We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s>1. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with s -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is c>0 such that for any K»1 we find a solution u and a time T such that ¿u(T)¿Hs=K¿u(0)¿Hs. Moreover, the time T satisfies the polynomial bound 0
Guardia, M., Kaloshin, V. Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. "Journal of the European Mathematical Society", 2015, vol. 17, núm. 1, p. 71-149.