Growth of Sobolev norms for $2d$ NLS with harmonic potential

Abstract

We prove polynomial upper bounds on the growth of solutions to the $2d$ cubic nonlinear Schrödinger equation where the Laplacian is confined by the harmonic potential. Due to better bilinear effects, our bounds improve on those available for the $2d$ cubic nonlinear Schrödinger equation in the periodic setting: our growth rate for a Sobolev norm of order $s$ is $t^{2(s-1)/3+\epsilon }$, for $s=2k$ and $k\ge 1$ integer. In the appendix we provide a direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.