Finite-time blow-up of solutions of an aggregation

Finite-timeblow-upofsolutionsofanaggregationequationinRnAndreaL.Bertozzia;bandThomasLaurentbaDepartmentofMathematics,UniversityofCalifornia,LosAngeles90095bDepartmentofMathematics,DukeUniversity,Durham,[e-mail],[e-mail]2,2006AbstractWeconsidertheaggregationequationut+r
(urKu)=0inRn,n2,whereKisarotationallysymmetric,nonnegativedecayingkernelwithaLipschitzpointattheorigin,e.g.K(x)=ejxj.Weprovenite-timeblow-upofsolutionsfromspecicsmoothinitialdata,forwhichtheproblemisknowntohaveshorttimeexistenceofsmoothsolutions.1IntroductionTheaggregationequationut+r
(urKu)=0(1)arisesinanumberofcontextofrecentinterestinthephysicsandbiologyliterature.Inbiology,aswarmingmechanism,inwhichindividualorganismssenseothersatadistance,andmovetowardsregionsinwhichtheysensethepresenceofothers,in-volvesacomplexneurologicalprocessattheindividuallevel.Theselocalindividualinteractionsleadtolargescalepatternsinnatureforwhichitisdesirabletohaveatractablemathematicalmodel.Themodelalsoarisesinthecontextofself-assemblyofnanoparticles(seee.g.HolmandPutkaradze[15,16]andreferencestherein).Theequation(1)withdierentclassesofpotentialsandwithadditionalregularizingtermsappearsinanumberofrecentandolderpapers.TopazandBertozzi[27]derivethemodelasmulti-dimensionalgeneralizationofone-dimensionalaggregationbehaviordiscussedinthebiologyliterature[13,23].BodnarandVelazquez[2]considerwell-posednessonRfordierenttypesofkernels.Burgerandcollaborators[5,4]considerwell-posednessofthemodelwithanadditional`porousmedia'typesmoothingterm.Thisclassofmodels,withdiusion,isalsoderivedbyTopazetal[28]whocitesomeearlierrigorousworkfromthe1980sinonespacedimension.Wespecicallyconsider1thecasewherethekernelKhasaLipschitzpointattheorigin,asinthecaseofejxjwhicharisesinboththebiologicalandnanoscienceapplications.Asamathematicsproblem,equation(1)isanactivescalarproblem[10]inwhichaquantityistransportedbyavectoreldobtainedbyanonlocaloperatorappliedtothescalareld.Suchproblemscommonlyariseinuiddynamics,forexample,thetwodimensionalvorticityequation@!@t+v
r!=0;v=r?Ne!whereNeistheNewtonianpotentialintwospacedimensions.Theseactivescalarequationssometimesserveasmodelproblemsforthestudyofnitetimeblowupofsolutionsofthe3DEulerequationsinwhichthevorticityvectorsatises@~!@t+~v
r~!=!
rv;~v=~K3!(2)whereK3isthe3DBiot-Savartkernel,homogeneousofdegree2in3D.Stretchingofvorticityisaccomplishedbytherighthandsideofof(2)inwhichthenonlocalamplicationofvorticityoccurs.Forthisproblem,nitetimeblowupofsolutionsisanopenproblem.The2D-quasi-geostrophicequationsalsoposeaninterestingfamilyofactivescalarequations[9,17].Animportantdistinction,betweenourproblemandthatoftheEuler-relatedproblems,isthatourtransporteld~visagradientowwhereasthe~vinuiddynamicsisdivergencefree.Neverthelessitisinterestingtonotethisanalogywhichalsoincludesthewell-knownonedimensionalmodelproblemstudiedbyConstantin,Lax,andMajda[8]ut=H(u)uwhereHdenotestheHilberttransform.Thereisnotransportinthisproblemandtheequationisknowntohavesolutionsthatblowupinnitetime.Bydierentiation,ourproblemcanbewrittenas@u@t+~v
ru=(
Ku)u;(3)anadvection-reactionequationinwhichthesolutionuisampliedbythenonlocaloperator(
Ku).AsintheEulerexamples,thereisaconservedquantity,namelytheL1normofthesolution.Inonespacedimension,(3)takesonaparticularlysimpleformforwhichitiseasytoshowsolutionsbecomesingularinnitetimeforpointypotentialssuchasejxj.Thatis,the
KuoperatorsplitsintoconvolutionwithaDiracmassplusconvolutionwithaboundedkernel.Thus,forasmoothsolutiononecanapplyamaximumprincipleargumenttoobtainanestimateoftheform(um)tCu2mDum(whereumisthemaximumvalueofu)whichimpliesnitetimeblowupofthesolutionprovidedasuitablecontinuationtheoremholds.Thisargumentisdescribedinanonrigorousfashionin[15].BodnarandVelazquezconsidertheone-dimensionalproblemin[2]andprovenitetimeblowupbydirectcomparisonwithaBurgers-likedynamics.Inhigherspacedimensionsonedoesnotobtainsuchastraightforwardblowupresult.Thisisbecause
K,forn2,in(3)doesnothaveaDiracmass,insteaditssingularpartisoftheform1jxj.Asaconvolutionoperator,
Kisincreasinglylesssingularly,as2thedimensionofspace,n,increases.NoteforexamplethattheNewtonianpotential,whichhastheformofcnjxj2ninRn,n3,introducesagainoftwoderivatives,asaconvolutionoperator.Indeed,ifKhasboundedsecondderivative,thereisnonitetimeblowup,sothetypeofkernelconsideredhereposesaninterestingproblemfornonlocalactivescalarequations.Anotherrelatedproblemofbiologicalrelevanceisthechemotaxismodelt+r
(
rc)=
;
c=;(4)whereisamassdensityofbacteriaandcisconcentrationofachemoattractant[3].Themodelisrelatedtothemuch-studiedKeller-Segelmodel[18]anditalsoarisesasanoverdampedversionoftheChandrasekharequationforthegravitationalequilibriumofisothermalstars[6].In(4)thelefthadsidehasthesamestructureas(1)wherethekernelKisnowtheNewtonianpotential,whichissignicantlymoresingularthantheaggregationkernelsconsideredinthispaper.Finitetimesingularitiesfor(4)areknowntooccurevenwiththelineardiusionontherighthandside.Thepaper[3]showsthatt