Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Abstract

In this paper we prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space $H^1(\mathbb T^3)$.

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set $\Sigma_T$, $\mathbb P(\Sigma_T) > 0$ such that any data $\phi^{\omega}(x) \in H^{\gamma}(\mathbb T^3), \gamma<1, \omega \in \Sigma_T$, evolves up to time $T$ into a solution $u(t)$ with $u(t) - e^{it\Delta} \phi^{\omega} \in C([0,T]; H^s(\mathbb T^3))$, $s=s(\gamma)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1(\mathbb T^3)$, that is in the supercritical scaling regime.