Remark on the semilinear ill-posedness for a periodic higher order KP-I equation

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Remark on the semilinear ill-posedness for a periodic higher order KP-I equation

Tristan Robert

To cite this version:

Tristan Robert. Remark on the semilinear ill-posedness for a periodic higher order KP-I equation. C.

R. Acad. Sci. Paris, Ser. I., 2018, 356 (8), pp.891 - 898. �10.1016/j.crma.2018.06.002�. �hal-01788098�

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C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Partial differential equations

Remark on the semilinear ill-posedness for a periodic higher-order KP-I equation

Remarque sur le caractère semi-linéairement mal posé pour une équation KP-I périodique d’ordre supérieur

Tristan Robert

UniversitédeCergy-Pontoise,LaboratoireAGM,2,av.Adolphe-Chauvin,95302Cergy-Pontoisecedex,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received5May2018 Accepted7June2018 Availableonline19June2018 PresentedbyHaïmBrézis

We prove that,for someirrational torus, the flowmap of theperiodic fifth-order KP-I equationisnotlocallyuniformlycontinuousontheenergyspace,evenonthehyperplanes offixedx-meanvalue.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Onmontreque,pouruntoreirrationnelbienchoisi,leflotpourl’équationKP-Id’ordre5 périodiquen’est paslocalementuniformémentcontinu surl’espace d’énergie,même sur leshyperplansdedonnéesinitialesàmoyenneenxfixée.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Thestudyofwell-posednessfornonlineardispersiveequationshasseenconstantprogressduringthelastfewdecades.

A cornerstoneinthelow-regularityCauchytheoryfortheseequationshasbeentheworkofBourgain [1],whodevelopeda remarkablyeffectivemethodtoprovelocalwell-posedness,basedonafixed-pointargumentinafunctionspacetailoredto thelinearpartoftheequation.A consequenceofthisapproachistoprovideaflowmapthatiscontinuousontheSobolev spaces,andevenlocallyLipschitzcontinuous.Inthatcase,wesaythattheproblemissemilinearlywell-posed(see [19]).

In the early 2000’s, though, some examples arose, showing that this behaviour is not universal. First, the failure of C^{2 regularityforthe KdVequation below} H˙^−3/4 [18] (which was alreadyknown forC^{3 [3])suggestedthat not onlythe} bilinear estimate inBourgain spacesmaynot holdallthe waydownto thescaling criticalregularity(which corresponds toacontrolonthesecond PicarditerateandthusonthelevelofregularityC^{2),butthatthesemilinear}ill-posednessmay appear.Then,thecaseoftheperiodicfifth-orderKP-Iequation [16] showedthatthebilinearestimateinstandardBourgain spaces can evenfail at anyregularity. Ofcourse, thisis apriori not an objection to still recover a smooth flow map by

E-mailaddress:[email protected].

https://doi.org/10.1016/j.crma.2018.06.002

1631-073X/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

892 T. Robert / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 891–898

performing aniterationprocedureinotherfunctionalspaces:fortheKP-IIequation,thebilinearestimateisnolongertrue instandardBourgainspacesmodeledontheanisotropicSobolevspaceH^s1,s2(R²)fors1<−¹/3 [17],yetaPicarditeration canbeperformedinwell-chosenspacesfordatainthescalingatthescalingcriticalregularitys1= −¹/2 [4].Nevertheless, combiningthetwoideasabove,Molinet,Saut,andTzvetkov [13] provedthat,asfarastheKP-Iequationisconcerned,the failure ofthebilinearestimateisnot justoftechnicalnature,sinceonceagaintheflow mapcannot beofclassC^{2 inany} Sobolevspace.Thenanothercounter-examplewasgivenbytheBenjamin–Onoequation [12].Totreatboththeseequations, oneisthenforcedtogiveupontheimplementationofthecontractionprinciple,whichmotivatedthedevelopmentofnew methodsbasedoncompactnessratherthancompletenesstoattackthoseproblems.

