02169nas a2200109   4500008004100000245012900041210006900170520172600239100002401965700002201989856004802011       2017                        en  d00aAlmost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions0 aAlmost global existence of solutions for capillaritygravity wate3 aThe goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size ϵ, is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude ϵ−N for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal forms procedure, in order to eliminate those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations are a quasi-linear system, usual normal forms approaches would face the well known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillarity-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization.The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.1 aBerti, Massimiliano1 aDelort, Jean-Marc  uhttp://preprints.sissa.it/handle/1963/3528501482nas a2200133   4500008004100000245013000041210007100171260001300242520098000255100002401235700001701259700002101276856005101297       2014                        en  d00aAn Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds0 aAbstract Nash–Moser Theorem and QuasiPeriodic Solutions for NLW   bSpringer3 aWe 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.1 aBerti, Massimiliano1 aCorsi, Livia1 aProcesi, Michela  uhttp://urania.sissa.it/xmlui/handle/1963/3465102010nas a2200385   4500008004100000022001300041245007600054210006900130300001200199490000700211520082800218653001601046653002101062653002301083653002101106653003001127653001901157653002201176653002501198653002701223653001801250653001801268653002401286653002801310653002201338653002201360653002401382653001901406653001201425653001901437100002401456700002001480700002101500856010301521       2010                        eng d  a0294144900aAn abstract Nash-Moser theorem with parameters and applications to PDEs0 aabstract NashMoser theorem with parameters and applications to P  a377-3990 v273 aWe 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.10aAbstracting10aAircraft engines10aFinite dimensional10aHamiltonian PDEs10aImplicit function theorem10aInvariant tori10aIterative schemes10aLinearized operators10aMathematical operators10aMoser theorem10aNon-Linearity10aNonlinear equations10aNonlinear wave equation10aPeriodic solution10aPoint of interest10aResonance phenomena10aSmall divisors10aSobolev10aWave equations1 aBerti, Massimiliano1 aBolle, Philippe1 aProcesi, Michela  uhttps://www.math.sissa.it/publication/abstract-nash-moser-theorem-parameters-and-applications-pdes00800nas a2200109   4500008004100000245005300041210005200094260003600146520039600182100002400578856008800602       2002                        en  d00aArnold diffusion: a functional analysis approach0 aArnold diffusion a functional analysis approach  bNatsīonal. Akad. Nauk Ukraïni3 aWe present, in the context of nearly integrable Hamiltonian systems, a functional analysis
approach to study the “splitting of the whiskers” and the “shadowing problem” developed
in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the
problem of Arnold diffusion for nearly integrable partially isochronous systems improving
known results.1 aBerti, Massimiliano  uhttps://www.math.sissa.it/publication/arnold-diffusion-functional-analysis-approach00394nas a2200109   4500008004100000245007600041210006900117260001800186100002400204700002000228856003600248       2000                        en  d00aArnold's Diffusion in nearly integrable isochronous Hamiltonian systems0 aArnolds Diffusion in nearly integrable isochronous Hamiltonian s  bSISSA Library1 aBerti, Massimiliano1 aBolle, Philippe  uhttp://hdl.handle.net/1963/1554