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

Joackim Bernier (1, 2) , Benoît Grébert (1) , Tristan Robert (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 CNRS - Centre National de la Recherche Scientifique
3 IECL - Institut Élie Cartan de Lorraine

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 $s > 2/5$). Note that since $2/5 < 1$, we cannot claim conservation of energy and, more importantly, since $2/5 < 1/2$, 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.

Mots clés: Birkhoff normal forms; low regularity; NLS equation; Strichartz estimates

Domaines: Equations aux dérivées partielles [math.AP]

Joackim Bernier, Benoît Grébert, Tristan Robert. Dynamics of quintic nonlinear Schrödinger equations in $H^{2/5+}(\mathbb{T})$. 2024. ⟨hal-04090717v2⟩