Consequently,theCauchy problemforboththeseequationshasbeenextensivelyinvestigated.The latterwas shownto be globallywellposedinboth L²(R)[6] andL²(T)[11].Ontherealline,itsquasilinearbehaviour(inthesense oflackof regularityoftheflowmap)wasfurtherexaminedin [9],whereitwasshownthatthetransporteffectduetothederivative inthe nonlinearityleads toachangeofspeedoftheplane waves,whichinturncontradicts thelocaluniformcontinuity.

Regarding the periodic Benjamin–Ono equation, it was also proved in [11] that the flow map is not locally uniformly continuous onthewhole spaceL²(T),yetitison thesubspaceofzeromeanvalue data.Fortheformer,whichisknown to begloballywell posedintheenergyspacesE(R²)[7] andE(R×T)[15] associatedwithitsHamiltonianstructure,the sameeffecthasbeenexploitedin [10] (alongwithatransverseeffect)toshowthelackoflocaluniformcontinuityofthe flowmap.

Allthesesemilinearill-posednessresultsthusrelyonthefailureofuniformcontinuityfortheGalileantransformation G_t^±:^u0∈^Hs(T)→^u0

⎛

⎝· ±^t

T

u0(x)dx

⎞

⎠∓

T

u0(x)dx, (1.1)

whichiswelldefinedfors^0.

Indeed,forn∈N^∗,take

u_n(x):=^n−scos(nx)+^n−1andvn(x):=^n−scos(nx),

thenun,vn areuniformlyboundedinH^s(T),satisfy

||^un−^vn||H^s∼ⁿ⁻¹ −→

n→+∞0,

butfort∈ ]⁰;¹]

_Gt⁺(un)−G_t⁺(vn)

H^s=^cn^−s cos

n(x+^n−1t) −^cos(nx)

H^s|^sin(t)|>0

whichshowsthenon-uniformcontinuityofGt⁺ fort∈ ]⁰;¹] ^{(whereas,fort}=^0,thenG^±₀ ^{isLipschitzcontinuousonHs}(T) foranys^0).

Inparticular,thegeneralstrategytostudytheCauchyproblemistoconstructaflowmap⁰_t :^u0→^u(t)onthesubspace L²₀(T)⊂^L2(T)ofzeromeanvaluedata,andthentoobtainaflowmaponthewholespaceviatheformula

t:=G_t⁻◦_t⁰◦G⁺₀. (1.2)

This ispossiblesincethemeanvalue isaconstant ofthemotion.Still,fortheperiodicBenjamin–Ono equation,themap ⁰_t itself is actually Lipschitz continuous [11]. The argument above is in fact quite general: it applies for anyperiodic Hamiltonianequationundertheform

∂tu=∂x∇H(u(t)) (1.3)

givena HamiltonianfunctionalH:^u0(x,y)∈^Hs(T^1+d)→H(u0)∈R,forwhichthe x-meanvalueisindependent oft (as can easilybe seenby integrating(1.3) in x). Forexample,thisisthecasefortheperiodicKdVequation andfortheKP-II equation onthetorus,whicharenonethelessgloballysemilinearlywellposedonL²₀(T)[1] and L²₀(T²)[2],respectively.As fortheBOandKP-IequationsonR andR² respectively,thefailure ofuniformcontinuityisobtainedbyusingaproperly localizedversionoftheGalileantransformationabove.

NotethatthisargumentisverysimilartotheoneobservedbyHerr [5] concerningthederivativenonlinearSchrödinger equation,withtheGalileantransformationGt^±beingreplacedbythegaugetransformation

G^±_t :=^u0∈^Hs(T)→^e±iI(u)u0(· ±^t

T

|^u|^2dx),

with

I(u)=

T

∂_x⁻¹

⎛

⎝|^u|²−

T

|^u|^2dx

⎞

⎠dx

whichleadstothefailureoflocaluniformcontinuityoftheflowmap,yetthelocaluniformcontinuityisrecoveredonthe spheresofdatawithprescribed L²(T)norm(whichreplacetheabovehyperplanesinthiscase).

