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 %D 2014 %T An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds %A Massimiliano Berti %A Livia Corsi %A Michela Procesi %X We prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor-like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups. %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34651 %1 34858 %2 Mathematics %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-10-20T12:19:54Z No. of bitstreams: 1 preprint2014.pdf: 549502 bytes, checksum: 4896c2df9fba6a09abb33941adb07837 (MD5) %R 10.1007/s00220-014-2128-4 %0 Journal Article %J Arch. Ration. Mech. Anal. %D 2014 %T KAM for Reversible Derivative Wave Equations %A Massimiliano Berti %A Luca Biasco %A Michela Procesi %XWe prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

%B Arch. Ration. Mech. Anal. %I Springer %V 212 %P 905-955 %G en %U http://urania.sissa.it/xmlui/handle/1963/34646 %N 3 %1 34850 %2 Mathematics %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-10-14T16:55:00Z No. of bitstreams: 1 preprint2014.pdf: 620515 bytes, checksum: d1d981f350e63c9906f793bcfe66e972 (MD5) %R 10.1007/s00205-014-0726-0 %0 Journal Article %J Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni %D 2013 %T Existence and stability of quasi-periodic solutions for derivative wave equations %A Massimiliano Berti %A Luca Biasco %A Michela Procesi %K Constant coefficients %K Dynamical systems %K Existence and stability %K Infinite dimensional %K KAM for PDEs %K Linearized equations %K Lyapunov exponent %K Lyapunov methods %K Quasi-periodic solution %K Small divisors %K Wave equations %X In this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*. %B Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni %V 24 %P 199-214 %G eng %R 10.4171/RLM/652 %0 Journal Article %J Annales Scientifiques de l'Ecole Normale Superieure %D 2013 %T KAM theory for the Hamiltonian derivative wave equation %A Massimiliano Berti %A Luca Biasco %A Michela Procesi %XWe prove an infinite dimensional KAM theorem which implies the existence of Can- tor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. © 2013 Société Mathématique de France.

%B Annales Scientifiques de l'Ecole Normale Superieure %V 46 %P 301-373 %G eng %0 Journal Article %J Duke Mathematical Journal %D 2011 %T Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces %A Massimiliano Berti %A Michela Procesi %X We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions. %B Duke Mathematical Journal %V 159 %8 2011 %G eng %N 3 %& 479 %R 10.1215/00127094-1433403 %0 Journal Article %J Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis %D 2010 %T An abstract Nash-Moser theorem with parameters and applications to PDEs %A Massimiliano Berti %A Philippe Bolle %A Michela Procesi %K Abstracting %K Aircraft engines %K Finite dimensional %K Hamiltonian PDEs %K Implicit function theorem %K Invariant tori %K Iterative schemes %K Linearized operators %K Mathematical operators %K Moser theorem %K Non-Linearity %K Nonlinear equations %K Nonlinear wave equation %K Periodic solution %K Point of interest %K Resonance phenomena %K Small divisors %K Sobolev %K Wave equations %X We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. © 2009 Elsevier Masson SAS. All rights reserved. %B Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis %V 27 %P 377-399 %G eng %R 10.1016/j.anihpc.2009.11.010 %0 Journal Article %J Comm. Partial Differential Equations 31 (2006) 959 - 985 %D 2006 %T Quasi-periodic solutions of completely resonant forced wave equations %A Massimiliano Berti %A Michela Procesi %X We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number. %B Comm. Partial Differential Equations 31 (2006) 959 - 985 %G en_US %U http://hdl.handle.net/1963/2234 %1 2010 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-16T07:48:45Z\\nNo. of bitstreams: 1\\n0504406v1.pdf: 330239 bytes, checksum: 5dbf59bdd590a6876ea206f70cf0ecc9 (MD5) %R 10.1080/03605300500358129 %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %D 2005 %T Quasi-periodic oscillations for wave equations under periodic forcing %A Massimiliano Berti %A Michela Procesi %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %I Accademia Nazionale dei Lincei %G en %U http://hdl.handle.net/1963/4583 %1 4350 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T10:27:05Z\\nNo. of bitstreams: 1\\nBertiProcesi05-1.pdf: 211758 bytes, checksum: b6c3ae059191cddb5c025aee61a23799 (MD5)