EMS LECTURES SUMMER SCHOOL - 16, 17, 18 July 2018


Gigliola Staffilani (MIT)
The many faces of dispersive equations

Abstract. In recent years great progress has been made in the study of dispersive and wave equations.  Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a variety of techniques from Fourier and harmonic analysis, analytic number theory, mathematical physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of problems connected with dispersive and wave equations, such as the derivation of a certain nonlinear Schrodinger equations from a quantum many-particles system, periodic Strichartz estimates, the concept of energy transfer, the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system and non-squeezing theorems for such systems when they also enjoy a symplectic structure.


Gigliola Staffilani (MIT)
On the long-time dynamics for the defocusing nonlinear Schrödinger equations

In these lectures I will use as a model  the cubic periodic nonlinear Schrodinger equation (NLS) in 2D, or equivalently on the 2D torus.  In the first part of the lectures I will  zoom into  the rich  mathematics behind the proofs of the Strichartz estimates. These estimates are fundamental in order to prove well-posedness of dispersive initial value problems by assuming only boundedness of the mass and energy of the initial data.  The periodic case is interesting since the boundary conditions prevents dispersion from kicking in for long times. In recent years, while looking for a proof of Strichartz estimates in the periodic setting, a much deeper connection than previously anticipated has been found between analytic number theory and harmonic analysis. I will  describe in general terms this connection throughout the lectures.  Once the general Strichartz estimates are set, we will  analyze more in details how the rationality or irrationality of the torus  influences the length of the time interval where the Strichartz estimates old.

In the second part of the lectures I will analyze the  concept of energy transfer for the  2D periodic defocusing cubic NLS. Also in this case I will  outline the differences between the dynamics on rational versus irrational tori.  In particular we will analyze how the the structure of the resonance sets in these two  different settings influences what we can say about growth of Fourier coefficients for the solutions.  Open problems will be stated throughout the lectures.

Nicola Visciglia (Pisa)
Scattering theory for the nonlinear Schrödinger equation in the euclidean setting

Abstract. I will discuss classical results about the long-time behavior of solutions to the nonlinear Schrödinger equation posed in the euclidean setting. As a consequence I will deduce that the transfer of energy from low to high energy modes, as well as the phenomena of the growth of Sobolev norms, cannot occur on Rn. On the contrary, this type of phenomena are well expected in the compact setting. The last one is the main topic of the lectures by G. Staffilani.

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