Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Abstract

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 $\mathcal{K}\gg 1$ we find a solution $u$ and a time $T$ such that $\Vert u(T){\Vert }_{H^s}\ge \mathcal{K}\Vert u(0){\Vert }_{H^s}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<{\mathcal{K}}^c$.