Quasi-Periodic Solutions of Nonlinear Wave Equations on the d - Dimensional Torus

376 Pages·2021·2.606 MB·English

by  Massimiliano Berti, Philippe Bolle

Massimiliano Berti Q M Monographs in Mathematics
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Quasi-Periodic Solutions of Nonlinear Wave P i
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Many partial differential equations (PDEs) arising in physics, such as the nonlinear N pp
wave equation and the Schrödinger equation, can be viewed as infinite-dimensional o e
Quasi-Periodic Solutions
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Hamiltonian systems. In the last thirty years, several existence results of time quasi- B
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periodic solutions have been proved adopting a “dynamical systems” point of view. Most n
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of them deal with equations in one space dimension, whereas for multidimensional a e of Nonlinear Wave Equations
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An updated introduction to the now rich subject of KAM theory for PDEs is provided in v on the d-Dimensional Torus
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the first part of this research monograph. The focus then moves to the nonlinear wave

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equation, endowed with periodic boundary conditions. The main result of the monograph
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proves the bifurcation of small amplitude finite-dimensional invariant tori for this u
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equation, in any space dimension. This is a difficult small divisor problem due to complex t
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resonance phenomena between the normal mode frequencies of oscillations. The proof o
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requires various mathematical methods, ranging from Nash–Moser and KAM theory to s

reduction techniques in Hamiltonian dynamics and multiscale analysis for quasi-periodic o
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linear operators, which are presented in a systematic and self-contained way. Some of

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the techniques introduced in this monograph have deep connections with those used in h
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Anderson localization theory.

