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Nonlinear PDE’s in Condensed Matter and Reactive Flows PDF

524 Pages·2002·19.77 MB·English
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Nonlinear PDE's in Condensed Matter and Reactive Flows NATO Science Series ASeriespresentingthe resultsofactivitiessponsoredby tneNATOScience Committee. TheSeriesispublishedby10S Press andKluwerAcademic Publishers, inconjunction with the NATOScientificAffairsDivision. A. LifeSciences las Press B. Physics Kluwer AcademicPublishers C. Mathematicaland PhysicalSciences KluwerAcademicPublishers D. BehaviouralandSocialSciences KluwerAcademicPublishers E. AppliedSciences KluwerAcademicPublishers F. Computerand SystemsSciences lasPress 1. DisarmamentTechnologies KluwerAcademicPublishers 2. EnvironmentalSecurity KluwerAcademicPublishers 3. HighTechnology KluwerAcademicPublishers 4. ScienceandTechnologyPolicy lasPress 5. ComputerNetworking lasPress Asaconsequence ofthe restructuringoftheNATOScience Programmein1999,theNATO ScienceSerieshasbeen re-organizedandnewvolumeswillbeincorporatedintothefollowing revised sub-seriesstructure: 1. Lifeand BehaviouralSciences lasPress II. Mathematics,PhysicsandChemistry KluwerAcademicPublishers III. Computerand SystemsSciences lasPress IV. Earth and EnvironmentalSciences KluwerAcademicPublishers V. ScienceandTechnologyPolicy lasPress NATO-PCO-DATABASE The NATOScienceSeriescontinuestheseriesofbooks publishedformerlyintheNATOASI Series.AnelectronicindextotheNATOASISeries providesfullbibliographicalreferences(with keywordsand/orabstracts)tomore than50000 contributionsfrom internationalscientists publishedinaIIsectionsoftheNATOASISeries. AccesstotheNATO-PCO-DATASASE ispossibleviaCD-ROM"NATO-PCO-DATASASE"with user-friendlyretrieval softwareinEnglish,FrenchandGerman(WTV GmbH andDATAWARE Technologies Inc.1989). The CD-ROMoftheNATOASISeriescanbeorderedfrom:PCO,Overijse, Selgium SeriesC:Mathematicaland PhysicalSciences- VoI.569 Nonlinear PDE's in Condensed Matter and Reactive Flows edited by Henri Berestycki Ecoledes hautesEtudesenSciencesSociales, Paris,France and Yves Porneau CNRS, Ecole NormaleSuperieure, Paris,France Springer Science+Business Media, B.V. Publishedincooperationwith NATOScientificAffairs Division ProceedingsoftheNATOAdvanced Study Instituteon PDE'sinModels ofSuperfluidity,SuperconductivityandReactive Flows Carqese,France 21June-3July 1999 AC.I.P.CataloguerecordforthisbookisavailablefromtheLibrary ofCongress. ISBN 978-1-4020-0973-0 ISBN 978-94-010-0307-0 (eBook) DOI 10.1007/978-94-010-0307-0 Printedonacid-freepaper AIIRights Reserved ©2002 SpringerScience+Business Media Oordrecht Originally published by KluwerAcademic Publishers in 2002 Softcover reprint of the hardcover tst edition2002 Nopart ofthiswork maybereproduced, stored inaretrievalsystem, ortransmitted inanyformorbyanymeans,electronic, mechanical,photocopying, microfilming, recording orotherwise,without writtenpermissionfromthe Publisher,withtheexception ofanymaterial supplied specificallyforthepurposeofbeing entered andexecuted onacomputersystem,forexclusiveusebythepurchaserofthework. Table ofContents Preface IX Acknowledgements Xl Part I M.Z. BAZANT, H.A. STONE A two-species reaction-diffusion problemwithone static reactant: thecaseofhigherorderkinetics 1 H. BERESTYCKI The influence ofadvection onthe propagationof fronts for reaction-diffusion equations 11 P.CLAVIN Instabilities and NonlinearPatterns ofOverdriven Detonations in Gases 49 l.W.DOLD Ends ofLaminarFlamelets: TheirStructure, Behaviourand Implications 99 E. FFIREISL, M. MIMURA, D. HILHORST, R. WEIDENFELD On some reaction-diffusion