Low-regularity global well-posedness for the one-dimensional, defocusing, septic nonlinear Schrödinger equation

Abstract

In this paper, we show that the one dimensional septic nonlinear Schrödinger equation is globally well-posed in $H^s(\mathbb{R})$ when $s>39/89$ with accompanying polynomial bounds in time. We utilize bilinear estimates to obtain improved bounds on the modified energy increment and prove new $L_{t,x}^{p+3}$ almost Morawetz estimates for the defocusing $p$-NLS adapted to the low-regularity setting, before specializing to the septic case $p=7$. These tools together allow us to reach Sobolev regularities $s<1/2$, showing for the first time a global existence result in the mass supercritical regime for a class of unbounded, discontinuous data.