Growth of Sobolev norms in the cubic defocusing nonlinear Schr\"odinger equation

We consider the cubic defocusing nonlinear Schr\"odinger equation in the two dimensional torus. Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} 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 $\mathcal{K}\gg 1$ we find a solution $u$ and a time $T$ such that $\| u(T)\|_{H^s}\geq\mathcal{K} \| u(0)\|_{H^s}$. Moreover, time $T$ satisfies polynomial bound $0<T<\mathcal{K}^c$.