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ISBN 978-3-03719-211-5
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Massimiliano Berti
Philippe Bolle
Quasi-Periodic Solutions
of Nonlinear Wave Equations
on the d-Dimensional Torus
Authors:
Massimiliano Berti Philippe Bolle
SISSA Laboratoire de Mathématiques d’Avignon, EA2151
Via Bonomea 265 Avignon Université
34136 Trieste 84140 Avignon
Italy France
e-mail: [email protected] e-mail: [email protected]
2020 Mathematics Subject Classification: Primary: 37K55, 37K50, 35L05; secondary: 35Q55
Key words: Infinite-dimensional Hamiltonian systems, nonlinear wave equation, KAM for PDEs,
quasi-periodic solutions and invariant tori, small divisors, Nash–Moser theory, multiscale analysis
ISBN 978-3-03719-211-5
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Preface
Manypartialdifferentialequations(PDEs)arisinginphysicscanbeseenasinfinite-
dimensionalHamiltoniansystems
@ z DJ.r H/.z/; z 2E; (0.0.1)
t z
where the Hamiltonian function HW E ! R is defined on an infinite-dimensional
Hilbert space E of functions z WD z.x/, and J is a nondegenerate antisymmetric
operator.
Somemainexamplesarethenonlinearwave(NLW)equation
u (cid:2)(cid:2)uCV.x/uCg.x;u/D0; (0.0.2)
tt
thenonlinearSchro¨dinger(NLS)equation,thebeamequation,thehigher-dimensional
membraneequation,thewater-wavesequations,i.e.,theEulerequationsofhydrody-
namicsdescribingtheevolutionofanincompressibleirrotationalfluidundertheac-
tionofgravityandsurfacetension,aswellasitsapproximatemodelsliketheKorte-
wegdeVries(KdV),Boussinesq,Benjamin–Ono,andKadomtsev–Petviashvili(KP)
equations,amongmanyothers. Wereferto[102]forageneralintroductiontoHamil-
tonianPDEs.
Inthismonographweshalladopta“dynamicalsystems”pointofviewregarding
the NLW equation (0.0.2) equipped with periodic boundary conditions x 2 Td WD
.R=2(cid:3)Z/d as an infinite-dimensional Hamiltonian system, and we shall prove the
existenceofCantorfamiliesoffinite-dimensionalinvarianttorifilledbyquasiperiodic
solutionsof(0.0.2). ThefirstresultsinthisdirectionwereobtainedbyBourgain[42].
The search for invariant sets for the flow is an essential change of paradigm in the
studyofhyperbolicequationswithrespecttothemoretraditionalpursuitoftheinitial
value problem. This perspective has allowed the discovery of many new results,
inspiredbyfinite-dimensionalHamiltoniansystems,forHamiltonianPDEs.
When the space variable x belongs to a compact manifold, say x 2 Œ0;(cid:3)(cid:4) with
Dirichlet boundary conditions or x 2 Td (periodic boundary conditions), the dy-
namicsofaHamiltonian PDE(0.0.1), like (0.0.2), is expectedto havea“recurrent”
behaviorintime,withmanyperiodicandquasiperiodicsolutions,i.e.,solutions(de-
finedforalltimes)oftheform
u.t/DU.!t/2E where T(cid:2) 3' 7!U.'/2E (0.0.3)
is continuous, 2(cid:3)-periodic in the angular variables ' WD .' ;:::;' / and the fre-
1 (cid:2)
quencyvector! 2R(cid:2) isnonresonant,namely!(cid:3)`¤0,8`2Z(cid:2)nf0g. When(cid:5) D1,
the solutionu.t/is periodicin time, with period2(cid:3)=!. IfU.!t/ is a quasiperiodic
solutionthen,sincetheorbitf!tgt2R isdenseonT(cid:2),thetorusmanifoldU.T(cid:2)/(cid:4)E
isinvariantundertheflowof(0.0.1).
vi
Notethatthelinearwaveequation(0.0.2)withg D0,
u (cid:2)(cid:2)uCV.x/uD0; x 2Td ; (0.0.4)
tt
possessesmanyquasiperiodicsolutions. Indeedtheself-adjointoperator(cid:2)(cid:2)CV.x/
hasacompleteL2orthonormalbasisofeigenfunctions‰ .x/,j 2N,witheigenval-
j
ues(cid:6) !C1,
j
.(cid:2)(cid:2)CV.x//‰ .x/D(cid:6) ‰ .x/; j 2N: (0.0.5)
j j j
Suppose for simplicity that (cid:2)(cid:2) C V.x/ > 0, so that the eigenvalues (cid:6) D (cid:7)2,
j j
(cid:7) >0,arepositive,andallthesolutionsof(0.0.4)are
j
X
˛ cos.(cid:7) t C(cid:8) /‰ .x/; ˛ ; (cid:8) 2R; (0.0.6)
j j j j j j
j2N
which,dependingontheresonancepropertiesofthelinearfrequencies(cid:7) D(cid:7) .V/,
j j
are periodic, quasiperiodic, or almost-periodic in time (i.e., quasiperiodic with in-
finitelymanyfrequencies).
Whathappenstothesesolutionsundertheeffectofthenonlinearityg.x;u/?
There exist special nonlinear equations for which all the solutions are still peri-
odic,quasiperiodic,oralmost-periodicintime,forexampletheKdV,Benjamin–Ono,
and1-dimensionaldefocusingcubicNLSequations. Thesearecompletelyintegrable
PDEs. However, for generic nonlinearities, one expects, in analogy with the cele-
bratedPoincare´nonexistencetheoremofprimeintegralsfornearlyintegrableHamil-
toniansystems,thatthisisnotthecase.
On the other hand, for sufficiently small Hamiltonian perturbations of a nonde-
generate integrable system in Tn (cid:5) Rn, the classical Kolmogorov–Arnold–Moser
(KAM) theoremprovesthe persistenceof quasiperiodicsolutions with Diophantine
frequencyvectors! 2Rn,i.e.,vectorssatisfyingforsome(cid:9) >0and(cid:10) (cid:6)n(cid:2)1,the
nonresonancecondition
(cid:9)
j!(cid:3)`j(cid:6) ; 8`2Znnf0g: (0.0.7)
j`j(cid:3)
Such frequenciesform a Cantor set of Rn of positive measure if (cid:10) > n(cid:2)1. These
quasiperiodicsolutions(whichdenselyfillinvariantLagrangiantori)wereconstructed
by Kolmogorov [99] and Arnold [2] for analytic systems using an iterative New-
tonscheme,andbyMoser[105–107]fordifferentiableperturbationsbyintroducing
smoothing operators. This scheme then gave rise to abstract Nash–Moser implicit
function theorems like the ones proved by Zehnder [128,129] (see also [109] and
Section2.1).
Whathappensforinfinite-dimensionalsystemslikePDEs?
(cid:7) The centralquestionofKAM theory forPDEs is: do“most” of the periodic,
quasiperiodic, or almost-periodic solutions of an integrable PDE (linear or
nonlinear)persist, just slightlydeformed, underthe effectof anonlinearper-
turbation?
vii
KAMtheoryforPDEsstartedabitmorethanthirtyyearsagowiththepioneering
worksofKuksin[100]andWayne[126]on theexistenceofquasiperiodicsolutions
forsemilinearperturbationsof1-dimensionallinearwaveandSchro¨dingerequations
intheintervalŒ0;(cid:3)(cid:4). TheseresultsarebasedonanextensionoftheKAMperturbative
approach developed for the search for lower-dimensional tori in finite-dimensional
systems (see [56,107,111]) and relies on the verification of the so-called second-
orderMelnikovnonresonanceconditions.
NowadaysKAMtheoryfor1-dimensionalPDEs hasreachedasatisfactorylevel
of comprehension concerning quasiperiodic solutions, while questions concerning
almost-periodicsolutionsremain quite open. The knownresults include bifurcation
ofsmall-amplitudesolutions[17,103,113],perturbationsoflargefinitegapsolutions
[29,34,97,101,102],extensionto periodicboundaryconditions[36,48,54,71],use
ofweaknondegeneracyconditions[12], nonlinearitieswith derivatives[19,93,130]
uptoquasilinearones[6–8,65],includingwater-wavesequations[5,32],applications
toquantumharmonicoscillators[10,11,82],andafewexamplesofalmost-periodic
solutions[43,114]. WedescribethesedevelopmentsinmoredetailinChapter2.
Also,KAMtheoryformultidimensionalPDEsstillcontainsfewresultsandasat-
isfactorypictureisunderconstruction.Ifthespacedimensiond istwoormore,major
difficultiesare:
1. Theeigenvalues(cid:7)2 oftheSchro¨dingeroperator(cid:2)(cid:2)CV.x/in(0.0.5)appear
j
in huge clusters of increasing size. For example, if V.x/ D 0, and x 2 Td,
theyare
jkj2 Dk2C(cid:3)(cid:3)(cid:3)Ck2; k D.k ;:::;k /2Zd:
1 d 1 d
2. Theeigenfunctions‰ .x/maybe“notlocalized”withrespecttotheexponen-
j
tials. Roughlyspeaking,thismeansthatthereisnoone-to-onecorrespondence
h W N ! Zd suchthatthe entries.‰j;eik(cid:2)x/L2 ofthechange-of-basismatrix
(between.‰j/j2Nand.eik(cid:2)x/k2Zd)decayrapidlytozeroasjk(cid:2)h.j/j !1.
Thefirstexistenceresultoftime-periodicsolutionsfortheNLWequation
u (cid:2)(cid:2)uCmu Du3Ch:o:t:; x 2Td; d (cid:6)2;
tt
wasprovedbyBourgainin[37],byextendingtheCraig–Wayneapproach[54],origi-
nallydevelopedifx 2T. Furtherexistenceofresultsofperiodicsolutionshavebeen
provedbyBerti andBolle [22]formerelydifferentiablenonlinearities, Berti, Bolle,
andProcesi[26]forZoll manifolds, Gentile andProcesi[76]usingLindstedtseries
techniques,andDelort[55]forNLSequationsusingparadifferentialcalculus.
The first breakthrough result about the existence of quasiperiodic solutions for
spacemultidimensionalPDEswasprovedbyBourgain[39]foranalyticHamiltonian
NLSequationsoftheform
iu D(cid:2)uCM uC"@ H.u;uN/ (0.0.8)
t (cid:4) uN
with x 2 T2, where M D Op.(cid:11) / is a Fourier multiplier supported on finitely
(cid:4) k
many sites E (cid:4) Z2, i.e., (cid:11) D 0, 8k 2 Z2 nE. The (cid:11) , k 2 E, play the role of
k k
viii
external parameters used to verify suitable nonresonance conditions. Note that the
eigenfunctions of (cid:2) C M are the exponentials eik(cid:2)x and so the above-mentioned
(cid:4)
problem2isnotpresent.
Later, using subharmonic analysis tools previously developed for quasiperiodic
AndersonlocalizationtheorybyBourgain,Goldstein,andSchlag[44],Bourgainwas
able in [41,42] to extend this result in any space dimension d, and also for NLW
equationsoftheform
u (cid:2)(cid:2)uCM uC"F0.u/D0; x 2Td ; (0.0.9)
tt (cid:4)
where F.u/ is a polynomial in u. Here F0.u/ denotes the derivative of F. We
