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

Abstract

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 (T³). We also prove a long time existence result; more precisely, we show that for fixed T > 0 there exists a set ∑T with P(∑T ) > 0 such that any data Φ[superscript ω] (x) ∈ Ha[superscript γ] (T³), γ < 1, ω ∈ ∑ [subscript T], evolves up to time T into a solution u(t) with u(t) - e [superscript itΔ]Φ[superscript ω] ∈ C([0, T]; H [superscript s](T³ )), s = s(γ) > 1. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H¹ (T³), that is, in the supercritical scaling regime. Keywords: Supercritical nonlinear Schrödinger equation; almost sure well-posedness; random data