systemswith nonlinear diffusion arising in biology 115 E. FERNANDEZ-CARA, E. ZUAZUA Control ofweakly blowing up semilinearheatequations 127 V. HAKIM Spirals in ExcitableMedia 149 F. HAMEL, R. MONNEAU Conical-shapedtravelling fronts allied to the mathematical analysis ofthe shape ofpremixed Bunsen flames 169 VI R. KLEIN Numerical Modelling ofHigh Speed and Low Speed Combustion 189 A NEWELL Lectures on Wave Turbulence and Intermittency 227 B. SIRAKOV Overdeterminedelliptic problems in physics 273 H.A STONE Partial Differential Equations in thin film Flows in Fluid dynamics and rivulets 297 Part II A AFfALION The Ginzburg-Landau system for superconducting thin films 313 S. ALAMA, L. BRONSARD Symmetric Vortex solutions in the U(I) and 50(5) Ginzburg-Landau Models ofSuperconductivity 323 F. BETHUEL, I.e. SAUT Vortices and sound waves for the Gross-Pitaevskii equation 339 e. BOLLEY, B. HELLFER A priori estimates for Ginzburg-Landau solutions 355 S.J. CHAPMAN Asymptotic Analysis ofModels ofSuperconductivity 375 P. COULLET, E. RISLER, N. VANDERBERGHE Spatial Unfolding ofHomoclinic Bifurcations 399 M.ESCOBEDO Existence and long timebehaviourofsolutions for ahomogeneous quantum Boltzmann equation 413 e. R. JORDAN, JOSSERAND Asymptotic behaviorin amodel ofdispersive wave turbulence 429 vii J.R. OCKENDON Some problems in superconductivity and reacting flow 449 Y. POMEAU Finite timeblow-up ofsolutions ofkinetic equations andformation ofBose-Einstein condensate 461 J. RUBINSTEIN, P. STERNBERG Secondorderphase transitions 473 E. SANDIER, S.SERFATY Vortex Analysis in the Ginzburg-Landau Model of Superconductivity 491 G. TARANTELLa On Chern-Simons vortextheory 507 Preface TheNATOAdvancedScientificInstitute(ASI)on "Partialdifferentialequa tionsinSuperfluidity,superconductivityandReactiveFlows"washeldinCargese (France),fromJune 21st toJuly3rd, 1999. Weco-organizeditwithJonChap man andDavidLevermore. Nonlinearpartial differential equations (PDE's) abound in modemphysics. Theproblemsarisinginthesefields leadtofascinating mathematicalquestions and, atthesametime,progressinunderstandingthemathematicalstructuresis ofgreatimportancetothe models. Thethrivingactivityinoneortheotherofthesecomplementaryapproaches howeveris notalways sufficiently in touch with the developmentsin the other field. OnegoaloftheASIwaspreciselytobringtogethermathematicianswork ingin thefield ofnonlinearPDE's andphysicistsinvolvedin the modellingof reactive flows andthephysicsofcondensedmatter,inparticularsuperconduc tivityandsuperfluidity. Indeed, modemmacroscopicphysicsbearonnonlinear PDE's which often are at the forefront of - if not beyond - the mathematical techniques to study these problems. Some phenomena, like the weak interac tion ofwaves, even lead to nonlinearintegralequations with arathercomplex structure, thathavealso beenconsideredduring this meeting. The goal of the ASI was twofold: to expose the state of the art in these domains, with a special emphasis on the new directions and outstanding open problems, and toexchangeideasacrossfields andacross variedapproaches. Thephysicalthemesofreactiveflows (inparticularcombustionphenomena) and ofcondensed matter lead to mathematical problems with a certain unity: nonlinearPDEtechniques, singularities, interfaces,etc. Reaction-diffusionequationsdescribefor instancethepropagationoffronts across which the state ofa system undergoes a radical change. This may be relatedtoachemicalreactionortosomephasechangeincondensedmatter. In combustion,anessentialaspectistheinterplaywithth~motionoffluids-hence thenameofreactiveflows-leadingtodelicateproblems. Thingsarealsomuch