Existence of travelling-wave solutions and local w

arXiv:0802.2673v2[math.AP]31Mar2008EXISTENCEOFTRAVELLING-WAVESOLUTIONSANDLOCALWELL-POSEDNESSOFTHEFOWLEREQUATIONBORYSALVAREZ-SAMANIEGOANDPASCALAZERADInstitutdeMath´ematiquesetMod´elisationdeMontpellier(I3M)-UMR5149CNRSUniversit´eMontpellier2CC051,PlaceEug`eneBataillon,34095MontpellierCedex5,FranceAbstract.Westudytheexistenceoftravelling-wavesandlocalwell-posednessinasubspaceofC1b(R)foranonlinearevolutionequationrecentlyproposedbyAndrewC.Fowlertodescribethedynamicsofdunes.1.Introduction1.1.Generalsetting.Dunesarelandformationsofsandwhicharesubjecttodifferentformsandsizesbasedontheirinteractionwiththewindorwaterorsomeothermobilemedium.Inthecaseofdunesinthedeserttheirshapesdependmainlyontheamountofsandavailableandonthechangeofthedirectionofthewindwithtime(seeHerrmannandSauermann[6]).Someexamplesofdunepatternsarelongitudinal,transverse,starandBarchandunes,however,therearemorethan100categoriesofdunes.Dunesalsooccurunderrivers,forsimilarreasons,buttheirshapesarelessexoticinthiscase,becausetheflowismainlyuni-directional.Aninterestingtopicistotrytounderstandiftheshapeofaduneismaintainedwhenitmoves.WithregardtoBarchandunes,forexample,HerrmannandSauer-mann[6]havegivensomeargumentsagainstthehypothesisthatBarchandunesaresolitarywaves,mainlybecausetheyconstantlylosesandatthetwohornsandtendtodisappearifnotsuppliedwithnewsand.Recently,Dur´an,Schw¨ammleandHerrmann[2]consideredaminimalmodelfordunesconsistingofthreecou-pledequationsofmotiontostudynumericallythemechanismsofduneinteractionsforthecasewhenasmallBarchandunecollideswithabiggerone;fourdifferentcaseswereobserved,dependingonlyontherelativesizesofthetwodunes,namely,coalescence,breeding,budding,andsolitarywavebehavior.Inthispaper,weareconcernedwiththefollowingevolutionequationproposedbyFowler(see[3],[4]and[5]formoredetails)tostudynonlinearduneformation:∂u∂t(x,t)+∂∂xhu22(x,t)−∂u∂x(x,t)+Z+∞0ξ−1/3∂u∂x(x−ξ,t)dξi=0,(1.1)whereu=u(x,t)representstheduneamplitude,x∈R,andt≥0.Thesecondandfourthtermsofequation(1.1)correspondtothenonlinearandnonlocaltermsrespectively,whilethethirdtermisthedissipativeterm.Letusgiveabriefdescriptionofthemodelderivation.Formoredetails,werefertoFowler[3,4,5],whichwefollowclosely.ThemodelstemsfromtheExnerlaw,2000MathematicsSubjectClassification.47J35;35G25;76B25.Keywordsandphrases.Nonlocalevolutionequation;travelling-wave.12BORYSALVAREZ-SAMANIEGOANDPASCALAZERADwhichistheconservationofmassforthesediment:∂u∂t+∂q∂x=0,wherethebedloadtransportq=q(τ)isassumed,inthecaseofdunes,todependonlyonthestressτexertedbythefluidontheerodiblebed.Thenonlocalterminequation(1.1)arisesfromasubtlemodellingofthebasalshearstressτb.Roughlyspeaking,theturbulentbottomshearstressisgivenbyτb≈fρv2,whereρisthefluiddensity,fisadimensionlessfrictioncoefficientandvisthemeanfluidvelocity(verticallyaveraged).Byperforminganasymptoticexpansionwithrespecttotheaspectratioǫoftheevolvingbedform,ǫ=bedthicknessfluiddepth≪1,andaperturbationanalysisofabasicPoiseuilleflow(Orr-Sommerfeldequation),Fowler[3,4,5]wasabletoobtainthefollowingexpression:τb≈fρv21−u+αZ+∞0ξ−1/3∂u∂x(x−ξ,t)dξ,whereαisapositiveconstantproportionaltoRe1/3,RebeingtheReynoldsnum-ber.Duetothebedslope∂u∂x,thereisanadditionalforcegeneratedbygravityg.Therefore,thenetstresscausingmotionisactuallyτ=τb−(ρs−ρ)gDs∂u∂x,whereρsisthesedimentdensityandDsthemeandiameterofasedimentparticle.Aslongasuissmall,theshallowwaterapproximationappliestothevelocityvand,forsmallFroudenumber,the(dimensionless)meanfluidvelocitycanbeapproximatedbyv≈11−u.Indimensionlessvariables,takingallphysicalconstantsequalto1,theresultingnetstressisthengivenbyτ≈1+u+u2+Z+∞0ξ−1/3∂u∂x(x−ξ,t)dξ−∂u∂x.Noticethatthenonlinearnonlocalterm2uR+∞0ξ−1/3∂u∂x(x−ξ,t)dξhasbeendis-carded.ByaTaylorexpansion,uptoorder2,wegetq(τ)≈q(1)+q′(1)(τ−1)+12q′′(1)(τ−1)2.Now,consideringamovingspatialcoordinate,i.e.replacingxbythenewvariablex−q′(1)t,pluggingqintotheExnerequation,afterasuitablerescaling,weobtainthecanonicalequation(1.1).SomenumericalcomputationshavebeenperformedbyFowler[4,5]andAl-ibaud,AzeradandIs`ebe[1].Fowlermentionsthefactthatthenumericalsolution,computedwithapseudo-spectralmethodinalargedomain,startingfromrandominitialdata,convergestoafinalstateconsistingofonetravelling-wave.Alibaudetal.,usingafinitedifferenceschemevalidforaboundedtimeinterval,startingfromacompactlysupportednonnegativeinitialdata,showedthatthenumericalsolutionoftheFowlerequation(1.1)quicklyevolvestoasolutionwithanonzeronegativepart,showingtheerosiveeffectofthenonlocalterm.Theyalsoestab-lishtheoreticallythenonmonotonepropertyof(1.1),namelytheviolationofthemaximumprinciple(seealsoRemark3.1below).Totheauthors’knowledge,oursisthefirststudytoreportarigorousmathe-maticalproofoftheexistenceoftravelling-wavesfordunemorphodynamics.Theauthorshopethattheseresultscouldbeofinterestforgeographers,geologists,oceanographersandothers.1.2.Organizationofthepaper.InSection2westudytheexistenceoftravelling-wavesolutionstoequation(1.1).ThemainresultofthissectionisTheorem2.1whichimpliesthatforeachwavespeedd0,andηinaneighborhoodofzero,THEFOWLEREQUATION3η∈R,theree