Periodic nonlinear Schr\"odinger equation in critical $H^s(\T^n)$ spaces

TL;DR

This paper establishes multilinear Strichartz estimates for linear Schrödinger equations on tori and applies them to prove local and global well-posedness results for the nonlinear Schrödinger equation in critical Sobolev spaces.

Contribution

It introduces new multilinear Strichartz estimates on tori and uses them to achieve well-posedness results in critical and subcritical Sobolev spaces.

Findings

Local well-posedness in critical $H^s$ spaces.

Global well-posedness for energy-critical cases.

Global results with small initial data in subcritical cases.

Abstract

In this paper we prove some multi-linear Strichartz estimates for solutions to the linear Schr\"odinger equations on torus $\T^n$. Then we apply it to get some local well-posed results for nonlinear Schr\"odinger equation in critical $H^{s}(\T^n)$ spaces. As by-products, the energy critical global well-posed results and energy subcritical global well-posed results with small initial data are also obtained.