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 bidimensional 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 fully nonlinear nature of the gravity water waves equations -the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin -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 a Nash-Moser iterative scheme, 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 which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

We consider the Cauchy problem for the Kirchhoff equation on T d with initial data of small amplitude ε in Sobolev class. We prove a lower bound ε −4 for the existence time, which improves the bound ε −2 given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.

We consider the completely resonant defocusing non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s > 1. We show the existence of solutions of this equation which achieve arbitrarily large growth of H s Sobolev norms. We also give estimates for the time required to attain this growth.

Abstract. We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash-MoserHörmander implicit function theorem as a black box. MSC2010: 35Q55, 35Q93.

We prove that there are no networks homeomorphic to the Greek "Theta" letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self-similarly shrinking. This fact completes the classification of the self-similarly shrinking networks in the plane with at most two triple junctions, see [5, 7, 18].