Partial Differential Equations: Theory, Control and Approximation: In Honor of the Scientific Heritage of Jacques-Louis Lions

431 Pages·2014·9.73 MB·English

by  Alain Bensoussan

Philippe G. Ciarlet · Tatsien Li
Editors
Yvon Maday
Partial Diff erential
Equations:
Theory, Control
and Approximation
In Honor of the Scientifi c Heritage
of Jacques-Louis Lions
Partial Differential Equations: Theory, Control and
Approximation
Jacques-LouisLionsin1998
Philippe G. Ciarlet (cid:2) Tatsien Li (cid:2) Yvon Maday
Editors
Partial Differential
Equations:
Theory, Control
and Approximation
In Honor of the Scientific Heritage
of Jacques-Louis Lions
Editors
PhilippeG.Ciarlet YvonMaday
Dept.Mathematics LaboratoireJacques-LouisLions
CityUniversityofHongKong UniversitéPierreetMarie
HongKong,People’sRepublicofChina Paris,France
TatsienLi
SchoolofMathematicalSciences
FudanUniversity
Shanghai,People’sRepublicofChina
ISBN978-3-642-41400-8 ISBN978-3-642-41401-5(eBook)
DOI10.1007/978-3-642-41401-5
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2013953853
MathematicsSubjectClassification: 35-XX,49-XX,65-XX
©Springer-VerlagBerlinHeidelberg2014
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Preface
Jacques-LouisLions(1928–2001)wasanexceptionalmathematician,whoselasting
influenceisstilldeeplyfeltallovertheworld.
Hewasauniversallyrecognizedandadmiredexpertinpartialdifferentialequa-
tions, to the study of which he has made outstanding contributions regarding not
onlythetheoreticalaspectssuchasexistenceanduniquenessofpartialdifferential
equations, regularity of the solutions, homogenization, and control, but also their
numerical analysis and applications to fluid and solid mechanics, oceanography,
climatology,etc.
TogetherwithEnricoMagenes,hefirstproducedanexhaustiveanalysisoflinear
boundary value problems posed in Sobolev spaces, which includes, in particular,
a remarkably elegant proof of Korn’s inequality. He then developed with Guido
Stampacchiathetheoryofvariationalinequalities,visco-elasticity,orplasticity.But
he is perhaps even more remembered for the manifold landmark contributions he
madetotheresearchofnonlinearpartialdifferentialequations,notablybyrecogniz-
ing the efficiency of compactness, monotony, regularization, and penalty methods
fortheiranalysis.
Withanincredibleintuition,Jacques-LouisLionsforesawveryearlytheadvan-
tageofGalerkinmethods,forinstance,howthefiniteelementmethodexceedsthe
moretraditionalfinite-differencemethods.Insodoing,hewashighlyinstrumental
inthecreationofaverypowerfulschoolofnumericalanalysts“withoutfrontiers”
(across national boundaries), who made many extraordinary breakthroughs to the
theoreticalunderstandingaswellastothepracticalimplementationofawidearray
ofmethodsforapproximatingthesolutionsofpartialdifferentialequations.Healso
made pioneering contributions to the analysis of problems with small parameters
and,moregenerally,ofsingularperturbationproblems.
Buthisever-favoritesubjectwascontroltheory,where,asfarbackasin1958,he
made milestone advances in the extension of optimal control to systems governed
bypartialdifferentialequations.Onehighlightofhiscontributionstothisfieldwas
theprestigious“JohnvonNeumannLecture”thathegaveattheSIAMCongressin
Bostonin1986,wherehelaidthefoundationsofhiswell-known“HUMmethod”.
v
vi Preface
One can only be impressed by his immense works, for the quality, diversity, or
noveltyofthemathematicsused,andforhispermanentquestfornewapplications
thathadpreviouslybeenbelievedtobeinaccessible.
Jacques-LouisLionswasavisionary,whoquicklyunderstoodthattheavailabil-
ity of ever-increasing computational power would revolutionize the modeling of
numerous phenomena, provided however that the required mathematics were si-
multaneouslycreatedanddeveloped.This istheessenceofhisimmensescientific
heritage.
Jacques-Louis Lions justly received numerous honors. In particular, he was a
memberoftwenty-twoacademies,whichincludedthemostprestigiousones,such
as the Royal Society, the USSR Academy of Sciences, the National Academy of
SciencesoftheUSA,theFrenchAcademyofSciences,theThirdWorldAcademy
ofSciences,theAccademiaNazionaledeiLincei,andtheChineseAcademyofSci-
ences.HewasalsoawardedsuchhighlyprestigiousprizesastheJohnvonNeumann
Prize,theLagrangePrizeoftheICIAM,andtheJapanPrize.
ItistohonorthescientificheritageofJacques-LouisLionsthatan“International
ConferenceonPartialDifferentialEquations:Theory,ControlandApproximation”
was organized and held at Fudan University in Shanghai from May 28th to June
1st,2012.Thisconferencebroughttogetherexpertsfromallovertheworld,whose
talkscoveredthefieldsofresearchthatJacques-LouisLionscreatedorcontributed
somuchtocreate.Thisbookgatherssomeofthemostrepresentativecontributions
to the Conference, which have been and will be separately published in Chinese
