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1、208DynamicsofpartialdifferentialequationsonthindomainsGenevieveRaugelIntroductionThepurposeofthiscourseistostudythedynamicsofPDEonthindomains.Themainmotivationofthiskindofproblemscomesfromtwosources.Thefirstsourceisphysics;thindomainsproblemsareencounteredinsolidmechanics(thinrods,plates,
2、shells),fluiddynamics(lubrication,meteorologyproblems,oceandynamics,etc...).Forinstanceinoceandynamics,oneisdealingwithlargescalefluidproblems,inwhichtilefluidregionsarethincomparedtothehorizontallengthscales(seetherecentworkofCamassaandHolm(1992),forexample).Somemeteorologyproblemsinvolv
3、efluidscirculatingbetweentwoconcentricspheresofnearlyequalradii,whichmodelstileatmosphere.Problemsinmechanicsandespeciallyinelasticityconcerningthinrods,platesandshellshaveah'eadybeenstudiedforoveracentury.Thestudyoftheequilibriumstatesofathinplateftx(-e,+e)underexternalforceswheref~isasm
4、oothdomaininIR2andeisa.smallparameter,leadstoatwo-dimensionalmodellingofplates(seeCiarletandDestuynder(1979),Destuynder(1981),forinstance).Inlubricationproblems,theReynoldsequationcanbejustifiedmathematicallyfromtheStokesequationonthindomains(BayadaandChambat(1986)).Inreaction-diffusioneq
5、uationswecanalsoconsidersomemodelsforwhichallthediffusioncoefficientsarelarge;afterrescaling,weobtainaPDEonathindomainaroundapoint.Theinodellingofmulti-structuresinvolvesmorecomplicatedthindomains.Roughlyspeaking,amulti-structureproblemisdescribedbyaPDEgivenonaconnecteddomaininR'~consisti
6、ngofseveralparts,someofwhichcanberelativelythininoneorseveraldirections.Typicalexamplesareencounteredinelasticity,inheatdiffusion,etc...andleadtoproblemswithjunctions(seeCiarlet,LeDretandNzengwa(1987,1989),Ciarlet(1990),LeDret(1989a,1989b,1991)).Forsakeofsimplicityandofloweringthecostsofc
7、omputation,onetriestoreplacethen-dimensionalmulti-structurebyamulti-structure,thedimensionofwhichissmallerinthethinparts(seethejustmentionedpapers).Thindomainsproblemshavebeenconsideredbymanyauthorsfromdifferentpointsofview,includingmodelling,control,homogenizationq
