简介:ToillustratethegeneralpropertiesoftheBeltramiflows,anewsufficientandnecessarycondi-tionofthegeneralsolutionoftheBeltramiflowshasbeengiven,onwhosebasissomesufficientandnecessaryconditionsandsufficientconditionsinliteratureareunified.TheadvantagesofthenewexpressionsarethatonemaygetsolutionsofBeltramiflowswithconstantproportionalfactorswhichsatisfysomegivenasymptoticdecayingconditionsatinfinity.Finally,wefindaclassofsphericalvorticesofBeltramiflows,wherethemo-tiontrajectoriesofthefluidparticlesareconfinedwithinasphereortwoadjacentconcentricsphericalshells.Wefindthatthesuperpositionsofthesphericalvorticesofthesameorderwithdifferentpolaraxesmayproducechaotictrajectories.
简介:§0.IntroductionLetXbearealseparableBanachspaceandX*beitsdualspace,LetB(X)betheBorelfield,i.e.,thetopologicalσ-field.Afunctionalu:X→R’iscalledaboundedsmoothfunctional,ifn∈N,f1,…,fn∈X*andφ∈Cb∞(Rn),suchthat