Reducibility of first order linear operators on tori via Moser's theorem

In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus

${?}_{t}u+\zeta ?{?}_{x}u+a(\omega t,x)?{?}_{x}u=0,x\in {\mathbb{T}}^d,\zeta \in {\mathbb{R}}^d,\omega \in {\mathbb{R}}^{\nu }.$

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 ${\mathbb{T}}^{\nu +d}$ 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.