Progress in Nonlinear Differential Equations and Their Applications Volume67 Editor HaimBrezis Universite´ PierreetMarieCurie Paris and RutgersUniversity NewBrunswick,N.J. EditorialBoard AntonioAmbrosetti,ScuolaInternationaleSuperiorediStudiAvanzati,Trieste A.Bahri,RutgersUniversity,NewBrunswick FelixBrowder,RutgersUniversity,NewBrunswick LuisCaffarelli,TheUniversityofTexas,Austin LawrenceC.Evans,UniversityofCalifornia,Berkeley MarianoGiaquinta,UniversityofPisa DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh SergiuKlainerman,PrincetonUniversity RobertKohn,NewYorkUniversity P.L.Lions,UniversityofParisIX JeanMawhin,Universite´ CatholiquedeLouvain LouisNirenberg,NewYorkUniversity LambertusPeletier,UniversityofLeiden PaulRabinowitz,UniversityofWisconsin,Madison JohnToland,UniversityofBath Amandine Aftalion Vortices in Bose–Einstein Condensates Birkha¨user Boston • Basel • Berlin AmandineAftalion CNRS,LaboratoireJacques-LouisLions Universite´PierreetMarieCurie(Paris6) 175rueduchevaleret F-75013Paris France AMSSubjectClassification(2000):35-02,35J60,35Q55 LibraryofCongressControlNumber:2006922937 ISBN-100-8176-4392-3 e-ISBN0-8176-4492-X ISBN-13978-0-8176-4392-8 Printedonacid-freepaper. 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(TXQ/IBT) 987654321 www.birkhauser.com Preface Vortices are often associated with dramatic circumstances such as hurricanes; this typeofvortexhasbeenstudiedextensivelyintheframeworkofclassicalfluids.Nev- ertheless,itsquantumcounterparthasgainedmajorinterestinthepastfewyearsdue totheexperimentalrealizationofBose–Einsteincondensates(BEC),anewstateof matterpredictedbyEinsteinin1925.VorticesinBECarequantized,andtheirsize, origin, and significance are quite different from those in normal fluids since they exemplify“superfluid”properties. SincethefirstexperimentalachievementofBose–Einsteincondensatesin1995 in alkali gases and the award of the Nobel Prize in Physics in 2001, the properties ofthesegaseousquantumfluidshavebeenthefocusofinternationalinterestincon- densedmatterphysics.Thisbookwasbothmotivatedbythisintenseactivity,espe- ciallyinthegroupofJeanDalibardattheEcolenormalesupe´rieure,butalsobythe constantdevelopmentofmathematicaltechniqueswhichcouldproveusefulintack- ling these problems, in particular in the group of Haim Brezis. This monograph is dedicatedtothemathematicalmodellingofsomespecificexperimentswhichdisplay vorticesandtoarigorousanalysisoffeaturesemergingexperimentally.Itcanserve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum condensates, and can also complement a graduate semi- narinellipticPDEsormodellingofphysicalexperiments.Therearetwointroductory chapters:thefirstisrelatedtothephysicsbackground,whilethesecondisdevoted tothepresentationofthemathematicalresultsdescribedinthebook. Vortices have been observed experimentally by rotating the trap holding the atoms in the condensate. In contrast to a classical fluid, for which the equilibrium velocitycorrespondstosolidbodyrotation,aquantumfluidsuchasaBose–Einstein condensatecanrotateonlythroughthenucleationofquantizedvorticesbeyondsome criticalvelocity.Therearetwointerestingregimes:oneclosetothecriticalvelocity wherethereisonlyonevortex,andanotherathighrotationvalues,forwhichadense lattice is observed. Another experiment consists of a superfluid flow around an ob- stacle: at low velocity, the flow is stationary; while at larger velocity, vortices are nucleatedfromtheboundaryoftheobstacle. vi Preface One of the key issues is thus the existence of these quantized vortices. We ad- dressthisissuemathematicallyandderiveinformationontheirshape,number,and location.Inthedilutelimitoftheexperiments,thecondensateiswelldescribedby ameanfieldtheoryandamacroscopicwavefunction,solvingtheso-calledGross– Pitaevskiiequation.Themathematicaltoolsemployedareenergyestimates,Gamma convergence,andhomogenizationtechniques.Weproveexistenceofsolutionswhich have properties consistent with the experimental observations. Open problems