Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Source 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 \| u(T)\|_{H^s}\geq\mathcal K \| u(0)\|_{H^s} . Moreover, the time T satisfies the polynomial bound 0 < T < \mathcal K^c .