
AtChristmasof1969,thefirstoilwasdiscoveredintheNorthSea–theEkofiskfield
–stilloneofthelargestintheNorthSea. Thequestionwashowtotakeadvantage
1Numbersrefertothecompletelistofpublications.
vi Preface
ofthistothebenefitofNorwegiansociety. Fortunately,thegovernmentdecidedto
buildupnationalcompetencyinNorway,resultinginadramaticimprovementin
thestandardoflivinginNorway. ThosewhovisitedNorwayintheearly’80sand
visitnow,willseeanextrememakeoverinNorway–itsimplyisadifferentcountry
now. In the early ’80s, the Norwegian oil companies Saga, Hydro, and Statoil
neededscientistswithagoodbackgroundinthesciencesandinmathematics,and
thecompaniessupportedmathematicalresearch. RaphaelandHelge,together
withNilsHenrikRisebro,LarsHolden,ToreGimse,KyrreandFrodeBratvedt,
andChristianBuchholz,embarkedontheprojecttodevelopafullscalepetroleum
reservoirsimulator. Theapproachwasbasedatfirstonlotsof enthusiasmand,
retrospectively,lessondeepknowledgeof reservoirsimulation;itwasalsothe
startofalife-longcollaborationbetweenHelgeandNilsHenrik. Unfortunately,
Raphaelsuddenlypassedawayin1988,andHelgetookoverasPhDadvisorof
NilsHenrikandLars. Thedevelopmentofthereservoirsimulatorsubsequently
changedfocus,butitistothisdaystillusedbySchlumberger–theworld’sleading
oilfieldservicescompany.
Atthistimetherewasessentiallynoactivityinpartialdifferentialequations
inNorway. TogetherwithProf.RagnarWintherinOslo,HelgeandRagnarhave,
togetherwiththeirsubstantialnumberof students,beeninstrumentalindevel-
opingthefieldofpartialdifferentialequationsintooneofthestrongestareasof
Norwegianmathematics. Theresearchisnolongerrestrictedtotopicsrelevantto
flowinporousmedia,andaspecialfocushasbeentounderstandtheinteraction
between theoretical results and numerical simulations. There are now strong
researchgroupsinpartialdifferentialequationsattheuniversitiesinOslo,Bergen,
andTrondheim,inadditiontothatatSINTEF.
AftercompletionofhisPhD,Helgewentfortheacademicyear1985–86tothe
CourantInstituteofMathematicalSciencesofNewYorkUniversityonaFulbright
scholarshiptojointhegroupofProf.JamesGlimm. Whilethere,hewasoffereda
permanentpositionattheNorwegianInstituteofTechnology(nowNTNU–the
NorwegianUniversityofScienceandTechnology),inTrondheim,Norway,and
hehasremainedatNTNUeversince. Hespentthespringsemester1989atthe
CaliforniaInstituteofTechnology,withProf. BarrySimon,andtheacademicyear
1996/97attheUniversityofMissouri,Columbia,withFritz.
Helge’smathematicalactivityspansseveralareas. Startingoutinmathemati-
calphysics,focusingonnonrelativisticquantummechanics,helaterturnedhis
interesttononlinearpartialdifferentialequations. Heretheinterestinitiallywas
concentratedonthestudyofflowofhydrocarbonsinporousmedia,andthemost
importantmathematicalresultscameinthetheoryofhyperbolicconservations
laws,where,togetherwithNilsHenrik,hewrotethebookβ[4]thatrepresents
thestandardpresentationof theso-calledfront-trackingapproachtohyperbolic
Preface vii
conservationlaws. Withthegroupof Prof.BerntØksendalinOslo,Helgeworked
onstochasticdifferentialequationsusingtheso-calledwhitenoiseapproach,and
theirresultswerepresentedinthebookβ[2]. WithFritz,Helge’sinterestturnedto
completelyintegrablesystems–wherethecelebratedKdVequationisoneof the
mostrecognizedexamples–andtheycollectedtheirresultsinthetwo-volume
treatiseβ[5,6],thesecondvolumewrittenjointlywithJohannaMichorandGerald
Teschl. Subsequently, Helge’s interest moved to the Camassa–Holm equation,
whichhehadalreadystudiedwithFritzinthealgebro-geometricsetting. Nowthe
problemturnedintothestudyofanevolutionequationwhosesolutionsencounter
wavebreaking,andthisstudyledtofurtherworkalsoonthenonlinearvariational
waveequation. InthisareaheworkedwithXavierRaynaudandKatrinGrunert.
Hismostrecentbookdiscussesso-calledoperatorsplittingmethodsfornonlinear
partialdifferentialequationswithroughsolutions,andisjointwithKnut-Andreas
Lie,KennethH.Karlsen,andNilsHenrikβ[7].
Acommonfeatureofallofhisworkisthedeepinteractionbetweenmathe-
maticsandphysics. Todate,hehasco-authoredwellover160publicationsand
7books(4of whicharealreadyin2ndedition,ararefeat). Helgetrulyenjoys
collaborationandtothisdayhasworkedwithwellover60collaborators. Hiswork
hasbeenextensivelycited,accordingtoMathSciNet,almost2491timesbymore
than1631researchers.2
Helgehasreceivednumeroushonors,amongwhichwementionthefollowing:
–ElectiontotheNorwegianAcademyofScienceandLetters
–ElectiontotheRoyalNorwegianSocietyofSciencesandLetters
–ElectiontotheNorwegianAcademyofTechnologicalSciences
–ElectiontotheEuropeanAcademyofSciences
–Fellow,AmericanMathematicalSociety
–Fellow,SocietyforIndustrialandAppliedMathematics(SIAM)
–HegaveaninvitedtalkattheEuropeanCongressofMathematicsinStockholm
in2004.
