27 pages
English

Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic parabolic balance law

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Niveau: Supérieur, Doctorat, Bac+8
Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law Blake Barker? Mathew A. Johnson† Pascal Noble‡ L.Miguel Rodrigues Kevin Zumbrun¶ September 2, 2010 Keywords: Periodic traveling waves; St. Venant equations; Spectral stability; Nonlinear stability. 2000 MR Subject Classification: 35B35. Abstract In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by con- siderations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spec- tral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate pe- riod. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed “metastability” of the the limiting homoclinic orbits. ?Indiana University, Bloomington, IN 47405; : Research of B.B. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745. †Indiana University, Bloomington, IN 47405; matjohn@indiana.

  • found within intermediate

  • has no nontrivial

  • bounded stability

  • stable periodic

  • strictly parabolic

  • stability assumptions

  • traveling wave


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Nombre de lectures 39
Langue English
Whithamaveragedequationsandmodulationalstabilityofperiodictravelingwavesofahyperbolic-parabolicbalancelawBlakeBarkerMathewA.JohnsonPascalNobleL.MiguelRodrigues§KevinZumbrunSeptember2,2010Keywords:Periodictravelingwaves;St.Venantequations;Spectralstability;Nonlinearstability.2000MRSubjectClassification:35B35.AbstractInthisnote,wereportonrecentfindingsconcerningthespectralandnonlinearstabilityofperiodictravelingwavesolutionsofhyperbolic-parabolicsystemsofbalancelaws,asappliedtotheSt.Venantequationsofshallowwaterflowdownanincline.Webeginbyintroducinganaturalsetofspectralstabilityassumptions,motivatedbycon-siderationsfromtheWhithamaveragedequations,andoutlinetherecentproofyieldingnonlinearstabilityundertheseconditions.Wethenturntoananalyticalandnumericalinvestigationoftheverificationofthesespectralstabilityassumptions.Whilespec-tralinstabilityisshownanalyticallytoholdinboththeHopfandhomocliniclimits,ournumericalstudiesindicatesspectrallystableperiodicsolutionsofintermediatepe-riod.Amechanismforthismoderate-amplitudestabilizationisproposedintermsofnumericallyobserved“metastability”ofthethelimitinghomoclinicorbits.IndianaUniversity,Bloomington,IN47405;bhbarker@indiana.edu:ResearchofB.B.waspartiallysupportedunderNSFgrantsno.DMS-0300487andDMS-0801745.IndianaUniversity,Bloomington,IN47405;matjohn@indiana.edu:ResearchofM.J.waspartiallysup-portedbyanNSFPostdoctoralFellowshipunderNSFgrantDMS-0902192.Universite´LyonI,Villeurbanne,France;noble@math.univ-lyon1.fr:ResearchofP.N.waspartiallysupportedbytheFrenchANRProjectno.ANR-09-JCJC-0103-01.§Universite´deLyon,Universite´Lyon1,InstitutCamilleJordan,UMRCNRS5208,43bddu11novembre1918,F-69622VilleurbanneCedex,France;rodrigues@math.univ-lyon1.frIndianaUniversity,Bloomington,IN47405;kzumbrun@indiana.edu:ResearchofK.Z.waspartiallysupportedunderNSFgrantsno.DMS-0300487andDMS-0801745.1
1INTRODUCTION21IntroductionNonclassicalviscousconservationorbalancelawsariseinmanyareasofmathematicalmod-elingincludingtheanalysisofmultiphasefluidsorsolidmechanics.Suchequationsareknowntoexhibitawidevarietyoftravelingwavephenomenasuchashomoclinicorhet-eroclincsolutions,correspondingtothestandardpulseandfrontorshocktypesolutions,respectively,aswellassolutionswhicharespatiallyperiodic.Historically,agreatdealofefforthasbeenappliedtounderstandingthetime-evolutionarystabilityofthehomo-clinic/heteroclinicsolutionsofsuchequation,andtheirspectralandnonlinearstabilitytheoriesarewellunderstood(ingeneral).Incontrast,untilrecentlytheanalogousstabilitytheoriesoftheperiodiccounterpartshavereceivedrelativelylittleattention.Thegoalofthispaperistopresentrecentprogresstowardstheunderstandingofperiodictraveling-wavesolutionswithinthecontextofaparticularphysicallyinterestinghyperbolic–parabolicsys-temofsecondorderPDE’swhichwedescribebelow.Webegin,however,bybrieflyrecallingtheknowntheoryinthecaseofastrictlyparabolicsystemofconservationlaws.Therehasbeenagreatdealofrecentprogresstowardstheunderstandingofthestabilitypropertiesofperiodictravelingwavesofviscousstrictlyparabolicsystemsofconservationlawsoftheform(1.1)ut+r∙f(u)=Δu,xRd,uRn;see[JZ4,JZ2,OZ1,OZ2,OZ3,OZ4,Se1].Inparticular,usingdelicateanalysisoftheresolventofthelinearizedoperatorithasbeenshownthatanyperiodictravelingwavesolutionof(1.1)thatisspectrallystablewithrespecttolocalized(L2)perturbationsis(time-evolutionary)nonlinearlystableinLporHsforappropriatevaluesofpands.Infact,suchsolutionsareasymptoticallystable(inanappropriatesense)fordimensionsd2,whiletheyareonlynonlinearlyboundedstablefordimensionsd=1.However,whiletheseresultsaremathematicallysatisfying,uptonownoexampleofaspectrallystableperiodicsolutionofequationsoftheform(1.1)hasyetbeenfound.Infact,indimensiononeitwasshownin[OZ1]byrigorousEvansfunctioncomputationsthatsuchsolutionscannotexistforcertainmodelsystemsofform(1.1)admittingaHamiltonianstructure,duetotheexistenceofaspectraldichotomy.Thoughtheseisolatedresultsmayseemdiscouraging,itshouldbenotedthattheexplicitexamplesofform(1.1)consideredsofarhavepossiblyhadtoomuchorthewrongsortofstructuretoadmitstableperiodicsolution.Inparticular,itmayverywellbethecasethatbyconsideringmoreexoticpotentials1itmaybepossibletofindastableperiodicsolutionevenwithintheclassofexamplesstudiedin[OZ1].And,forhigherdimensional(n3)ormoregeneralsystems,onemayexpecttofindricherbehavior,including,possibly,stableperiodicwaves.Whetherornotthisoccursforsystemsofform(1.1)remainsaninterestingopenproblem.However,withintheslightlywiderclassofnonstrictlyparabolicsystemsofbalancelaws,ithasrecentlybeenshownthatstableperiodicwavescanindeedoccur,whichleadsustotheinvestigationspresentedinthispaper.1Inparticular,potentialsforwhichtheperiodisdecreasingwithrespecttoamplitudeinsomeregime.
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