involvedwhensuperfluidity, superconductivityandevenmoreexoticstatesare present. Then,thedynamicsisconstrainedbythesymmetriesoftheunderlying physicalsystem, asourceofrich phenomenology. lX x A trait ofthe conference was its aiming at a double mixing: of advanced results and more classical ones and ofphysicists and mathematicians. It has been very successful on both endeavors. The wide range oftopics has been a sourceofmutualenrichment,particularlyfortheyoungerparticipantswhohave seen several different aspects ofthese theories. Various collaborations began at this ASI, following discussions betweenparticipants. New topics have also emerged in this meeting. Onecan mention thestudyofrotatingBose-Einstein condensates, that really started as a new field at this meeting, although no experimenthadbeendoneatthattime,andverylittletheoryexisted. Likewise, in this meeting, some presentations were madeon weak turbulence equations. Recentprogresshas beenachievedontheunderstandingoftheseequationsfor which it now appears that the Kolmogoroff-Zakharov solutions playarole far morerestrictedthanexpectedbefore. Anotherimportantexampleistheintense activitywhichhasbeengeneratedonthepropagationoffrontsinheterogeneous media. Whilecompletingthepreparation oftheseproceedings, itis withgreatsad ness that we have learned of the passing away of Guy Dewell, following a longand cruel illness. Guy has been one ofour main lecturers, buthis illness preventedhimtowrite uphisremarkablecontributiononthestabilityofBose Einsteincondensatewithattractiveinteractions. Weallmournagreatcolleague andfriend. Theseproceedings area tribute tohis memory. Henri Berestycki andYves Pomeau Paris, Spring2002 Acknowledgements Support from various institutions has been essential to make this program possible. Firstandforemost, themeetingcouldnothavebeenheldwithoutthe generous supportoftheNATO scientificprogram. Thestaffofthe Division of ScientificAffairsofNATOinBrussels,inparticularitsDirector,hasprovideda veryprofessional,openandsupportivehelp. WeareespeciallygratefultoMrs. Trapp for herconstantreadiness to kindlyhelp us throughouttheprojectin its (attimes daunting) administrativeaspects. Wehavebenefitedformthesupportofseveralotherorganizations: theFrench DGA, theFrenchMinistry ofEducation and Research, the ContinuingEduca tion Program and SPM Department ofCNRS in France, and the NSF (USA). Wegratefully acknowledge theirsupport. Severalcolleaguesgaveusahelpinghandatvariousstagesintheorganization of the conference. We are thankful to all ofthem, with a special mention to Amandine Aftalion. We thank Kluwer, the publisherofthese proceedings, and in particularMs. Wil Bruins,whohavepatientlywaitedfor afinal versionofthis manuscript. It hasbeentypesetusingLaTeX.BoyanSirakov'shelpinthetechnicalpreparation wasofgreatimportanceinthefinal stagesandisgratefullyacknowledged. We alsothankthestaffofCentred'AnalyseetdeMathematiquesSocialesatEcole des hautes etudes in Paris, particularly Annie Gille and Catherine Gueraud for their great and kind help. Finally our warm thanks go to the whole team of the Institut Scientifique de Cargese (including the chef) under Elizabeth Dubois-Violette. They have provided us the atmosphere for a verycongenial andfriendly meeting. Henri BerestyckiandYves Pomeau Paris, Spring2002 XI

Nonlinear partial differential equations abound in modern physics. The problems arising in these fields lead to fascinating questions and, at the same time, progress in understanding the mathematical structures is of great importance to the models. Nevertheless, activity in one of the approaches is
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