Critical local well-posedness of the nonlinear Schrödinger equation on the torus

Abstract

In this paper, we study the local well-posedness of nonlinear Schrödinger equations on tori ${\mathbb{T}}^d$ at the critical regularity. We focus on cases where the nonlinearity $|u{|}^{a}u$ is nonalgebraic 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 Schrödinger 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.