Almost Sure Global Well-posedness for Fractional Cubic Schrödinger equation on torus

Abstract

In [12], we proved that $1$-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s>\frac{1-\alpha }{2}$ and globally well-posed for $s>\frac{5\alpha -1}{6}$. In this paper we define an invariant probability measure $\mu$ on $H^s$ for $s<\alpha -\frac{1}{2}$, so that for any $\epsilon >0$ there is a set $\Omega \subset H^s$ such that $\mu ({\Omega }^c)<\epsilon$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for $\frac{1-\alpha }{2}<\alpha -\frac{1}{2}$, i.e. $\alpha >\frac{2}{3}$ in almost sure sense.