Global well-posedness of one-dimensional cubic fractional nonlinear Schr\"odinger equations in negative Sobolev spaces

We study the Cauchy problem for the cubic fractional nonlinear Schr\"odinger equation (fNLS) on the real line and on the circle. In particular, we prove global well-posedness of the cubic fNLS with all orders of dispersion higher than the usual Schr\"odinger equation in negative Sobolev spaces. On the real line, our well-posedness result is sharp in the sense that a contraction argument does not work below the threshold regularity. On the circle, due to ill-posedness of the cubic fNLS in negative Sobolev spaces, we study the renormalized cubic fNLS. In order to overcome the failure of local uniform continuity of the solution map in negative Sobolev spaces, by applying a gauge transform and partially iterating the Duhamel formulation, we study the resulting equation with a cubic-quintic nonlinearity. In proving uniqueness, we present full details justifying the use of the normal form reduction for rough solutions, which seem to be missing from the existing literature. Our well-posedness result on the circle extends those in Miyaji-Tsutsumi (2018) and Oh-Wang (2018) to the endpoint regularity.