In this paper, we show that the one dimensional septic nonlinear Schr\"odinger equation is globally well-posed and scatters in $H^s (\mathbb{ R})$ when $s > 19/54$. We prove new smoothing estimates on the nonlinear Duhamel part of the solution and utilize a linear-nonlinear decomposition to take advantage of the gained regularity. We also prove new $L^{ p+3}_{ t,x}$ almost Morawetz estimates for the defocusing $p-$NLS adapted to the low-regularity setting, before specializing to the septic case $p=7$.
Categories: math.AP