Global Well-posedness and Scattering for Stochastic generalized KdV Equations with additive noise

TL;DR

This paper proves local and global well-posedness, as well as scattering, for stochastic generalized KdV equations with additive noise, focusing on mass-critical and supercritical cases, using novel oscillatory integral estimates.

Contribution

It introduces new oscillatory integral estimates for general dispersion relations and applies them to establish scattering for stochastic sgKdV equations.

Findings

Global well-posedness and scattering for small initial data in $L^2$.

Local well-posedness almost surely up to critical regularity.

Novel oscillatory integral estimates for general dispersion relations.

Abstract

We study the defocusing stochastic generalized Korteweg-de Vries equations (sgKdV) driven by additive noise, with a focus on mass-critical and supercritical nonlinearities. For integers $k \geq 4$, we establish local well-posedness almost surely up to scaling critical regularity. We also prove global well-posedness and scattering in $L^{2}_{x}(\mathbb{R})$ for the mass-critical equation with small initial data; also in $H^{1}_{x}(\mathbb{R})$ for the mass supercritical equation. In particular, we prove oscillatory integral estimates associated with more general dispersion relations, which are of independent interest; and we make use of a special case of these estimates as a main ingredient for the necessary bounds on the tail of the stochastic convolution for sgKdV, which is crucial to conclude scattering results.