Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2\mathrm{/}5+}(\mathbb{T})$

In this paper, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on the circle for small but very irregular initial data (in $H^s$ for $s>2\mathrm{/}5$). Note that since $2\mathrm{/}5<1$, we cannot claim conservation of energy and, more importantly, since $2\mathrm{/}5<1\mathrm{/}2$, we must dispense with the algebra property of $H^s$. This is the first dynamical result where we use the dispersive properties of NLS in a context of Birkhoff normal form.