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SUMMARY:Stable and Unstable Manifolds for Capillary Gravity Water Waves an
 d a Class of Nonlinear PDEs
DTSTART:20260522T133000Z
DTEND:20260522T150000Z
DTSTAMP:20260523T230800Z
UID:indico-event-16595@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Chongchun Zeng (Georgia Institute of Technology)\n\n
 Séminaire d'Analyse\nInvariant manifold theory is a fundamental tool in t
 he study of local dynamics near invariant structures in smooth evolution s
 ystems. It ensures the existence of nonlinearly invariant structures from 
 linear ones. The theory has been well developed for diffeomorphisms\, ODEs
 \, semilinear PDEs\, and some quasilinear parabolic PDEs. However\, it bec
 omes subtle for quasilinear or more nonlinear PDEs due to regularity issue
 s when there is no smoothing effect. In this talk\, we consider a class of
  nonlinear PDEs whose linearizations satisfy certain energy estimates. We 
 prove that the linear exponential dichotomy implies the existence of local
  stable/unstable manifolds of the equilibria. In particular the result app
 lies to a class of nonlinear Hamiltonian PDEs including the capillary grav
 ity water waves of one or two fluids\, quasilinear wave and Schrödinger e
 quations\, KdV type equations\, etc.\, for which the linear analysis is al
 so discussed. Basically\, for such systems under certain conditions\, spec
 tral instability implies the existence of stable and unstable manifolds\, 
 which in particular yields the nonlinear instability in rough Sobolev norm
 s and/or the existence of solutions decaying in high Sobolev norms. This i
 s a joint work with Jalal Shatah. \n \n========\nPour être informé des
  prochains séminaires vous pouvez vous abonner à la liste de diffusion e
 n écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: "subscr
 ibe seminaire_mathematique PRENOM NOM"(indiquez vos propres prénom et nom
 ) et laissez le corps du message vide.\n\nhttps://indico.math.cnrs.fr/even
 t/16595/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/16595/
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