re- latedtorecentexperimentsarealsopresented.Theywillrequirethedevelopmentof newtoolsrelatedforinstance,tomicrolocalanalysisortime–dependentproblems. ThesuggestionforsettingdowntheseimportantideascamefromHaimBrezis, andIwouldliketothankhimwarmlyforhisconstantenthusiasmandsupportwhile Iwasworkingonit.Manytoolsusedherehavebeendevelopedbyeitherhimorhis school.Iamgladtobeabletopresentanapplicationofthisbeautifulmathematics totoday’sphysics. I am also extremely grateful to Jean Dalibard, who has always been willing to take time to share his experiments, his ideas, and his interests in how mathematics cancontributetophysics.Workingtogetherwithhimandwritingajointpaperwasa realpleasureandasourceofmathematicalproblemsformanyyearstocome.Iwould alsoliketothankhimforhiscarefulreadingofthismanuscript.Beforeworkingwith Jean,Ihadtheopportunityofmanyfruitfuldiscussionswithmembersofhisgroup, in particular Vincent Bretin, Yvan Castin, and David Gue´ry-Odelin. I have always appreciatedtheiropenmindsandinterestinmathematics.IwouldliketothankDavid inparticularforhiscommentsleadingtoimprovementsintheintroductorypartsof thisbook. IowemypersonalinterestininterdisciplinarytopicstothejointeffortsofEtienne Guyon,YvesPomeau,andHenriBerestycki,wholaunchedaprogramforstudents attheEcolenormalesupe´rieuretosparkinterestinproblemsontheborderbetween mathematicsandphysics.Thiseffortwasarealsuccess,aswerethevariousmaths- physics meetings in Foljuif, a property of the Ecole normale supe´rieure. At one of them,ImetYvanCastinandrealizedthatwehadmathematicaltoolsthatcouldhelp inunderstandingproblemsemerginginrotatingBose–Einsteincondensates.Iwould liketoagainmydeepgratitudetoEtienneGuyon,YvesPomeau,andmysupervisor HenriBerestycki,forallthatIdiscoveredhasbeenthankstothem. I would like to also thank, of course, all my collaborators on these topics, in particular: Tristan Rivie`re, with whom this huge program started and the evidence of vortex bending occurred; Bob Jerrard, whose involvement in understanding the shape of vortices was quite influential and with whom it was a pleasure to work in Vancouver,Milan,Istanbul,andMinneapolis(Ithankallthehostinginstitutions)and whohasundertakenaverycarefulreadingofthebook;QiangDuandIonutDanaila, who have performed, on different topics, beautiful numerical computations; Stan Alama and Lia Bronsard, who came to Paris and became interested in these topics when they discovered them; Xavier Blanc, with whom I am very happy to have worked with on many projects; and very recently Francis Nier, who has allowed metodiscovermicrolocalanalysisandBargmanntransforms,whichhaveprovedto Preface vii bequiteusefulintacklingtheseproblems.Itwasalsoveryrewardingtoworkwith outstanding physicists, Jean Dalibard and Yves Pomeau. Many colleagues all over theworldhavementionedthatIwasquiteluckytohavehadthisopportunityandto havefoundacommonlanguagetospeak.Icertainlybelieveit. Part of this monograph was taught as a Ph.D. course at Paris 6 in 2003–2004. OneoftheresultspresentedherewasobtainedbytwoofthesePh.D.students,Radu IgnatandVincentMillot,whomIjointlysupervisedwithHaimBrezis.Iampleased todescribetheirworkinoneofthechapters. The quality of the presentation of this book was greatly improved thanks to all the lectures given in various universities or summer schools. I would like to thank in particular: Luis Caffarelli and Irene Gamba, Peter Constantin, Peter Sternberg, Miguel Escobedo, Gero Friesecke, Fang-Hua Lin, Stefan Muller, Tristan Rivie`re, andJuan-LuisVazquez.IhavealsobenefitedfrominformaldiscussionswithFabrice Bethuel,PetruMironescu,SylviaSerfaty,EtienneSandier,andDidierSmets. I take the opportunity here to express my gratitude to all my colleagues in the Laboratoire Jacques-Louis Lions, in particular: to Yvon Maday, who is a very en- thusiasticheadofdepartment;tomyofficemateXavierBlanc;tomyofficeneigh- bours, Edwige Godlewski and Francois Murat; to Olivier Glass, Fre´de´ric Hecht, Simon Masnou, and the staff in the laboratory, Danielle Boulic, Michel