On the growth of Sobolev norm for the cubic NLS on two dimensional product space

TL;DR

This paper establishes polynomial bounds on the growth of Sobolev norms for solutions to the cubic defocusing nonlinear Schrödinger equation on a two-dimensional product space, using improved bilinear Strichartz estimates.

Contribution

It introduces angular improved bilinear Strichartz estimates and applies them to enhance smoothing estimates, providing new bounds on energy transfer in the equation.

Findings

Polynomial bounds on Sobolev norm growth are proven.

Angular improved bilinear Strichartz estimates are developed.

Enhanced smoothing estimates are achieved.

Abstract

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for frequency localized functions, which estimates are used for enhancement of a smoothing estimates. Such upper bounds for the growth of Sobolev norms measure the transfer of energy from low to high modes as time grows on.