(J.P.Puel)Contrallability of Navier Stokes

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1、ControllabilityofNavier-StokesEquationsJean-PierrePuel1November22.20121J.-P.Puel:(jppuel@math.uvsq.fr)LaboratoiredeMathematiquesdeVersailles,Uni-versitedeVersaillesSaint-QuentinThesenotescorrespondtoacoursetaughtinBCAM,BilbaoinMarch-April2012andinCMM,Santiag

2、odeChileinNovember2012.Mostofthemfollowthelinesofpublishedarticlesonthesubject,essentiallythebasicarticlebyE.Fernandez-Cara,S.Guerrero,O.Yu.ImanuvilovandJ.-P.PuelinJ.Math.PuresetAppl.[3]andthearticlebyO.Yu.Imanuvilov,J.-P.PuelandM.YamamotoinChineseAnn.ofMath.[

3、8],andalsoanarticlebyO.Yu.Imanuvilov,J.-P.PuelandM.Yamamoto[9]whichisnotyetpublished.But,strictlyspeaking,theresultspresentedherearenewinthesensethattheyarenotwrittenanywhereinthisgenerality.Moreoverthemethodofproofisdierentfromtheonesgivenintheabovearticles.

4、ThepresentmonographintendstogiveaprecisedescriptionofthelatestmethodusedtostudythelocalexactcontrollabilityofNavier-Stokesequations.ItreliesonanewglobalCarlemanestimatefortheStokesequationsforthelinearizedproblemanditfollowsessentiallythelinesof[3]inaslightlyd

5、ierentfunctionalclassforthenonlinearproblem.Theresultobtainedhereseemsquiteoptimalandisalittlestrongerthantheoneobtainedin[3].Arelevantobjectiveforthecontrolofaviscous uidisnotstraightforwardasthesystemisdissipativeandnotreversible.ThenotionofExactControllabi

6、litytoTrajectoriesseemstobethemostinterestingoneanditisdescribedintherstchapter.Inthesecondchapterwetreatthelinearizedproblem,whichcorrespondstoanullcontrollabilityproblemandrequiresthemainmathematicaltools.Thethirdchapterdealswiththefullnonlinearproblem,whic

7、hisstudiedinaquitecomplexfunctionalclassevenifitissimpliedfromtheoneconsideredin[3].TheargumentusesregularitypropertiesfortheStokessystemandsomeneinterpolationresults.1Chapter1Exactcontrollabilitytotrajectories1.1IntroductionThephysicaldomainisaregularbounde

8、dconnectedopensetofIRNwithN=2orN=3,whoseboundaryisdenotedby.Wedenotebytheunitexteriornormalvectoratapointof.Thetimevariabletwillbetakenintheinterval(0;T)withT>0.Weconsiderthefol