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EMS LECTURES SUMMER SCHOOL - 16, 17, 18 July 2018 Colloquium 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. Courses Gigliola Staffilani (MIT) On the long-time dynamics for the defocusing nonlinear Schrödinger equations Abstract. 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. back to the home page |