Global Well-posedness for the periodic fractional cubic NLS in 1D
Alexandre Megretski, Nikolaos Skouloudis
TL;DR
This work addresses global well-posedness for the defocusing periodic fractional NLS $i\partial_t u + (-\Delta)^{\alpha}u = -|u|^2 u$ on $\mathbb{T}$ with $\frac{1}{2}<\alpha<1$, showing global well-posedness in $H^s(\mathbb{T})$ for $s \ge \frac{1-\alpha}{2}$. The authors implement the $I$-method, constructing modified energies $E^1$ and $E^2$ on a rescaled torus and proving that $E^2$ is a small perturbation of $E^1$, with growth controlled by long-time Strichartz estimates. Central to the argument are linear $L^6_{t,x}$ and improved long-time bilinear Strichartz estimates on $\mathbb{T}_\lambda$, derived via decoupling and a counting lemma, which yield polynomial-in-time bounds for the $H^s$ norm and thus global existence. They also prove ill-posedness below the threshold, showing the data-to-solution map cannot be $C^3$ for $s < \frac{1-\alpha}{2}$ via a Galilean-type construction, highlighting the sharpness of the threshold. Together, these results extend global control for infinite-energy data in the subcritical regime and align with the predicted pseudo-Galilean invariance threshold.
Abstract
We consider the defocusing periodic fractional nonlinear Schrödinger equation $$ i \partial_t u +\left(-Δ\right)^αu=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< α< 1$ and the operator $(-Δ)^α$ is the fractional Laplacian with symbol $\lvert k \rvert ^{2α}$. We establish global well-posedness in $H^s(\mathbb{T})$ for $s\geq \frac{1-α}{2}$ 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.