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

### Andrea R. Nahmod

University of Massachusetts, Amherst, USA### Gigliola Staffilani

Massachusetts Institute of Technology, Cambridge, USA

## 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 $H_{1}(T_{3})$.

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set $Σ_{T}$, $P(Σ_{T})>0$ such that any data $ϕ_{ω}(x)∈H_{γ}(T_{3}),γ<1,ω∈Σ_{T}$, evolves up to time $T$ into a solution $u(t)$ with $u(t)−e_{itΔ}ϕ_{ω}∈C([0,T];H_{s}(T_{3}))$, $s=s(γ)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H_{1}(T_{3})$, that is in the supercritical scaling regime.

## Cite this article

Andrea R. Nahmod, Gigliola Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space. J. Eur. Math. Soc. 17 (2015), no. 7, pp. 1687–1759

DOI 10.4171/JEMS/543