Critical local well-posedness of the nonlinear Schr\"odinger equation on the torus

TL;DR

This paper establishes local well-posedness for nonlinear Schrödinger equations on tori at critical regularity, especially for non-algebraic nonlinearities with small exponents, by developing new bilinear estimates and function spaces.

Contribution

It introduces novel bilinear estimates and a tailored function space to prove well-posedness for non-algebraic nonlinearities on tori, extending previous results.

Findings

Proved local well-posedness in the mass-supercritical regime.

Developed a new bilinear estimate for Schrödinger operators on tori.

Designed a function space based on new bilinear and Strichartz estimates.

Abstract

In this paper, we study the local well-posedness of nonlinear Schr\"odinger equations on tori $\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering $a$, we prove a bilinear estimate for the Schr\"odinger operator on tori $\mathbb{T}^{d}$, which enhances previously known multilinear estimates. We design a function space adapted to the new bilinear estimate and a package of Strichartz estimates, which is not based on conventional atomic spaces.