• Second school and workshop: 15-19 May 2023, Rome 3

Minicourses:

• M. Berti: Long time dynamics of water waves
  Abstract: I will present some long time existence results for the initial value problem of the water waves equations, with space periodic boundary conditions. In view of the quasi-linearity of the equations also the local existence theory is subtle and no global in time existence results are known so far. The long time existence results are based on a paradifferential Birkhoff normal form reduction of the equations which allow to derive deterministic energy estimates.
  Slides lecture 1, lecture 2, lecture 3

• N. Burq: Deterministic and Probabilistic Nonlinear Scattering Theory
  Abstract: we first present the classical scattering results for nonlinear dispersive equations going back to the classical papers by Ginibre and Velo, Yajima and Tsutsumi etc. Then we show how the randomization of the initial datum allows for improvements of the classical results in several directions.

• K. Nakanishi:  Global dynamics and threshold manifolds around solitons
  Abstract: Solitons are important objects both practically and theoretically in the dynamics of nonlinear dispersive equations, but they are often unstable for small perturbations, including those with the least energy, namely the ground states. In the unstable case, however, we may construct an invariant manifold of initial data for which the solutions stay close to the solitons. Such manifolds around the ground states often appear as threshold sets to distinguish different types of solutions, such as scattering and blow-up, so they play crucial roles in classifying and predicting the global behavior of solutions from the initial data. The course will describe the main ideas and steps in the analysis, as well as some technical ingredients, for the nonlinear Klein-Gordon equations, including the case of multi-solitons.

Seminars: 

• D. Bambusi (Milano): Almost global existence for some nonlinear Hamiltonian PDEs on Toric Manifolds
  Abstract: A toric manifold is a manifold with integrable geodesic flow which, in some sense, admits global action variables. Some examples are flat tori, Zoll Manifolds, Lie Groups and their homogeneous spaces, rotation invariant surfaces. I will present a result of almost global existence for some abstract nonlinear PDEs on toric manifolds and apply it to some concrete equations, namely a nonlinear Schrodinger equation with a convolution potential and a beam equation. The abstract theorem can also be used to ensure effective stability of the plane waves in NLS. This is joint work with Roberto Feola, Beatrice Langella and Francesco Monzani

• P. D'Ancona (La Sapienza): Global almost radial solutions to supercritical dispersive equations outside the unit ball
  Abstract: In this talk I consider the two most important dispersive models i.e. defocusing NLS and NLW with power nonlinearity, on the exterior of the unit ball of Rn, with Dirichlet conditions at the boundary. The power is assumed to be sufficiently large, p > O(n), and the space dimension is 3 or larger. Even in the radial case, the corresponding problem on Rn is essentially open. I prove that the problem on the exterior of the ball can be globally and uniquely solved, provided the initial data are quasi-radial, namely, sufficiently close to radial initial data in a suitable weighted Sobolev norms of high order. Proofs are based on Strichartz estimates with larg time--dependent potentials, combined with the Penrose resp. pseudoconformal transform on exterior domains.

• J. Massetti (Roma 3): Long time behavior of Sobolev norms: Normal Form and Energy methods

Abstract: We discuss the problem of long time behavior of solutions of two given PDEs defined on a compact manifold. This talk will be twofold.

On the one hand, I shall discuss exponential type stability times in the degenerate context of the beam equation with mass in 1 space dimension.  A key ingredient is a suitable Diophantine condition (weaker than the original one proposed by Bourgain) that enables one to perform a Birkhoff Normal Form procedure.  

On the other hand, with the aim of relaxing the requirement on the size/regularity of initial data arising from the BNF, we discuss a different approach on the completely resonant NLS on tori. A key ingredient is some energy method based on paradifferential calculus and suitable tame estimates. The control over finite but long times on high Sobolev norms requires only conditions on the low ones.

This discussion is based on recent results in collaboration with Roberto Feola.

• N. Visciglia (Pisa): Probabilistic Nonlinear Scattering for NLS

Abstract: We discuss a result about probabilistic nonlinear scattering for the cubic NLS on R^4 at a level of regularity forbidden by the deterministic theory. The proof is based on a combination of Lens transform, energy estimates and stochastic estimates.

• P. Ventura (SISSA): Modulational instability in water waves

Abstract: An important problem in fluid dynamics regards stability/instability of Stokes waves, namely steady solutions of the water waves system, with respect to longitudinal space periodic long-wave perturbations. After producing a global picture of the problem at its linear level, I will describe, both in the finite and infinite-depth cases, the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. In particular it will be justified the conjecture that a pair of eigenvalues with non-zero real part depicts a closed figure “8”, parameterized by the Floquet exponent, in full agreement with numerical simulations.

List of recommended hotels:

This is a list of possible hotels recommended by the organizers. Everyone attending the conference should reserve a room.

• Duca di Cavour   https://sites.google.com/view/duca-di-cavour/

• Hotel Centro Cavour  https://hotelcentrocavour.it

• Madonna de’ Monti Suites - Daplace Collection https://daplacecollection.kross.travel/roma/madonna-de-monti-suites-daplace-collection?guests_rooms=0,0

• B&B Matisse  http://www.matissebb.com

• Hotel Mondial https://www.hotelmondialrome.com

• Hotel Impero https://www.hotelimperorome.com/?utm_source=gbp&utm_medium=organic

• Hotel Laurentia https://www.hotellaurentia.com

• Villa San Lorenzo http://www.villasanlorenzo.com/index.html

• The Roman Empire Guesthouse https://www.theromanempire.it

• Santa Sofia Hotel   https://www.santasofiahotel.com

• Hotel Santa Prisca https://www.hotelsantaprisca.it/

Schedule: