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Time quasi-periodic gravity water waves in finite depth

Show simple item record Baldi, Pietro Berti, Massimiliano Haus, Emanuele Montalto, Riccardo 2017-09-27T15:46:10Z 2017-09-27T15:46:10Z 2017
dc.description.abstract We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. en_US
dc.language.iso en en_US
dc.relation.ispartofseries arXiv;1708.01517
dc.subject Water waves en_US
dc.subject KAM for PDEs en_US
dc.subject uasi-periodic solutions en_US
dc.subject standing waves en_US
dc.title Time quasi-periodic gravity water waves in finite depth en_US
dc.type Preprint en_US
dc.contributor.area Mathematics en_US
dc.relation.firstpage 1 en_US
dc.relation.lastpage 127 en_US

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