Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Abstract

We consider the cubic defocusing nonlinear Schrodinger equation on 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 < T < Kc.