Probabilistic local wellposedness of 1D quintic NLS below [formula omitted

Abstract

We consider the Cauchy problem of the nonlinear Schrödinger equation i∂tu+∂x2u±u5=0 on the real line, which is L2-critical. We prove the local well-posedness of the initial value problem (IVP) for the scaling supercritical regularity regime −110<s<0 in probabilistic manner. One of the main ingredient is to establish the probabilistic bilinear Strichartz estimate.