We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size $$ \varepsilon $$in a sufficiently smooth Sobolev space leads to a solution that remains in an $$ \varepsilon $$-ball of the same Sobolev space up times of order $$ \varepsilon ^{-2}$$. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.

}, isbn = {2523-3688}, url = {https://doi.org/10.1007/s42286-020-00036-8}, author = {Massimiliano Berti and Roberto Feola and Luca Franzoi} } @article {FEOLA2019119, title = {Local well-posedness for quasi-linear NLS with large Cauchy data on the circle}, journal = {Annales de l{\textquoteright}Institut Henri Poincar{\'e} C, Analyse non lin{\'e}aire}, volume = {36}, number = {1}, year = {2019}, pages = {119 - 164}, abstract = {We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schr{\"o}dinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.

}, keywords = {Dispersive equations, Energy method, Local wellposedness, NLS, Para-differential calculus, Quasi-linear PDEs}, issn = {0294-1449}, doi = {https://doi.org/10.1016/j.anihpc.2018.04.003}, url = {http://www.sciencedirect.com/science/article/pii/S0294144918300428}, author = {Roberto Feola and Felice Iandoli} } @article {FEOLA2019932, title = {Reducibility of first order linear operators on tori via Moser{\textquoteright}s theorem}, journal = {Journal of Functional Analysis}, volume = {276}, number = {3}, year = {2019}, pages = {932 - 970}, abstract = {In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ.∂xu+a(ωt,x).∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser{\textquoteright}s straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

}, keywords = {Hyperbolic PDEs, KAM theory, Nash{\textendash}Moser, Reducibility}, issn = {0022-1236}, doi = {https://doi.org/10.1016/j.jfa.2018.10.009}, url = {http://www.sciencedirect.com/science/article/pii/S0022123618303793}, author = {Roberto Feola and Filippo Giuliani and Riccardo Montalto and Michela Procesi} } @article {1806.06604, title = {Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation}, year = {2018}, author = {Roberto Feola and Filippo Giuliani and Michela Procesi} }