EMS LECTURES SUMMER SCHOOL - 16, 17, 18 July 2018 ColloquiumGigliola Staffilani (MIT) The many faces of dispersive equations
CoursesGigliola
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. Abstract. 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 settingAbstract.
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 R^{n}. 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 |