First School & Workshop: 12-16 September 2022

Aula Magna del Dipartimento di Matematica, Largo B. Pontecorvo, Pisa



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The first school and workshop of the project will be held in Pisa, from 12 to 16 September 2022.
There will be 3 minicourses of 4.5 hours each, and 7 seminars.
In the afternoons there will be seminars announced by the local units.

Minicourses:

  • D. Bambusi: Normal form methods for Hamiltonian PDEs
    Normal form theory consists of several methods which allow to transform a system of differential equations in a new one which in some sense is simpler to analyze or for which some dynamical features are almost evident.

I will start by presenting the general technique that allows to put systems in normal form.

Then I plan to present the kind of normal form used in order prove almost global existence of smooth solutions in nonlinear wave and Schroedinger equations in one space dimension. I will try to present in a unified way results going back to 2006 and results of this year which allow to deal also with non smooth solutions.

Then I will move to the kind of normal form used in the context of KAM theory for PDEs, which is also very similar to the one used in a recent results by Bernier-Faou-Grebert allowing to control high Sobolev norms in the wave equation on $T^d$.

The last section will be devoted to recent works on Pseudodifferential normal form. Pseudodifferential normal form was introduced by Iooss Plotnikov and Toland and extended by Baldi Berti Montalto in order to deal with quite general quasilinear problems in one space dimensions. Here I will present a generalization of this method to some PDEs in higher dimension.

Suggested background material:

-- Introduction to classical Birkhoff normal form: http://users.mat.unimi.it/users/bambusi/pedagogical.pdf

-- Classical result for PDEs in 1-d: https://arxiv.org/abs/math-ph/0411011

-- Birkhoff normal form in low regularity: https://arxiv.org/abs/2102.09852

-- A different normal form: almost global existence with regularity loss: https://arxiv.org/abs/1906.05107

-- Pseudodifferential normal form: https://arxiv.org/abs/1606.04494 (a paper in 1-d)
 https://arxiv.org/abs/1702.05274 (a simple case in more dimensions)

https://arxiv.org/abs/1903.09449 (a spectral problem)

https://arxiv.org/abs/2202.04505 (a general scheme)


Lecture notes: link


  • P. Gerard: Integrability of the Benjamin-Ono equation on the circle and applications
    I will construct the nonlinear Fourier transform for the Benjamin-Ono equation with periodic boundary conditions, starting from the Lax pair structure. Various applications will be discussed, among the following list:

-low regularity wellposedness and almost periodicity of trajectories,

-characterization and stability of traveling waves, existence of non smooth,

-periodic orbits and quasi-periodic trajectories, high frequency approximation and Tao's gauge transform.

These lectures are based on joint works in collaboration with Thomas Kappeler and Peter Topalov.

Suggested background material: the first 4 paragraph of the paper "On the integrability of the Benjamin-Ono equation on the torus" by P. Gerard and T. Kappeler, link.




  • N. Tzvetkov: On the propagation of gaussian measures under the flow of Hamiltonian PDE's
    The invariance of the Gibbs measure under the flow of a Hamiltonian PDE has been first established by L. Friedlander (Comm. Math. Phys. 1985) and later developed in the works by P. Zhidkov, J. Bourgain and many others. In more recent years, the activity around the invariance of Gibbs measures has led to two other type of results. The first type of results is the global well-posedness of the corresponding PDE with rough initial data, distributed according to probability measures much more general than the Gibbs measure. The second type of results concerns the description of the propagation, by the flow of the corresponding PDE, of gaussian measures much more general than the gaussian measure related to the Gibbs measure (out of equilibrium statistical dynamics). In these lectures we plan to review these two type of developments. In particular, we will discuss the relevance in these developments of quantities "close" to conservation laws and of the resonant interactions.

Suggested background material: Tzvetkov's CIME lectures "Random data wave equation", link.

Lecture notes: link



Seminars:

  • J. Bellazzini (Pisa): Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime.
    In the talk we discuss the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Maris in 2013. However, such result is valid only in dimension 3 and higher. We prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3. Joint work with D. Ruiz (Granada).



