Water waves as a spatial dynamical system Frederic Dias Gerard Iooss

Description

Water-waves as a spatial dynamical system Frederic Dias & Gerard Iooss F.D.: CMLA UMR CNRS 8536, ENS Cachan 61 Ave du President Wilson, F-94235 Cachan e-mail: G.I.: Institut Universitaire de France INLN UMR CNRS-UNSA 6618 1361 route des Lucioles, F-06560 Valbonne e-mail: Abstract The mathematical study of travelling waves, in the context of two dimensional potential flows in one or several layers of perfect fluid(s) and in the presence of free surface and interfaces, can be formulated as an ill-posed evolution problem, where the horizontal space variable plays the role of “time”. In the finite depth case, the study of near equilibria waves reduces to a low dimensional reversible ordinary differential equation. In most cases, it appears that the problem is a perturbation of an integrable system, where all types of solutions are known. We describe the method of study and review typical results. In addition, we study the infinite depth limit, which is indeed a case of physical interest. In such a case, the above reduction technique fails because the linearized operator possesses an essential spectrum filling the whole real axis, and new adapted tools are necessary. We also discuss the latest results on the existence of travelling waves in stratified fluids and on three dimensional travelling waves, in the same spirit of reversible dynamical systems.

• wave problem


• problem can


• waves


• basic undisturbed free


• dimensional travelling


• existence proofs


• surfaces tension


• results

Sujets

• Wilson
• Results

Informations

• Publié par: pefav
• Nombre de lectures: 6
• Langue: English

Extrait

Water-wavesasaspatialdynamicalsystem

Fre´de´ricDias&Ge´rardIooss
F.D.:CMLAUMRCNRS8536,ENSCachan
61AveduPre´sidentWilson,F-94235Cachan
e-mail:[e-mail]
G.I.:InstitutUniversitairedeFrance
INLNUMRCNRS-UNSA6618
1361routedesLucioles,F-06560Valbonne
e-mail:[e-mail]

Abstract
Themathematicalstudyoftravellingwaves,inthecontextoftwo
dimensionalpotentialflowsinoneorseverallayersofperfectfluid(s)and
inthepresenceoffreesurfaceandinterfaces,canbeformulatedasan
ill-posedevolutionproblem,wherethehorizontalspacevariableplaysthe
roleof“time”.Inthefinitedepthcase,thestudyofnearequilibriawaves
reducestoalowdimensional
reversibleordinarydifferentialequation.
In
mostcases,itappearsthattheproblemisa
perturbationofanintegrable
system
,wherealltypesofsolutionsareknown.Wedescribethemethod
ofstudyandreviewtypicalresults.Inaddition,westudytheinfinite
depthlimit,whichisindeedacaseofphysicalinterest.Insuchacase,the
abovereductiontechniquefailsbecausethelinearizedoperatorpossesses
an
essentialspectrum
fillingthewholerealaxis,andnewadaptedtoolsare
necessary.Wealsodiscussthelatestresultsontheexistenceoftravelling
wavesinstratifiedfluidsandonthreedimensionaltravellingwaves,inthe
samespiritofreversibledynamicalsystems.Finally,wereviewtherecent
resultsontheclassicaltwo-dimensionalstandingwaveproblem.

Contents
1Introduction3
2Two-dimensionaltravellingwaterwavesasareversibledynam-
icalsystem5
2.1Caseofonelayerwithorwithoutsurfacetensionactingonthe
freesurface..............................5
2.1.1Casewithoutsurfacetension(
b
=0)............6
2.1.2Casewithsurfacetension(
b
6
=0)..............7
2.2Caseofafluidlayerbelowanelasticiceplate...........8

1

2.3Caseoftwolayerswithoutsurfacetension.............9
2.4Caseofonelayerofinfinitedepth,withsurfacetension......11
2.5Caseoftwolayers,onebeinginfinitelydeep,withoutsurfacenor
interfacialtension...........................12
3Thelinearizedproblem13
3.1Spectrumofthelinearizedoperator.................13
3.2Physicalsituations..........................16
3.3Centermanifoldreduction(finitedepthcase)...........19
4Finitedepthcaseviareversiblenormalforms20
4.1Caseofonelayerwithoutsurfacetension.............20
4.2Caseofonelayerwithsurfacetension...............24
4.2.1Case(i):0
2+
resonance...................24
4.2.2Case(ii):0
2+
(
iq
)resonance.................26
4.2.3Case(iii):(
iq
)
2
resonance..................29
4.2.4Bifurcationofplethoraofsolitarywaves..........32
4.3Caseoftwolayerswithoutsurfacetension.............33
5Caseofinfinitedepth35
5.1Periodicwaves............................36
5.2Normalformsininfinitedimensions................38
5.3Resultsforthe(
iq
)
2
resonancewithcontinuousspectrum....39
5.4Anewreversiblebifurcation:pairofeigenvaluesdivinginthe
essentialspectrumthroughtheorigin................40
6Stratifiedfluids42
7Three-dimensionaltravellingwaterwaves43
7.1Formulationasadynamicalsystem.................44
7.2Spectrumofthelinearizedoperator.................47
7.3Three-dimensionaltravellingwavesperiodicinthedirectionof
propagation..............................48
7.4Dimensionbreakingbifurcation...................49
8Two-dimensionalstandingwaveproblem49

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1Introduction
Thepresentarticlefocusesonclassicalproblemsinthetheoryofwaterwaves.
Thetopicofwaterwavesisanoldoneandonecansaythatthetheoryofwater
waveswasinitiatedbyStokes[108]in1847.Whatdowemeanbytheclassical
problemofwaterwaves?Wemeantheproblemconsistinginsolvingthein-
compressibleEulerequationsinadomainboundedabovebyafreesurface(the
interfacebetweenairandwater)andbelowbyasolidboundary(thebottom).
Thebottomcanbeatanydepth(eveninfinite).Thedrivingforceisdueto
gravity.Theeffectsofsurfacetensionmightbeequallyimportantandcanbe
includedintheanalysis.Theremaybeseveralsuperposedlayersofimmiscible
fluids,withfreeinterfacesbetweenthem,andwithorwithoutinterfacialten-
sionthere.Whatmakesthewater-waveproblemsodifficultisnotitsgoverning
equationwhichislinear(Laplace’sequation),butitstwononlinearboundary
conditionsoneachfreesurfaceandinterface.Foralotofcoastalengineering
applications,solutionsgivenbythelinearizedwater-waveproblemareaccurate
enough(see[105]),butforanumberofpracticalapplicationsthefullynonlin-
earproblemmustbesolved.Moreover,thewater-waveproblemhasattracted
mathematiciansforalmostacenturybecauseofitsextremelyrichstructure.
Itisimportanttoemphasizeherethatthepresentreviewisrestrictedtoa
mathematicalpointofview.Themostrecentreviewsonwaterwaves,whichare
lessmathematical,arethoseofHammack&Henderson[49]onresonantinter-
actionsamongsurfacewaterwaves,Banner&Peregrine[12]onwavebreaking
indeepwater,Dias&Kharif[36]onthebifurcation,stabilityandevolutionof
waterwaves,Perlin&Schultz[98]oncapillaryeffectsonsurfacewaves,Duncan
[39]onspillingbreakers.RecentreviewsonnumericalaspectsarethoseofTsai
&Yue[117]andScardovelli&Zaleski[102]onthedirectnumericalsimulationof
free-surfaceandinterfacialflows.Foracompletebibliographyonthenumerical
computationofthree-dimensionalwaterwaves,onecanrefertothepaper[44].
Ourreviewismoreinthespiritofthesectionentitled“Existencetheo-
rems”inWehausen&Laitone’scontributiontotheEncyclopediaofPhysics
[120].Sincethewater-waveproblemisadifficultnonlinearproblemtosolve,
approximatetheorieshavebeendeveloped.Mostoftheseapproximatetheories
arebasedonperturbationexpansionsandtodayperturbationexpansionsare
stillcommonlyused.Whensuchapproximatetheoriesareused,itistacitly
assumedthatthereisanexactsolutionwhichisbeingapproximated.Therefore
existenceanduniquenessproofsareanessentialpartofexactwater-wavetheory.
Butsuchproofshavegenerallybeendifficulttoestablish,andhaveusuallybeen
obtainedforonlyratherrestricted,althoughphysicallyimportant,situations.
Noattemptwillbemadetogiveanexpositionofallmathematicalmethods
whichhavebeenusedinestablishingthevariousexistingtheorems.
Insteadthepurposeofthepresentreviewistoshowhow
dynamicalsystems
methods
canbeusedtoobtainresultsonthe
spatialbehavioroftravellingwaves
nearthebasicundisturbedfreesurfacestate
.Thewater-waveproblemcanbe
viewedasabifurcationproblemwhichoverlapsseveralimportantsubjects:i)
ellipticpartialdifferentialequationsinunboundeddomainslikestrips,ii)the