Ourpurposehereisthentocomebacktothefirstknownexampleoffailureofthebilinearestimate,namelytheperiodic fifth-orderKP-Iequation

∂tu−∂_x⁵u−∂_x⁻¹∂²_yu+^u∂xu=⁰, (t,x,y)∈R×T². (1.4) ThismodeladmitstheHamiltonianstructure(1.3) givenbytheHamiltonian

H(u0):=¹ 2

∂_x²u0²

L²+¹

2∂_x⁻¹∂yu0²

L²−¹ 6 ˆ

T²

u0(x,y)³dxdy, (1.5)

wheretheoperator∂_x⁻¹∂yiswelldefinedon

D₀(T²):=

u₀∈D(T²),u₀(0,n)=⁰∀ⁿ=⁰

astheFouriermultiplierwithsymboln/m.

The localwell-posedness forthisequation was first studied in [8] fordatain H^s(T²)∩D0(T²)for s>2. In [14], we constructedaglobalflowmapontheenergyspace

E²(T²):=

u₀∈^L2(T²)∩D₀(T²),H(u₀) <+∞

, (1.6)

endowedwiththenorm

||^u0||²_E2:= ||^u||²_L2+∂_x²u₀²

L²+∂_x⁻¹∂_yu₀²

L², (1.7)

andprovedpersistenceofregularityintheBanachscaleEσ, σ^{2 offunctionswithfinitenorm}

||^u0||²_E^σ := ||^u||²_L2+∂_x^σu₀²

L²+∂_x⁻¹∂yu₀²

L²+∂_x^σ−3∂yu₀²

L². (1.8)

Thisflowmapwasconstructedonthewholeenergyspacebytheproceduredescribedabove(1.2) andassuchisindeednot uniformlycontinuous onEσ(T²).Notethat,fromthedefinitionofD₀(T²)andtheHamiltonianstructure(1.3),thex-mean valueisactuallyaconstantofthemotion,independentofbothtand y,thusGt^±iswelldefined.Theaimofthisnoteisto showthatthequasilinearbehaviourofequation(1.4) isactuallymoreinvolved,byprovingthefailureofuniformcontinuity oftheflowmap⁰_t definedonthehyperplaneEσ

0(T²)⊂Eσ(T²), σ2,ofzerox-meanvaluedata:

Theorem1.1.There existsλ>0suchthat,forany σ^{2,thereexiststwopositiveconstantsc andC andtwosequences}(u_n)and(v_n) ofsolutionsto(1.4) inC([⁰;¹],Eσ

0(T²_λ))suchthat sup

t∈[0;1]||^un(t)||Eσ+ ||^vn(t)||Eσ^C, (1.9)

andsatisfyinginitially

n→+∞lim ||^un(0)−^vn(0)||Eσ=⁰, (1.10)

butsuchthatforeveryt∈ [⁰;¹]^, lim inf

n→+∞||^un(t)−^vn(t)||Eσ^c|^t|. ^(1.11)

Here,we willworkonan irrationaltorusT²_λ:=T×λ⁻¹Tforsome λ>0.Notethattheconstruction oftheflowmap in [14] isperformedforasquaretorus,butiscompletelyinsensitivetothechoiceoftheperiodsoftheinitialdata,thusit canbeadaptedonT²_λinastraightforwardmanner.

2. Outlineoftheproof

Let us now discuss the strategy of the proof of Theorem 1.1. As explained above, working on Eσ

0(T²_λ) rules out the transporteffectduetotheBurgers-typenonlinearity.Here,themainnonlinearphenomenoncomesfromtheresonantlow (butnonzero)–highfrequencyinteraction.