  • P. Baldi (Naples): Resonant and non-resonant dynamics of the Kirchhoff equation: longer lifespan and chaotic behaviour.
    We consider the Kirchhoff equation on $\mathbb{T}^d$ (a Hamiltonian PDE with cubic quasi-linear nonlinearity) and its Cauchy problem with small initial data. After two steps of a quasi-linear normal form procedure, and taking sums over Fourier spheres, we obtain an effective system for the dynamics of the problem, with nonzero, but harmless, resonant cubic terms, resonant quintic terms giving nontrivial contributions to the energy estimates, and higher order remainders. We use the effective system to study non-resonance $(i)$ and resonance $(ii)$ situations.
    $(i)$ We introduce some ad hoc non-resonance conditions on the initial data of the Cauchy problem and prove a longer lifespan of the corresponding solutions. The mechanism at the base of this improvement is an averaging effect, which reduces the growth rate of the superactions of the effective equations.
    $(ii)$ We construct special solutions, Fourier supported on four spheres of $\mathbb{Z}^d$ forming two resonant triplets. The main part of the dynamics is conjugated to a classical nearly integrable Hamiltonian system of two pendulums. From the persistence of a hyperbolic periodic orbit of that system and the transversal splitting of its invariant manifolds, using Gronwall estimates, we obtain solutions for the original Kirchhoff equation that present an interesting chaotic behaviour.
    Joint works with E. Haus, F. Giuliani, M. Guardia.


  • F. Cacciafesta (Padua): Dispersive estimates for the Dirac-Coulomb equation.
    The Dirac-Coulomb equation represents a relevant model in relativistic quantum mechanics, as well as a challenging one from the point of view of dispersive PDEs: indeed, the Coulomb potential is critical with respect to the scaling of the (massless) Dirac operator, and therefore proving dispersive estimates is a complicated task. In this talk, after introducing the main features of the operator in the setting of dispersive PDEs, I will present some families of dispersive estimates (namely local smoothing and Strichartz estimates with loss of angular derivatives) that we recently obtained, trying to highlight the main differences and difficulties with respect to its non-relativistic counterpart, the Schroedinger equation.

The talk is based on works in collaboration with E. Séré (Paris CEREMADE) and J. Zhang (Beijing Institute of Technology).



  • R. Feola (Rome 3): Long time NLS approximation for a quasilinear Klein-gordon equation on large compact domains.
    We consider a class of Klein-Gordon equations with quasilinear, Hamiltonian and quadratic nonlinearities posed on a large box with periodic boundary conditions. We discuss how the cubic NLS equation can be derived to describe, approximately, the evolution of slow modulations in time and space of a spatially and temporarily oscillating wave packet. We show that the approximation is valid over a time scale which goes beyond the natural quadratic lifespan of solutions of cubic equations. We provide error estimates in Sobolev spaces. The proof is based on a combination of normal form techniques and energy methods.


  • A. Maspero (SISSA): Growth of Sobolev norms on linear Schrodinger equations as a dispersive phenomenon.
    We consider linear, time dependent Schrödinger equations of the form i∂tψ=(H+V(t))ψ, where H is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V(t) a time periodic potential. We give sufficient conditions on the potential ensuring that the Hamiltonian generates unbounded orbits. The main condition is that the resonant average of V(t) has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations. The proof combines pseudodifferential normal form with dispersive estimates in the form of local energy decay.


  • R. Montalto (Milan): Quasi-periodic solutions and inviscid limit for Euler and Navier Stokes equations via KAM methods.
    Abstract: In this talk I will discuss some recent results on Euler and Navier Stokes equations concerning the construction of quasi-periodic solutions and the problem of the invscid limit for the Navier Stokes equation. I will discuss the following two results:

1) Construction of quasi-periodic solutions for the Euler equation with a time quasi-periodic external force, bifurcating from a constant, diophantine velocity field

2) I shall discuss the inviscid limit problem from the perspective of KAM theory, namely I shall prove the existence of quasi-periodic solutions of the Navier Stokes equation converging to the one of the Euler equation constructed in 1) .

The main difficulty is that this is a singular limit problem. We overcome this difficulty by implementing a normal form methods which allow to prove sharp estimates (global in time) w.r. to the viscosity parameter.



  • O. Ivanovici (Sorbonne): Dispersive estimates for the wave flow with Dirichlet boundary condition outside a general strictly convex obstacle in 3D.
    We prove that the wave flow with Dirichlet boundary conditions outside a general strictly convex obstacle in 3D satisfies the same dispersive bounds as in R^3. Notice that in the general case the so called "Melrose and Taylor" parametrix doesn't allow to obtain dispersion estimates (but only Strichartz type bounds as had been done by Smith and Sogge in '95).


  • F. Planchon (Sorbonne): Bilinear virial estimates for (N)LS, old and new
    We will present how bilinear estimates from Planchon-Vega, or variation thereof, can be used to (re/im)prove on known results for linear and nonlinear Schrodinger equations in various settings.