3

theoryofreversiblesystemsininfinitedimensions,iii)thenormalformtechnique
andiv)themethodsofanalysisavailableforsystemsclosetointegrableones.
Theideatouse“dynamical”argumentsforsolvingnonlinearellipticproblems
inastripwasdevelopedinthe1980s,pioneeredbyK.Kirchgassner[73](see[74]
forareviewonthewater-waveproblem).
Wesaidabovethatourpointofviewismathematical.Butevenwithinthis
pointofview,ourreviewisratherrestricted.Indeedonlydynamicalsystemsas-
pectsofthe“smallsolutions”ofthesteadywater-waveproblemareconsidered,
exceptforthesectiononstandingwaves,wheretheflowisunsteady.When
dealingwithprogressivewaves,wedonotreviewthefollowingaspects:
•
Existenceproofsbasedonmethodsoflocalanalysisofthetypeofimplicit
functionstheorem,includingconformalmappingsforreducingtheproblem
tooneofexistenceofaharmonicfunctionsatisfyingnonlinearboundary
conditions(seeforexample[81]and[94]),integralequationformulation
(seeforexample[43]),Lyapunov-Schmidtmethod(seeforexample[29]);
•
Existenceproofsbasedonvariationalformulations(seeforexample[6],[48]);
•
Mathematicalresultsonapproximatemodelsoramplitudeequationsfor
waterwaves(seeforexample[18]forareviewonmodelequations);
•
Resultsonlarge-amplitudewaves(seeforexample[7]),andresultswhich
relyonnumericalarguments;
•
Globalresults(seeforexample[77],[116],[5]);
•
Stabilityresults(seeforexample[88],[21]);
•
Existence(ornon-existence)ofsolutionstotheCauchyproblemforwater
waves(seetheworkofWu[122],[123]).Thisisastillwidelyopenproblem.
Existingresultsarevalidonafinite(small)timeintervalandimpose
restrictionseitheronthesizeoronthesmoothnessoftheinitialdata;
•
Fullyrigorousderivationsofnonlinearamplitudeequations(seeforexam-
ple[103]).
Finallywedeliberatelypresenttheprobleminthe
frameworkofreversible
vectorfields
,thusnottakingadvantageoftheHamiltonianstructureofthe
spatialwater-wave
problem.ThisHamiltonianstructurehasbeenfullyjustified
andismoreandmoreusedinthemathematicaltheoryofwaterwaves(seefor
example[47],[48]andtheirbibliography).Ourpointofviewallowsasimpler
presentationandthemathematicalargumentsweprovidecanbeusedinmore
generalsettingsthanjusttheHamiltonianones.

4

Figure1:Sketchofawavetravellingalongthefreesurfaceofafluidlayerof
thickness
h

2Two-dimensionaltravellingwaterwavesasa
reversibledynamicalsystem
2.1Caseofonelayerwithorwithoutsurfacetensionact-
ingonthefreesurface
Considerfirstthecaseofonelayer(thickness
h
)ofaninviscidfluidofdensity
ρ
undertheinfluenceofgravity
g,
withorwithoutsurfacetension
T
acting
onthefreesurface(seefig