AtNTNUHelgehasbeeninvolvedinseveralactivities–hehashadastaggering
numberof almost90masterstudentsand24PhDstudents(3currently),most
of whom are active researchers with successful careers of their own, and he
is currently serving on the Board of NTNU. In 1993 he took the initiative to
organizeanannualLarsOnsagerLectureandLarsOnsagerProfessorshipatNTNU
tocommemoratetheiconicLarsOnsager(1903–76),NobelLaureateinChemistry
in1967,whoseAlmaMaterwastheNorwegianInstituteofTechnology;25years
laterthisactivityisstillgoingstrong.
2MathSciNet,April1,2017.No,notanAprilfoolsjoke!
viii Preface
Historically, NTNU has had a strong focus on engineering with a constant
sourceofexcellentstudentscomingfromtheprograminIndustrialMathematics,
developedtogetherwithpartnersinECMI–EuropeanConsortiumofMathematics
inIndustry. HelgegotinvolvedwiththisactivitythroughthelateProf.Henrik
Martens(1927–93)atNTNU,whowasoneofthefoundersofECMI.Helgeserved
asPresidentofECMIduringtheperiod2004–06.
Onthenationalscene,Helgehasbeenexceptionallyactive. Hechairedseveral
key panels of the Research Council of Norway for extended periods, the most
importantbeingthepanelthatdecidedonallnationalindividualresearchgrants
inmathematicsandthenaturalsciences. IntheScandinaviancountries,Helge
has chaired evaluations of research and education in mathematics at several
universitiesinDenmarkandSweden. Similarly,ontheEuropeanscene,Helgehas
chairedthepanelof theEuropeanResearchCouncilthatawardstheprestigious
ConsolidatorGrantsinMathematics.
HelgehaswrittennumerousarticlesinNorwegiannewspapersandjournals
onvariousresearchpoliticaltopicsandthepopularizationofmathematics. Hehas
beenveryactivelyinvolvedintheAbelPrizeinMathematicssinceitsstartin2003.
Inparticular,heservedasChairoftheAbelBoardoftheNorwegianAcademyof
ScienceandLetters,fortheperiodof2010–14. Duringhischairmanshipseveral
novelactivitiesstarted,forinstance,thecompetitionUngeAbel,theHeidelberg
LaureateForum,andthecollaborationwith PetroleumGeo-Services. Healsotook
theinitiativetoeditbooksabouttheAbelLaureates,andwithRagniPienetwo
volumeshavebeencompletedε[7,12],withathirdvolumecurrentlyinpreparation.
Helgehasalwaysbeenastrongproponentof internationalcollaboration,and
heservedasSecretary(2003–06)andVicePresident(2007–10)of theEuropean
MathematicalSociety. CurrentlyheisservingasSecretaryof theInternational
MathematicalUnion(IMU),withheadquartersinBerlin,Germany.
He has served on the Board of the Norwegian Academy of Technological
SciencesandtheRoyalNorwegianSocietyofSciencesandLetters,andduringthe
period2014–16heservedasPresidentofthelatter.
WhilethishintsatsomeaccountofHelgeasthescientistandhissubstantial
servicetoourprofessioninthenationalaswellasinternationalarena,wewould
beamissnottocommentonsomepersonalaspectsof ourlongstandingfriend-
shipwithhim. Hiskind,yetfirm,demeanor,hisintegrityandsenseoffairness,
hisdedicationtoscience, hisintellectualcuriosityabouttheworldasawhole,
substantiallytranscendingthenaturalsciences,hisdeepinterestinthearts,and
especially,hisloveofliterature,showshimtobethecomplexandmulti-faceted
personalityweallcametoappreciatesomuchovertheyears. Severaloftheeditors
arecollaboratorsofHelge’s,someforupto35yearsnow,allofusviewhimasa
dearandtrustedfriend.
Preface ix
The volume at hand is based to some extent on the conference, Non-linear
PDEs,MathematicalPhysicsandStochasticAnalysis,heldatNTNU,Trondheim,
July4–7,2016(https://wiki.math.ntnu.no/holden60). Thefieldsrepresentedin
the contributions to this volume reflect to a large degree Helge’s longstanding
researchinterests. Theycenteraroundinfinite-dimensionalanalysis(integralsof
probabilisticandoscillatorytype),linearandnonlinearpartialdifferentialequa-
tions(includingdiscreteevolutionequations,Ostrovsky–Hunter-typeequations,
modelingcrowddynamics,porousmediumtypeequations,nonlineardegener-
ateanisotropichyperbolic-parabolicequations,Riemannproblemsformodelsof
polymerflooding,systemsofconservationlaws,nonlineardispersivePDEs,com-
pensatedcompactnessandisometricimmersionsofmanifolds),stochasticanalysis
(optimalcontrolforasystemof stochasticVolterraequations), spectraltheory
(includingspectraofleakysurfaces,Hardy–Rellich-typeinequalities,dispersion
estimateforone-dimensionalSchrödingeroperators,Schrödingeroperatorsin-
volvingtheHeisenbergsub-Laplacian),completelyintegrablesystems(including
themodifiedtwo-componentCamassa–Holmsystem),randommatrixtheory,and
chaoticdynamicsandsestinapoetry.
Finally,weexpressoursinceregratitudetothestaffattheEMS,particularly,
Thomas Hintermann, for their help, support, and expertise in producing this
volume. Wealsothankallauthorsfortheircontributionsandtherefereesfortheir
invaluableassistance.
HappyBirthday,Helge,wehopethisvolumebringssomefondmemoriesandjoy!
FritzGesztesy
HaraldHanche-Olsen
EspenR.Jacobsen
YuriiLyubarskii
NilsHenrikRisebro
KristianSeip