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'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 %0 Journal Article %J Journal of Differential Equations %D 2017 %T Quasi-periodic solutions for quasi-linear generalized KdV equations %A Filippo Giuliani %K KAM for PDE's %K KdV %K Nash–Moser theory %K Quasi-linear PDE's %K Quasi-periodic solutions %XWe prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash–Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.

%B Journal of Differential Equations %V 262 %P 5052 - 5132 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039617300487 %R https://doi.org/10.1016/j.jde.2017.01.021