Legendre, JacquesPortes,andLilianeRuprecht. The writing of this book was made possible by my position at CNRS, which should be naturally associated with the outcome of such interdisciplinary effort. My research during this period was supported by a CNRS grant for young re- searchers and a French ministry of research grant, ACI “Nouvelles interfaces des mathe´matiques.” Some of the open problems were derived during a maths–physics meetingorganizedwithDavidGue´ry-Odelinatthe“FondationdesTreilles”inTour- tour,andIwouldliketoacknowledgetheirwelcome. Finally,Iwouldliketothankmyfamilyandfriendsfortheirconstantsupportin thepreparationofthemanuscriptandallthepeopleatBirkha¨user,inparticularAnn Kostant,fortheirhelp. Paris,October31,2005, AmandineAftalion Contents Preface ......................................................... v 1 ThePhysicalExperimentsandTheirMathematicalModelling ...... 1 1.1 Ahintontheexperiments.................................... 1 1.1.1 Briefhistoricalsummary.............................. 1 1.1.2 HowtoachieveaBECexperimentally................... 2 1.1.3 Experimentalobservations ............................ 3 1.2 Themathematicalframework................................. 5 1.2.1 TheGross–Pitaevskiienergy........................... 5 1.2.2 TheThomas–Fermiregime............................ 7 1.2.3 Remarksontheoriginalproblem ....................... 12 1.2.4 Mean-fieldquantumHallregime ....................... 13 1.2.5 Flowaroundanobstacle .............................. 16 2 TheMathematicalSetting:ASurveyoftheMainTheorems ........ 19 2.1 Smallεproblem ........................................... 19 2.1.1 Thetwo-dimensionalsetting........................... 20 2.1.2 Thethree-dimensionalsetting ......................... 23 2.2 Vortexlattice .............................................. 24 2.3 Flowaroundanobstacle .................................... 27 3 Two-DimensionalModelforaRotatingCondensate ............... 29 3.1 Mainresults ............................................... 30 3.1.1 Single-vortexsolutionandlocationofvortices............ 31 3.1.2 Ideasoftheproof .................................... 32 3.2 Preliminaries .............................................. 35 3.2.1 Determiningthedensityprofile......................... 35 3.2.2 Existenceofaminimizerof Eε......................... 36 3.2.3 Splittingtheenergy .................................. 37 3.3 Boundednumberofvortices ................................. 38 3.3.1 Firstenergybound ................................... 39 x Contents 3.3.2 Vortexballs ......................................... 40 3.3.3 Therotationterm .................................... 41 3.3.4 Alowerboundexpansion ............................. 43 3.4 Refinedstructureofvortices ................................. 46 3.4.1 Somelocalestimates ................................. 47 3.4.2 Baddiscs........................................... 49 3.4.3 Nodegree-zerovortex ................................ 50 3.4.4 ProofofProposition3.12.............................. 53 3.5 Lowerbound .............................................. 55 3.6 Upperbound .............................................. 61 3.7 Finalexpansionandpropertiesofvortices ...................... 66 3.7.1 Vorticeshavedegreeone .............................. 66 3.7.2 Thesubcriticalcase .................................. 67 3.7.3 Thesupercriticalcase................................. 68 3.8 OpenQuestions ............................................ 75 3.8.1 Vorticesintheregionoflowdensity .................... 75 3.8.2 Othertrappingpotentials.............................. 75 3.8.3 Intermediate(cid:2) ...................................... 76 3.8.4 Time-dependentproblem.............................. 77 4 OtherTrappingPotentials..................................... 79 4.1 Nonradialharmonicpotential ................................ 80 4.2 Quarticpotential ........................................... 82 4.2.1 Giantvortex ........................................ 