Global Well-posedness for the periodic fractional cubic NLS in 1D

Abstract

We consider the defocusing periodic fractional nonlinear Schrödinger equation $$i{\partial }_{t}u+{\left(-\hat{I}”\right)}^{\hat{I}}\pm u=-|u{|}^{2}u,$$ where $\frac{1}{2}<\hat{I}\pm <1$ and the operator $(-\hat{I}”{)}^{\hat{I}}\pm$ is the fractional Laplacian with symbol $|k{|}^{2\hat{I}\pm }$. We establish global well-posedness in $H^s(\mathbb{T})$ for $s\ge \frac{1-\hat{I}\pm }{2}$ and we conjecture this threshold to be sharp as it corresponds to the pseudo-Galilean symmetry exponent. Our proof uses the $I$-method to control the $H^s(\mathbb{T})$-norm of solutions with infinite energy initial data. A key component of our approach is a set of improved long-time bilinear Strichartz estimates on the rescaled torus, which allow us to exploit the subcritical nature of the equation.