WELL-POSEDNESS AND ILL-POSEDNESS FOR THE CUBIC FRACTIONAL SCHRODINGER EQUATIONS

Abstract

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrodinger equations with Levy indices 1 < alpha < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in H-S for s >= 2-alpha/4. This is shown via a trilinear estimate in Bourgain's X-s,X-b space. We also show that non-periodic equations are ill-posed in H-S for 2-3 alpha/4(alpha+1) < 2-alpha/4 in the sense that the flow map is not locally uniformly continuous.