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1、J.Math.PuresAppl.86(2006)471491www.elsevier.com/locate/matpurConvergencetoequilibriaandblowupbehaviorofglobalstrongsolutionstotheStokesapproximationequationsfortwo-dimensionalcompressibleflowswithlargedata✩FeiminHuanga,JingLia,b,∗,ZhoupingXinbaInstituteofApp
2、liedMathematics,AMSS,AcademiaSinica,Beijing100080,PRChinabInstituteofMathematicalSciencesandDepartmentofMathematics,TheChineseUniversityofHongKong,Shatin,N.T.,HongKongReceived24March2006AbstractThispaperconcernsthelargetimebehaviorofstrongandclassicalsoluti
3、onstothetwo-dimensionalStokesapproximationequationsforthecompressibleflows.Weconsidertheuniqueglobalstrongsolutionorclassicalsolutiontothetwo-dimensionalStokesapproximationequationsforthecompressibleflowswithlargeexternalpotentialforce,togetherwithaNavier-sli
4、pboundarycondition,forarbitrarilylargeinitialdata.Undertheconditionsthatthecorrespondingsteadystateexistsuniquelywiththesteadystatedensityawayfromvacuum,weprovethatthedensityisboundedfromaboveindependentlyoftime,consequently,itconvergestothesteadystatedensi
5、tyinLpandthevelocityuconvergestothesteadystatevelocityinW1,pforany1p<∞astimegoestoinfinity;furthermore,weshowthatiftheinitialdensitycontainsvacuumatleastatonepoint,thenthederivativesofthedensitymustblowupastimegoestoinfinity.©2006ElsevierMassonSAS.Allrightsr
6、eserved.RésuméCetarticleconcernelecomportemententempslongdessolutionsfortesetdessolutionsclassiquesdelapproximationdeStokesbi-dimensionnelledunécoulementcompressible.OnconsidèreluniquesolutionforteousolutionclassiquedelapproximationdeStokesbi-dimensionnelle
7、pourunécoulementcompressibleavecunegrandeforceextérieuredetypepotentielenchoisissantuneconditionauxbordsdeNavieretdesdonnéesinitialesarbitrairementgrandes.Sousleshypothèsesquelétatstationnairecorrespondantexistedemanièreuniqueavecdensitééloignéeduvide,onmon
8、trequeladensitéduproblèmededépartrestebornéinférieurementpourtouttempsetparconséquencequelleconvergeentempslongverslétatstationnaireennormeLp,pourtout1p<+∞.Deplusonmontrequesiladensitéinitialesannulea
