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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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