Third School & Workshop: 9-13 September 2024
University of Milan, via Saldini
There will be 3 minicourses of 4.5 hours each, and 5 seminars by the local units.
Registration form: if you wish to attend the school, it is mandatory to register HERE
This is a list of recommended hotels
Hotel Dieci
Hotel Kleos
Hotel Lombardia
Hotel Susa
Hotel Montecarlo
Minicourses:
J. Bernier (Nantes University): Typical dynamics of some nonlinear resonant Hamiltonian PDEs
Abstract: In this course, I will describe and compare two methods allowing to prove long time stability of most of the small and smooth solutions to some nonlinear resonant Hamiltonians PDEs. We will mainly focus on the case of the nonlinear Schrödinger equations on generic flat tori. The first method, which is closer to Birkhoff normal forms techniques, is the one we introduced with E. Faou and B. Grébert in 2020 and called "rational normal forms". The other method, which is more in the spirit of KAM techniques, is the one introduced by J. Bourgain in 2000.
P. D’Ancona (Roma La Sapienza): Asymptotic analysis of nonlinear dispersive equations
Abstract: The course will provide an understanding of the fundamental concepts and methods used in the study of Nonlinear Dispersive Equations, including: decay properties of linear operators and applications to nonlinear scattering; the dichotomy dispersion vs. blow up; the role of non dispersive solutions in the global analysis of the nonlinear problems.
A. Debussche (Rennes): Introduction to stochastic PDEs
Abstract: The aim of this course is to introduce the objects of stochastic calculus necessary to study stochastic PDEs: Wiener processes, Ito integral. Then I will explain the classical methods to solve SPDEs and give an insight of the theory of singular SPDEs.
Seminars:
A. Belloni (Unimi): Non-relativistic limit of the KAM tori for the Klein-Gordon equation
Abstract: It is heuristically well known that the non-relativistic limit of the nonlinear Klein-Gordon equation is the nonlinear Schrodinger equation. Several authors have proved rigorous results ensuring that solutions of the nonlinear Klein Gordon equation, after a Gauge transformation, converge to solutions the nonlinear Schrodinger equation uniformly on compact intervals of time. I will present a result proving existence of quasiperiodic solutions of the nonlinear Klein Gordon equation on the one dimensional torus uniformly as c → ∞. I will also prove that, after a Gauge transformation, such solutions converge uniformly with respect to t ∈ R to solutions of the nonlinear Schrodinger equation.
V. Georgiev (Pisa): Modified scattering for 2d NLS with contact interaction
Abstract: We study 2d NLS with quadratic defocussing nonlinearity and contact interaction. The modified wave operators have simple explicit representation. The main result guarantees the convergence of the global solution to a modified dynamics in $L^2$ sense.B. Langella (SISSA)
F. Cacciafesta (Roma 1): Dispersive dynamics of the Dirac equation on curved spaces
Abstract: In the last years a lot of effort has been devoted to investigate the behavior of the solutions to dispersive equations on curved spaces: in this talk, I will survey some recent results concerning dispersive estimates (Strichartz, local smoothing) for the Dirac equation in some different non-flat settings.
E. Haus (Roma 3): Reducibility and nonlinear stability for a quasi-periodically forced NLS
Abstract: We prove reducibility and nonlinear stability for a quasi-periodically forced NLS on T^2, arising from the linearization of the autonomous cubic NLS at a KAM torus. In order to prove such a result, we obtain a precise asymptotic expansion of the frequencies, which allows us to impose Melnikov conditions at arbitrary order. This work, motivated by the question of stability/instability of KAM tori for the cubic NLS on T^2, is in collaboration with B. Langella, A. Maspero, and M. Procesi.P. Baldi (Napoli):
Special seminar:
M. Procesi (Rome 3): session of open problems
Schedule:
