NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES

In this thesis an approach to linear PDEs on higher dimensional spatial domains is proposed. I prove two kinds of results: first I develop an algorithm which enables to obtain reducibility for linear PDEs which depend quasi-periodically on time, and I apply it to a quasilinear transport equation of the form ∂_t u= ν•∇u+ ε P (ωt)u on the d-dimensional torus T^d, where ε is a small parameter, ν and ω are Diophantine vectors, P (ωt)=V(x,ωt)•∇+W(ωt), V is a smooth function on T^(d+n) and W(ωt) is an unbounded pseudo-differential operator of order strictly less than 1. The strategy is an extension of the methods originally developed in the context of quasilinear one dimensional equations. It consists in first using quantum normal form techniques in order to conjugate the original system to a new one with a smoothing perturbation, and then exploiting the smoothing nature of the new perturbation in order to balance the effects of the small denominators, which in this problem accumulate very fast to 0. The quantum normal form procedure developed in order to obtain reducibility for the above transport equation is global in phase space. In order to overcome such a limitation, the second problem I tackle in this thesis is that of developing a local quantum normal form procedure, which could be applied to much more general systems. As the simplest relevant model containing all the difficulties of the general case, I consider the operator H=-∆+V(x) with Floquet boundary conditions on the flat torus T^d_Γ, where T^d_Γ is the manifold obtained as quotient between the d-dimensional space R^d and an arbitrary d-dimensional lattice Γ, with the purpose of adapting the quantum normal form procedure to deal with this operator. As a result, I prove for the operator H a Structure Theorem à la Nekhoroshev, and I characterize the asymptotic behavior of all its eigenvalues. The asymptotic expansion is in |λ|^{-δ}, with δ ∊ (0, 1) for most of the eigenvalues λ (stable eigenvalues), while it is a "directional expansion" for the remaining eigenvalues (unstable eigenvalues).