Abstract view
Almost Sure Global Wellposedness for the Fractional Cubic Schrödinger Equation on Torus


Published:20150505
Printed: Sep 2015
Seckin Demirbas,
Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.
Abstract
In a previous paper, we proved that $1$d periodic fractional
Schrödinger equation with cubic nonlinearity is locally wellposed
in $H^s$ for $s\gt \frac{1\alpha}{2}$ and globally wellposed for
$s\gt \frac{10\alpha1}{12}$. In this paper we define an invariant
probability measure $\mu$ on $H^s$ for $s\lt \alpha\frac{1}{2}$,
so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$
such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally
wellposed for initial data in $\Omega$. We see that this fills
the gap between the local wellposedness and the global wellposedness
range in almost sure sense for $\frac{1\alpha}{2}\lt \alpha\frac{1}{2}$,
i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense.