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SUMMARY:Time quasi-periodic gravity water waves in finite depth - Pietro B
 aldi (Università degli Studi di Napoli Federico II  )
DTSTART:20171005T080000Z
DTEND:20171005T084500Z
UID:TALK84121@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We consider the water wave equations for a 2D ocean of finite 
 depth under the action of&nbsp\;gravity. We present a recent existence and
  linear stability result&nbsp\;for small amplitude standing wave solutions
  that are periodic in space and&nbsp\;quasi-periodic in time.&nbsp\;The re
 sult holds for values of a normalized depth parameter in a Cantor-like set
 &nbsp\;of asymptotically full measure.&nbsp\;<br>The main difficulties of 
 the problem are the presence of derivatives in the nonlinearity (the syste
 m is quasi-linear)\,&nbsp\;and a small divisors problem where the frequenc
 ies of the linear part grow in a&nbsp\;sublinear way at infinity (like the
  square root of integers).&nbsp\;To overcome these problems we first reduc
 e the linearized operators (which are obtained at each approximate quasi-p
 eriodic solution&nbsp\;along a Nash-Moser iteration) to constant coefficie
 nts up to smoothing operators\,&nbsp\;using pseudo-differential changes of
  variables that are quasi-periodic in time.&nbsp\;Then we apply a KAM redu
 cibility scheme which requires very weak second Melnikov non-resonance con
 ditions (losing derivatives both in time and space).&nbsp\;Such non-resona
 nce conditions are sufficiently weak to be satisfied&nbsp\;for most values
  of the normalized depth parameter\,&nbsp\;thanks to arguments from degene
 rate KAM theory.<br>Joint work with Massimiliano Berti\, Emanuele Haus and
  Riccardo Montalto.
LOCATION:Seminar Room 1\, Newton Institute
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