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 ${\varphi }^{\omega }(x)\in H^{\gamma }({\mathbb{T}}^3),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u(t)$ with $u(t)-e^{it\Delta }{\varphi }^{\omega }\in C([0,T];H^s({\mathbb{T}}^3))$, $s=s(\gamma )>1$. 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.