Multilinear estimates for periodic KdV equations and applications

arXiv:math/0110049v3 [math.AP] 14 Jul 2006MULTILINEARESTIMATESFORPERIODICKDVEQUATIONS,

ANDAPPLICATIONS

J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO

Abstract.WeproveanendpointmultilinearestimatefortheXs,bspaces

associatedtotheperiodicAiryequation.Asaconsequenceweobtainsharp

localwell-posednessresultsforperiodicgeneralizedKdVequations,aswellas

someglobalwell-posednessresultsbelowtheenergynorm.

Contents

1.Introduction2

1.1.MainResults3

1.2.Remarksandpossibleextensions4

1.3.Outline5

2.Acounter-example6

3.LinearEstimates7

4.Reductiontoamultiplierbound10

5.Thenon-endpointcase12

6.Someelementarynumbertheory13

7.Theproofof(4.4)intheendpointcase.152J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO

8.Theproofof(4.5)intheendpointcase.17

9.ProofofProposition121

10.Localwell-posednessforsmalldata24

11.Largeperiods25

12.Aninterpolationlemma29

13.Globalwell-posedness31

14.ProofofLemma13.133

References38

1.Introduction

ThispaperstudiestheCauchyproblemforperiodicgeneralizedKdVequations

oftheform

(1.1)󰀉

∂

tu+1

4π2isconvenientinordertomakethe

dispersionrelationτ=ξ3,butitisinessentialandwerecommendthatthereader

ignoreallpowersof2πwhichappearinthesequel.WecanassumethatFhasno

constantorlineartermsincethesecanberemovedbyaGallileantransformation.

Themainresultestablishedinthispaperisasharpmultilinearestimatewhich

allowsustoshowtheinitialvalueproblem(1.1)islocallywell-posedinHs(T)for

s≥1MULTILINEARESTIMATESANDAPPLICATIONS3

Thelow-regularitystudyoftheequation(1.1)onThasbeenbasedaround

iterationinthespacesXs,1

2+󰀕󰀓ξ󰀔sˆu󰀕

L2

ξL1

τ.

WeshallalsoneedthecompanionspacesZsdefinedby

(1.3)󰀕u󰀕

Zs:=󰀕u󰀕

s,−1󰀓τ−ξ3󰀔󰀆

󰀆

󰀆

󰀆

L2

ξL1

τ.

1.1.MainResults.Themaintechnicalresultofthispaperisthefollowingmul-

tilinearestimate.

Theorem1.Foranys≥1

2󰀃k󰀈

i=1󰀕u

i󰀕

Ys.

Thisimprovesontheresultsin[29],wherethisestimatewasprovenfors≥1.

BycombiningthisestimatewithanestimateofKenig,Ponce,andVega[24]we

shallobtain

Proposition1.Foranys≥1

2.

In[24]therestrictions≥1

2fork≥4,andalsoforthek=3caseifoneallowsthemeanofthe

u

itobenon-zero.Inthek=3casewithmeanzerothecounter-exampleisalittle

trickier,andisdiscussedinSection2.

Thesharpestimate(1.5)ofProposition1willbeusefulinstudyingthedynami-

calbehaviorofsolutionsofpolynomialgeneralizationsoftheKdVequation.Inthis

paper,weuseProposition1toobtainsomelocalandglobalwell-posednessresults

fortheCauchyproblem(1.1).In[29](seealso[5],[22]),suchequationswereshown

tobegloballywell-posedforH1data.4J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,ANDT.TAO

Theorem2.IfF(u)isapolynomial,thentheCauchyproblemfortheperiodicgeneralizedKdVequation(1.1)islocallywell-posedinHs(T)foralls≥12[24](seealso[5],[7]).WhenFiscubicwehavethemodifiedKdVequation,for

whichlocalwell-posednesswasalreadyobtainedfors≥1

2;thisexamplecan

beextendedtogeneralpolynomialsFofdegreeatleast3.

OnthereallinewithF(u)=uk+1,theequation(1.1)isknowntobelocally

well-poseddowntothescalingexponents≥1

kfork≥4[22](seealso[4];

earlierresultsarein[18]).Thiswasrecentlyextendedtok=3(exceptatthe

endpoints=1

k)in[19].Thusthereisalossof2

4lossonehasinthek=1,2

cases.Theseobservationsresolveaproblem2posedbyCarlosKenig.

TherearesomeotherconsequencesofTheorem1.Itallowsustocompletethe

proof(in[11])ofglobalwell-posednessoftheKdVandmKdVequationsdownto

s≥−1

2.Inparticularinthesymplecticspace˙H−1

4π2u

xxx+u3u

x=0

u(x,0)=u

0(x),x∈T

islocallywell-posedforlargeHs(T)datafors≥1

6.Moreover,theinitialvalueproblem(1.6)isanalytically

ill-posedinHs(T)fors<1

6forglobalwell-posedness

fallsquiteshortofthelocalconditions≥1

2SeetheproblemposedafterTheorem5.3inKenig’slecturenotesat

http://www.msri.org/publications/ln/msri/1997/ha/kenig/5/banner/03.html.MULTILINEARESTIMATESANDAPPLICATIONS5

onthefluctuationofourmodifiedHamiltonian.FromourexperiencewiththeKdV

andmKdVequationsin[11]howeverwebelieveitisreasonabletohopethatglobal

well-posednessshouldholdforalls≥1

14−2

8π2u2

x−α

2ands≥1

2.Section3recordslinearesti-

matesbetweenspacesassociatedtotheAiryequation.Section4reducesTheorem

1toamultiplierboundwhichweestablishinthenon-endpointcase,s>1

2.Section9

establishesProposition1.Theorem2isproveninSection10.Section11rescales

variousestimatestothesettingoflargeperiods4.Section12containsageneral

interpolationresultrevealingacertainflexibilityinproofsoflocalwell-posedness.

TheglobalresultofTheorem3isproveninSections13and14.

TheauthorsthankJorgeSilvaforacorrection.