Bounds on Sobolev norms for the nonlinear Schr\"odinger equation on general tori

TL;DR

This paper establishes Strichartz estimates on general tori, enabling local well-posedness results for nonlinear Schrödinger equations and polynomial bounds on Sobolev norm growth in low dimensions.

Contribution

It provides the first Strichartz estimates on general flat tori and applies them to prove well-posedness and Sobolev norm bounds for NLS.

Findings

Strichartz estimates on general tori for arbitrary dimensions

Local well-posedness of cubic NLS in Sobolev spaces

Polynomial bounds on Sobolev norm growth in dimensions 2 and 3

Abstract

We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions.