Global well-posedness of the energy-critical stochastic nonlinear Schrödinger equation on the three-dimensional torus

TL;DR

This work addresses the global well-posedness of the defocusing energy-critical SNLS on the three-torus with additive noise, focusing on the quintic nonlinearity $|u|^4u$ and a stochastic forcing controlled by a smoothing operator $\phi$. The authors decompose the solution as $u=v+\Psi$, with $\Psi$ the stochastic convolution, and treat the resulting equation for $v$ as an energy-critical NLS with a perturbation derived from $\Psi$. They develop a deterministic local theory in the atomic spaces framework, and then combine energy estimates (via Itô calculus) with a probabilistic perturbation lemma to extend to global times, proving global well-posedness in $H^1(\mathbb{T}^3)$ for $\phi\in\mathrm{HS}(L^2;H^1)$. The result is the first global well-posedness for a periodic energy-critical SNLS, and it paves the way for similar results on higher-dimensional tori with adjusted estimates. Overall, the paper advances stochastic dispersive PDEs on compact manifolds by incorporating stochastic perturbations into the critical NLS theory through a robust atomic-spaces approach.

Abstract

We study the Cauchy problem of the defocusing energy-critical stochastic nonlinear Schrödinger equation (SNLS) on the three dimensional torus, forced by an additive noise. We adapt the atomic spaces framework in the context of the energy-critical nonlinear Schrödinger equation, and employ probabilistic perturbation arguments in the context of stochastic PDEs, establishing the global well-posedness of the defocusing energy-critical quintic SNLS in the energy space. It is the first global well-posedness result for the periodic SNLS in a critical space.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Corollary 2.6
• Proposition 2.7
• Proposition 2.8