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Teaching

Nonlinear Schrödinger equation with time dependent potential: large time properties By Prof. Jorge Drumond Silva (Instituto Superior Tecnico, Lisbon, Portugal)

  • Class schedule:  Thursday, Mar. 13th, 2014  from  01:30 pm to 02:30 pm
  • Location: Building 1, Room 4214

Abstract

We present recent results on large time behaviour of solutions to the nonlinear Schrödinger equation with time dependent external potential and defocusing nonlinearity. The potential is assumed to grow at most quadratically in space, uniformly for all time, for which a typical example is a (possibly anisotropic) harmonic potential with bounded coefficients in time.We start by presenting a global in time well posedness result without further assumptions on the potential, with a general exponential growth control of its first order derivatives and momenta. As a consequence, these yield a double exponential growth rate of the higher Sobolev norms and momenta. We also show that if the potential is harmonic and isotropic, with coefficients decaying sufficiently fast in time, then there is scatteting, the Sobolev norms remain  ounded and momenta grow polinomially in time.This is joint work with Rémi Carles.


Biography
Jorge Drumond Silva is an Assistant Professor at the Department of Mathematics of Instituto Superior Tecnico, in Lisbon, Portugal.
He obtained his undergraduate degree in Mechanical Engineering from Instituto Superior Tecnico, in 1994, and a PhD degree in Applied and Computational Mathematics, in 2001, from Princeton University, under the supervision of Charles Fefferman. After which he also held a Visiting Fellow position there for 10 months, pursuing research in microlocal analysis, that he had started in his PhD thesis, before returning to Portugal at the end of 2001 to take the faculty position that he has held, since then, at IST. Jorge Drumond Silva's current research interests focus on real variable methods of harmonic analysis as well as their applications to the study of nonlinear dispersive partial differential equations. More recently he has also been interested on mathematical general relativity problems, in particular the initial value problem for the Einstein equations and the study of the structure and collapse of its solutions, as they relate to the cosmic censorship conjecture.