82 4.2.2 Circleofvortices .................................... 87 4.3 Openquestions ............................................ 98 4.3.1 Circleofvortices .................................... 98 4.3.2 Giantvortexorisolatedvortices........................ 98 5 High-VelocityandQuantumHallRegime ........................ 99 5.1 Introduction ............................................... 100 5.1.1 LowestLandaulevel ................................. 100 5.1.2 Constructionofanupperbound ........................ 101 5.1.3 Propertiesoftheminimizer............................ 106 5.1.4 Othertrappingpotentials.............................. 107 5.2 Regularlattice ............................................. 107 5.3 Distortedlattice ............................................ 111 5.4 Infinitenumberofzeros ..................................... 117 5.5 Othertrappingpotentials .................................... 119 5.6 Openquestions ............................................ 119 5.6.1 Lowerboundand(cid:3)convergence ....................... 119 5.6.2 RestrictiontotheLLL ................................ 120 5.6.3 Reductiontoatwo-dimensionalproblem ................ 121 5.6.4 Meanfieldmodel .................................... 121 Contents xi 6 Three-DimensionalRotatingCondensate ........................ 123 6.1 Numericalsimulations ...................................... 124 6.2 Formalderivationofthereducedenergy E[γ]................... 126 6.2.1 Thesolutionwithoutvortices .......................... 126 6.2.2 Decouplingtheenergy................................ 127 6.2.3 EstimateofGε(vε)................................... 128 6.2.4 Estimateof Iε(vε).................................... 131 6.2.5 Finalestimatefortheenergy........................... 131 6.3 (cid:3)convergenceresults ....................................... 132 6.3.1 Mainresults......................................... 132 6.3.2 Mainideasintheproof ............................... 134 6.4 SingleVortexline,studyof E[γ] ............................. 142 6.4.1 Settingofminimizationof E[γ]........................ 142 6.4.2 Thebentvortex...................................... 144 6.4.3 Propertiesofcriticalpoints ............................ 152 6.5 Afewopenquestions ....................................... 154 6.5.1 Smallvelocity....................................... 154 6.5.2 Criticalpointsof E(cid:2)[χ] .............................. 155 6.5.3 Finitenumberofvortices.............................. 155 6.5.4 Othertrappingpotentials.............................. 155 6.5.5 Wholespaceproblem................................. 155 6.5.6 Decayofvortices .................................... 156 7 SuperfluidFlowAroundanObstacle ............................ 157 7.1 Mathematicalsetting........................................ 158 7.1.1 Two-dimensionalflow ................................ 158 7.1.2 Three-dimensionalflowaroundacondensate ............. 162 7.2 ProofofTheorem7.1 ....................................... 167 7.2.1 Solutionsatc=0.................................... 169 7.2.2 Existenceofasolutionto I ........................... 171 R 7.2.3 Boundsonthesolutionsof I .......................... 172 R 7.2.4 Estimatingthemomentum............................. 174 7.2.5 ProofofTheorem7.4................................. 177 7.2.6 Limitatinfinity...................................... 177 7.3 ProofofTheorem7.2 ....................................... 180 7.3.1 ProofofTheorem7.14................................ 181 7.3.2 ProofofTheorem7.16................................ 185 8 FurtherOpenProblems....................................... 195 8.1 SettinginthewholespacefortheThomas–Fermiregime ......... 195 8.1.1 Three-dimensionalproblem ........................... 195 8.1.2 Two-dimensionalproblem............................. 195 8.1.3 Painleve´ boundarylayer............................... 196 8.1.4 Vorticesinthehole................................... 196 8.2 Otherscalings ............................................. 196