TY - JOUR
T1 - Existence and stability of quasi-periodic solutions for derivative wave equations
JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni
Y1 - 2013
A1 - Massimiliano Berti
A1 - Luca Biasco
A1 - Michela Procesi
KW - Constant coefficients
KW - Dynamical systems
KW - Existence and stability
KW - Infinite dimensional
KW - KAM for PDEs
KW - Linearized equations
KW - Lyapunov exponent
KW - Lyapunov methods
KW - Quasi-periodic solution
KW - Small divisors
KW - Wave equations
AB - 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*.
VL - 24
N1 - cited By (since 1996)0
ER -
TY - JOUR
T1 - An abstract Nash-Moser theorem with parameters and applications to PDEs
JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
Y1 - 2010
A1 - Massimiliano Berti
A1 - Philippe Bolle
A1 - Michela Procesi
KW - Abstracting
KW - Aircraft engines
KW - Finite dimensional
KW - Hamiltonian PDEs
KW - Implicit function theorem
KW - Invariant tori
KW - Iterative schemes
KW - Linearized operators
KW - Mathematical operators
KW - Moser theorem
KW - Non-Linearity
KW - Nonlinear equations
KW - Nonlinear wave equation
KW - Periodic solution
KW - Point of interest
KW - Resonance phenomena
KW - Small divisors
KW - Sobolev
KW - Wave equations
AB - 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.
VL - 27
N1 - cited By (since 1996)9
ER -