Local wellposedness for the critical nonlinear Schrödinger equation on ${\mathbb{T}}^3$

Abstract

For $p\ge 2$, we prove local wellposedness for the nonlinear Schrödinger equation $(i{\partial }_t+\Delta )u=\pm |u{|}^{p}u$ on ${\mathbb{T}}^3$ with initial data in $H^{s_c}({\mathbb{T}}^3)$, where ${\mathbb{T}}^3$ is a rectangular irrational $3$-torus and $s_c=\frac{3}{2}-\frac{2}{p}$ is the scaling-critical regularity. This extends work of earlier authors on the local Cauchy theory for NLS on ${\mathbb{T}}^3$ with power nonlinearities where $p$ is an even integer.