Annals of Mathematics in 2013 and 2014. We thank Ms. Wei Wu of the Editorial
Board Office of Chinese Annals of Mathematics for her enthusiastic and effective
workineditingthiscollectionofpapers.
All those who approached Jacques-Louis Lions will cherish the memory of his
warmpersonality,thevisionthathesowellconveyed,andhisprofoundintelligence.
PhilippeG.Ciarlet
TatsienLi
YvonMaday
Contents
ControlandNashGameswithMeanFieldEffect . . . . . . . . . . . . . 1
AlainBensoussanandJensFrehse
TheRainonUndergroundPorousMedia . . . . . . . . . . . . . . . . . 41
ChristineBernardi,AdelBlouza,andLindaElAlaoui
FiniteVolumeMultilevelApproximationoftheShallowWaterEquations 67
ArthurBousquet,MartineMarion,andRogerTemam
Non-Gaussian Test Models for Prediction and State Estimation
withModelErrors . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
MichalBranicki,NanChen,andAndrewJ.Majda
AsymptoticAnalysisinaGas-SolidCombustionModelwithPattern
Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Claude-MichelBrauner,LinaHu,andLucaLorenzi
ImplicitSampling,withApplicationtoDataAssimilation . . . . . . . . 171
AlexandreJ.Chorin,MatthiasMorzfeld,andXueminTu
PeriodicHomogenizationforInnerBoundaryConditionswithEqui-
valuedSurfaces:TheUnfoldingApproach . . . . . . . . . . . . . . 183
DoinaCioranescu,AlainDamlamian,andTatsienLi
Global Null Controllability of the 1-Dimensional Nonlinear Slow
DiffusionEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Jean-MichelCoron,JesúsIldefonsoDíaz,AbdelmalekDrici,and
TommasoMingazzini
Sharp Interpolation Inequalities on the Sphere: New Methods
andConsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
JeanDolbeault,MariaJ.Esteban,MichalKowalczyk,andMichaelLoss
vii
viii Contents
OntheNumericalSolutiontoaNonlinearWaveEquationAssociated
withtheFirstPainlevéEquation:AnOperator-SplittingApproach 243
RolandGlowinskiandAnnalisaQuaini
MsFEMàlaCrouzeix-RaviartforHighlyOscillatoryEllipticProblems 265
ClaudeLeBris,FrédéricLegoll,andAlexeiLozinski
Exact Synchronization for a Coupled System of Wave Equations
withDirichletBoundaryControls . . . . . . . . . . . . . . . . . . . 295
TatsienLiandBopengRao
MixingMonte-CarloandPartialDifferentialEquationsforPricing
Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
TobiasLipp,GrégoireLoeper,andOlivierPironneau
h−P FiniteElementApproximationforFull-PotentialElectronic
StructureCalculations . . . . . . . . . . . . . . . . . . . . . . . . . 349
YvonMaday
IncreasingPowersinaDegenerateParabolicLogisticEquation . . . . . 379
JoséFranciscoRodriguesandHugoTavares
Composite Waves for a Cell Population System Modeling Tumor
GrowthandInvasion . . . . . . . . . . . . . . . . . . . . . . . . . . 401
MinTang,NicolasVauchelet,IbrahimCheddadi,
IreneVignon-Clementel,DirkDrasdo,andBenoîtPerthame
Control and Nash Games with Mean Field Effect
AlainBensoussanandJensFrehse
Abstract Mean field theory has raised a lot of interest in the recent years (see in
particular the results of Lasry-Lions in 2006 and 2007, of Gueant-Lasry-Lions in
2011, of Huang-Caines-Malham in 2007 and many others). There are a lot of ap-
plications.Ingeneral,theapplicationsconcernapproximatinganinfinitenumberof
playerswithcommonbehaviorbyarepresentativeagent.Thisagenthastosolvea
controlproblemperturbedbyafieldequation,representinginsomewaythebehav-
ioroftheaverageinfinitenumberofagents.Thisapproachdoesnotleadeasilyto
theproblemsofNashequilibriumforafinitenumberofplayers,perturbedbyfield
equations, unless one considers averaging within different groups, which has not
beendonein theliterature,andseems quitechallenging.Inthis paper,the authors
approachsimilarproblemswithadifferentmotivationwhichmakessenseforcon-
trolandalsofordifferentialgames.Thusthesystemsofnonlinearpartialdifferential
equationswithmeanfieldterms,whichhavenotbeenaddressedintheliteratureso
far,areconsideredhere.
Keywords Meanfield·Dynamicprogramming·Nashgames·Equilibrium·
Calculusofvariations
MathematicsSubjectClassification 49L20
ProjectsupportedbytheWCUWorldClassUniversityprogramthroughtheNationalResearch
FoundationofKoreafundedbytheMinistryofEducation,ScienceandTechnology(No.
R31-20007)andtheResearchGrantsCouncilofHKSAR(No.PolyU5001/11P).
A.Bensoussan(B)
InternationalCenterforDecisionandRiskAnalysis,SchoolofManagement,University
ofTexas-Dallas,Richardson,TX,USA
e-mail:[email protected]
A.Bensoussan
SchoolofBusiness,TheHongKongPolytechnicUniversity,HongKong,China
A.Bensoussan
GraduateDepartmentofFinancialEngineering,AjouUniversity,Suwon,Korea
J.Frehse
InstituteforAppliedMathematics,UniversityofBonn,Bonn,Germany
P.G.Ciarletetal.(eds.),PartialDifferentialEquations:Theory,Controland 1
Approximation,DOI10.1007/978-3-642-41401-5_1,
©Springer-VerlagBerlinHeidelberg2014