%0 Report %D 2017 %T Time quasi-periodic gravity water waves in finite depth %A P Baldi %A Massimiliano Berti %A Emanuele Haus %A Riccardo Montalto %X 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. %G en %U http://preprints.sissa.it/handle/1963/35296 %1 35602 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-09-27T15:46:10Z No. of bitstreams: 1 1708.01517.pdf: 1909445 bytes, checksum: 71fc741593666cacf41187c00b092503 (MD5)