Polynomial growth of Sobolev norms of solutions of the fractional NLS equation on \T^d

TL;DR

This paper establishes polynomial bounds for the growth of Sobolev norms in solutions to the fractional nonlinear Schrödinger equation on multi-dimensional tori, extending previous results through new Strichartz estimates.

Contribution

It introduces novel Strichartz estimates for fractional Schrödinger equations on ^d, enabling polynomial growth bounds for Sobolev norms in higher dimensions.

Findings

Polynomial growth bounds for Sobolev norms established

Strichartz estimates for fractional Schrödinger on ^d derived

Overcomes uniformity issues in higher dimensions

Abstract

In this paper, we prove polynomial growth bounds for the Sobolev norms of solutions to the fractional nonlinear Schr\"odinger equation on the torus \T^d (d \ge 2), following and extending a result of Joseph Thirouin on \T [Thi17]. The key ingredient is the establishment of Strichartz estimates for the fractional Schr\"odinger equation on \T^d. To this end, we employ uniform estimates for oscillatory integrals to overcome the lack of uniformity that arises in higher dimensions.