On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^{𝕕}, 𝕕≥3

Abstract

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) : i ∂ t u + Δ u = ± | u | 2 u : i \partial _t u + \Delta u = \pm |u|^{2}u on R d \mathbb {R}^d , d ≥ 3 d \geq 3 , with random initial data and prove almost sure well-posedness results below the scaling-critical regularity s c r i t = d − 2 2 s_\mathrm {crit} = \frac {d-2}{2} . More precisely, given a function on R d \mathbb {R}^d , we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove ‘conditional’ almost sure global well-posedness for d = 4 d = 4 in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when d ≠ 4 d \ne 4 , we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.