Nonlinear PDE’s in Condensed Matter and Reactive Flows

524 Pages·2002·19.77 MB·English

by  M. Z. Bazant

Nonlinear PDE's in Condensed Matter and
Reactive Flows
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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
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Originally published by KluwerAcademic Publishers in 2002
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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