Indeed,fortheKP-II equation,theproof ofglobalsemilinearwell-posedness of [2] heavilyusesan algebraicfeature of theequation:thesymbolω(m,n)=^m3−ⁿ²/msatisfiesthenonresonantrelation

894 T. Robert / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 891–898

(m₁,n₁,m₂,n₂):= |ω(m₁+^m2,n₁+ⁿ2)−ω(m₁,n₁)−ω(m₂,n₂)|

=

m1m2

m1+m2

3(m1+m2)²+ n1

m1− ⁿ² m2

2

|^m1m₂(m₁+^m2)|, (2.1)

whichprovidesasmoothingeffectinthenonlinearinteractionwhichcompensatesforthederivativeloss.

Forequation(1.4),thesymbolreads

ω(m,n)=^m5+ⁿ²

m, (2.2)

sothattheresonantfunctionbecomes

(m₁,n₁,m₂,n₂)= ^m1m2 m₁+^m2

5(m₁+^m2)²(m²₁+^m1m₂+^m2₂)− n₁

m₁− ⁿ² m₂

₂

, (2.3)

whichnowenjoysalargesetofresonantfrequencies(m₁,n₁,m₂,n₂)whichannul.Inparticular,forn∈Nwecanchoose

α(n)∈λZsuchthat

(1,0,n,α(n))=⁰. (2.4)

This means that thenonlinear interaction betweenthe linear solutionwith frequency (1,0) and theone with(n,α(n)) produces a linearsolutionwithfrequency(n+¹,α(n)).Thisparticularinteraction was alreadyexploitedin [16] toprove thefailureofthebilinearestimatesinBourgainspaces.Togetthefailureoflocaluniformcontinuity,wewillanalyzemore closelyhowtheinitialdataconsideredin [16] evolvesunderthenonlinearflowconstructedin [14].

More precisely, insection 3 we constructa familyof functionsthat agreeattime zerowiththe initial datagivenby the two modesconsidered above, andwho solve the equation (1.4) up to a sufficiently small error. The ansatz forthis construction istocomputethefirstPicarditerates.Ofcourse,theargumentin [16] showsthat thisiterationschemedoes not converge, yet the analysis in [14] relies on this iteration on small times oforder O(n⁻²). Here, in order to have a goodapproximationuptotime O(1)weslightlydampthelowfrequencycomponent.Thefirstiterateissimplythelinear evolution, and the second iterate describesthe nonlinear interaction betweenthe linear solutions ofeach frequency, in particularthelow–highfrequencyinteractionproducesanewlinearsolutionamplifiedlinearlyintime(whichisthereason forthe divergenceofthe iterationscheme). Wewill actuallytake forthelow-frequency componentthegenuine solution emanating fromthe low-frequency mode to kill the low–lowinteraction, which remains ofthe same order asthe main nonlineareffect.Forcomplexvaluedsolutions,wecanstoptheapproximationhere,butinordertoworkwithrealvalued solutions,we modulatethehigh-frequencymodewithanoscillatingtermintimetoabsorbsomeerrortermsthatappear withthislatterconstraint,andwewillalsotakeintoaccountthemaincomponentofthethirditerate,inordertohavean appropriate errorafterpluggingthisapproximatesolutionintheequation.Insection 4,wethen comparethesefunctions withthe nonlinearflow of [14] appliedtothe samedataby usingastandard energymethod,andthenwe concludethe proofofTheorem1.1insection5.

3. Constructionofafamilyofapproximatesolutions

Letusfix σ^2.Forθ∈ [−¹;¹]^andn∈N^∗,letusdefinethefamilyoffunctionson[⁰;¹]×T²_λby u_θ,n(t,x,y):=_t

θn⁻¹cos(x) +^cos

θ 2t

n^−σcos(ϕ_n(t,x,y)) +^sin

θ 2t

n^−σsin(ϕn+¹(t,x,y))+^Rθ,n(t,x,y), (3.1) wherethephasefunctionsaregivenby

ϕ1:=x+t,ϕn:=nx+α(n)y+ω(n,α(n))t and

ϕn+¹:=(n+¹)x+α(n)y+ω(n+¹,α(n))t.

