Growth of Sobolev Norms for 2d NLS with harmonic potential

TL;DR

This paper establishes polynomial bounds on the growth of Sobolev norms for solutions to the 2d cubic nonlinear Schrödinger equation with a harmonic potential, improving previous results in the periodic case.

Contribution

It provides new polynomial growth bounds for Sobolev norms in 2d NLS with harmonic potential, utilizing bilinear effects to enhance existing estimates.

Findings

Polynomial growth bounds for Sobolev norms are established.

Growth rate for Sobolev norm of order 2k is t^{2(s-1)/3+ε}.

Bilinear estimates for the harmonic oscillator are proved via integration by parts.

Abstract

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