Lower Bounds on the Blow Up Rate of the Axisymmetric Navier Stokes Equations II

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1、CommunicationsinPartialDifferentialEquations,34:203–232,2009Copyright©Taylor&FrancisGroup,LLCISSN0360-5302print/1532-4133onlineDOI:10.1080/03605300902793956LowerBoundsontheBlow-UpRateoftheAxisymmetricNavier–StokesEquationsIICHIUN-CHUANCHEN1,ROBERTM.STRAIN2,TAI-PENGTS

2、AI3,ANDHORNG-TZERYAU21DepartmentofMathematicsandTaidaInstituteforMathematicalSciences,NationalTaiwanUniversityandNationalCenterforTheoreticalSciences,TaipeiOffice,Taipei,Taiwan2DepartmentofMathematics,HarvardUniversity,Cambridge,Massachusetts,USA3DepartmentofMathemati

3、cs,UniversityofBritishColumbia,Vancouver,BritishColumbia,CanadaConsideraxisymmetricstrongsolutionsoftheincompressibleNavier–Stokesequationsin
3withnon-trivialswirl.Letzdenotetheaxisofsymmetryandrmeasurethedistancetothez-axis.Supposethesolutionsatisfies,forsome0≤
≤1,v

4、xt≤Cr−1+
t−
/2for−T≤t<0and0

5、artesiancoordinatesaregivenbytv+v·v+p=vdivv=0(N–S)Thevelocityfieldisvxt=vvv
3×−T0→
3andpxt
3×1230−T00→
isthepressure.Itisalongstandingopenquestiontodetermineifsolutionswithlargesmoothinitialdataoffiniteenergyremainregularforalltime.Inthispaperweco

6、nsiderthespecialclassofsolutionswhichareaxisymmetric.Thismeans,incylindricalcoordinatesrzwithx1x2x3=rcosrsinz,ReceivedOctober1,2007;AcceptedDecember15,2008AddresscorrespondencetoTai-PengTsai,DepartmentofMathematics,UniversityofBritishColumbia,1984Mathema

7、ticsRoad,Vancouver,BCV6T1Z2,Canada;E-mail:ttsai@math.ubc.ca203204Chenetal.thatthesolutionisoftheformvxt=vrrzter+vrzte+vzrztez(1.1)
Inthiscoordinatesystemr=x2+x2.Thecomponentsvvvdonotdepend12rzuponandthebasisvectorsereezarex1x2x2x1er=0e=−0

8、ez=001rrrrThemainresultofourpapershowsthataxisymmetricsolutionsmustblowupfasterthanthescaleinvariantrateswhichappearinTheorem