We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger 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.

%B Annales de l'Institut Henri Poincaré C, Analyse non linéaire %V 36 %P 119 - 164 %G eng %U http://www.sciencedirect.com/science/article/pii/S0294144918300428 %R https://doi.org/10.1016/j.anihpc.2018.04.003 %0 Journal Article %J Journal of Functional Analysis %D 2019 %T Reducibility of first order linear operators on tori via Moser's theorem %A Roberto Feola %A Filippo Giuliani %A Riccardo Montalto %A Michela Procesi %K Hyperbolic PDEs %K KAM theory %K Nash–Moser %K Reducibility %XIn 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'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.

%B Journal of Functional Analysis %V 276 %P 932 - 970 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022123618303793 %R https://doi.org/10.1016/j.jfa.2018.10.009 %0 Report %D 2018 %T Reducibility for a class of weakly dispersive linear operators arising from the Degasperis Procesi equation %A Roberto Feola %A Filippo Giuliani %A Michela Procesi %G eng