Theyarethe phasefunctionsoflinearsolutions,thatis,cos(ϕk)solvestheequation

(∂_t−L)cosϕ_k=0,

whereL= −∂_x⁵−∂⁻_x¹∂²_y isthelinearoperatorin(1.4).

Therestisgivenby R_θ,n=^n−σ

cos

θ 2t

⁻_n−¹₁cos(ϕn−ϕ1)+^sin θ

2t

⁻_n+¹₁sin(ϕn+¹+ϕ1)

, (3.2)

where

_n±1:= ±(1,0,n±¹,α(n)).

Notethatastraightforwardcomputation(seebelowforthedefinitionof α(n))gives

|n±¹| ∼n³, (3.3)

provided α(n)satisfies(2.4).Ofcourse,inthiscase,wealsohave

ϕ1+ϕn=ϕn+1. (3.4)

3.1. Onthechoiceoftheperiod

Inordertoannultheresonant function,wehavetheansatz

α(n)=ⁿ(n+¹)

5n²+⁵ⁿ+⁵∈λZ

Wearethuslookingforaλ>0 suchthatforn∈Nthen√

5n²+⁵ⁿ+⁵=λn₁,withn₁∈Z.Ifwetakeλ=√

5with∈N, settingthen X=²ⁿ+^{1 andY}=²ⁿ1,wearethusleftwithfindingtheintegersolutionsto

X²−Y²= −^{3 (3.5)}

Notethat wewantn→ +∞^{inthefollowing,so wehaveto choose}∈Nsuch thatthe abovehyperbola hasaninfinite numberofintegerpoints.Now,itiswellknownthatthesolutions(X_k,Y_k)k∈Nto(3.5) aregivenby

X_k+Y_k√

=(Y0+√

X0)(u_k+√ v_k),

where(X0,Y0)isa particularsolutionand(u_k,v_k) isasolution toPell’sequation u²−v²=^1.If issquare free,Pell’s equation hasan infinitenumberofsolutions uk+√

vk=(u0+√

v0)^k forall k∈N,where(u0,v0)isthe fundamental solution,thusitisenoughtofindasquarefreeintegersuchthat(3.5) hasatleastonesolution.Forexample,wecantake =^{7 and}(X0,Y0)=(2,1).Thischoiceofλ=√

35 providesaninfinitesetofnumbers{^Nk}⊂N^N suchthat

α(N_k):=^Nk(N_k+¹)

5N_k²+^5Nk+⁵∈λZ. (3.6)

Thus,inthefollowing,wewillworkwiththefunctionsu_θ,n (3.1) forn∈ {^Nk}^. 3.2. Estimatesontheapproximatesolutions

First,letusrecalltheprecisestatementofthedefinitionofthenonlinearflow [14,Theorem 1.1 (a),Proposition 6.2]:

Theorem3.1.Foranyu₀∈^E∞(T²_λ):=

σ²

Eσ(T²_λ),thereexistsauniqueglobalsmoothsolution

u=:^∞(u0)∈C(R,E^∞(T²_λ))

to(1.4),andmoreoverthereexistsapositive T=^T(||^u0||E²)∼

||^u0||E²

₋μ

(3.7) forsome μ >0suchthatforany σ2,wehave

^∞(u0)

L^∞_TEσ ^Cσ||^u0||Eσ. (3.8)

Next, we prove several boundson uθ,n. First, asexplained above, for the low-frequency part,we took the nonlinear solutioninsteadofthelinearoneto avoidthe contributionofthe low–lowinteraction.The nextlemma showsthat inall theothernonlinearinteractions,wecanreplacetheformerbythelatteruptoamanageableerror: