  We prove a global well-posedness result for defocusing nonlinear Schrodinger
equations with time dependent potential. We then focus on time dependent
harmonic potentials. This aspect is motivated by Physics (Bose--Einstein
condensation), and appears also as a preparation for the analysis of the
propagation of wave packets in a nonlinear context. The main aspect considered
here is the growth of high Sobolev norms of the solution.