also mentiontheexistenceresults ofquasiperiodicsolutionsbyBourgainandWang
[45,46] for NLS and NLW equations under a random perturbation. The stochastic
case is a priori easier than the deterministic one because it is simpler to verify the
nonresonanceconditionswitharandomvariable.
Quasiperiodic solutions u.t;x/ D U.!t;x/ of (0.0.9) with a frequency vector
! 2R(cid:2),namelysolutionsU.';x/,' 2T(cid:2),of
.!(cid:3)@ /2U (cid:2)(cid:2)U CM U C"F0.U/D0; (0.0.10)
' (cid:4)
areconstructedbyaNewtonscheme. Themainanalysisconcernsfinite-dimensional
restrictionsofthequasiperiodicoperatorsobtainedbylinearizing(0.0.10)ateachstep
oftheNewtoniteration,
(cid:2) (cid:3)
˘ .!(cid:3)@ /2(cid:2)(cid:2)CM C"b.';x/ ; (0.0.11)
N ' (cid:4) jHN
where b.';x/ D F00.U.';x// and ˘ denotes the projection on the finite-
N
dimensionalsubspace
( )
X
H WD hD h ei.`(cid:2)'Ck(cid:2)x/;`2 Z(cid:2);k 2Zd :
N `;k
j.`;k/j(cid:3)N
The matrix that represents (0.0.11) in the exponential basis is a perturbation of the
diagonalmatrixDiag.(cid:2).!(cid:3)`/2Cjkj2C(cid:11)k/withoff-diagonalentries".bb`(cid:4)`0;k(cid:4)k0/
that decay exponentially to zero as j.` (cid:2) `0;k (cid:2) k0/j ! C1, assuming that b is
analytic (or subexponentially, if b is Gevrey). The goal is to prove that such a ma-
trix is invertible for most values of the external parameters, and that its inverse has
an exponential (or Gevrey) off-diagonal decay. It is not difficult to impose lower
boundsfortheeigenvaluesoftheself-adjointoperator(0.0.11)formostvaluesofthe
parameters. These first-order Melnikov nonresonanceconditions are essentially the
minimalassumptionsforprovingthepersistenceofquasiperiodicsolutionsof(0.0.9),
andprovideestimatesoftheinverseoftheoperator(0.0.11)inL2-norm. Inorderto
provefastoff-diagonaldecayestimatesfortheinversematrix,Bourgain’stechniqueis
ix
a“multiscale”inductiveanalysisbasedontherepeateduseofthe“resolventidentity.”
Anessentialingredientisthatthe“singular”sites
ˇ ˇ
.`;k/2Z(cid:2) (cid:5)Zd suchthat ˇ(cid:2).!(cid:3)`/2Cjkj2C(cid:11) ˇ(cid:8)1 (0.0.12)
k
are separated into clusters that are sufficiently distant from one another. However,
the information (0.0.12) about just the linear frequencies of (0.0.9) is not sufficient
(unlikefortime-periodicsolutions[37])inordertoprovethattheinversematrixhas
an exponential(orGevrey)off-diagonaldecay. Also, finernonresonanceconditions
at each scale along the induction need to be verified. We describe the multiscale
approachinSection2.4andweprovenovelmultiscaleresultsinChapter5.
These techniques have been extended in the recent work of Wang [125] for the
nonlinearKlein–Gordonequation
u (cid:2)(cid:2)uCuCupC1 D0; p 2N; x 2Td ;
tt
that, unlike (0.0.9), is parameter independent. A key step is to verify that suitable
nonresonance conditions are fulfilled for most “initial data.” We refer to [124] for
acorrespondingresultfortheNLSequation.
AnotherstreamofimportantresultsformultidimensionalPDEswereinaugurated
inthebreakthroughpaperbyEliassonandKuksin[61]fortheNLSequation(0.0.8).
Inthispapertheauthorsareabletoblockdiagonalize,andreducetoconstantcoeffi-
cients,thequasiperiodicHamiltonianoperatorobtainedateachstepoftheiteration.
This KAM reducibility approach extends the perturbative theory developed for 1-
dimensionalPDEs, by verifyingthe so-calledsecond-orderMelnikov nonresonance
conditions. It allows one to also prove directly the linear stability of the quasiperi-
odicsolutions.Otherresultsinthisdirectionhavebeenprovedforthe2-dimensional-
cubicNLSequationbyGeng,Xu,andYou[75],inanyspacedimensionandarbitrary
polynomialnonlinearitiesbyProcesiandProcesi[117,118],andforbeamequations
by Geng and You [72] and Eliasson, Gre´bert, and Kuksin [58]. Unfortunately, the
second-orderMelnikovconditionsareviolatedforNLWequationsforwhichananal-
ogousreducibilityresultdoesnothold. WedescribetheKAMreducibilityapproach
withPDEapplicationsinSections2.2and2.3.
Wenowpresentmoreindetailthegoalofthisresearchmonograph.Themainre-
sultistheexistenceofsmall-amplitudetime-quasiperiodicsolutionsforautonomous
NLWequationsoftheform
u (cid:2)(cid:2)uCV.x/uCg.x;u/D0;
tt
x 2 Td; g.x;u/Da.x/u3CO.u4/; (0.0.13)
inanyspacedimensiond (cid:6) 1,whereV.x/isasmoothmultiplicativepotentialsuch
that(cid:2)(cid:2)CV.x/>0,andthenonlinearityisC1. GivenafinitesetS(cid:4) N(tangential
sites),weconstructquasiperiodicsolutionsu.!t;x/withfrequencyvector.!j/j2S,
oftheform
X
u.!t;x/D ˛ cos.! t/‰ .x/Cr.!t;x/; ! D(cid:7) CO.j˛j/; (0.0.14)
j j j j j
j2S