Regularization by noise for some strongly non-resonant modulated dispersive PDEs

TL;DR

This paper demonstrates that adding irregular noise to certain dispersive PDEs with non-resonant conditions ensures well-posedness at various regularities, extending previous results to broader classes of equations.

Contribution

It generalizes earlier work by establishing well-posedness for a wider class of dispersive PDEs with rough noise, under strong non-resonance conditions.

Findings

Irregular noise guarantees well-posedness at any regularity index.

Extension of previous results to more general dispersive PDEs.

Quantification of noise irregularity needed based on occupation measure.

Abstract

In this work, we pursue our investigations on the Cauchy problem for a class of dispersive PDEs where a rough time coefficient is present in front of the dispersion. We show that if the PDE satisfies a strong non-resonance condition (Theorem 1.6), eventually up to a completely resonant term (Theorem 1.9), then the modulated PDE is well-posed at any regularity index provided that the noise term in front of the dispersion is irregular enough. This extends earlier pioneering work of Chouk-Gubinelli and Chouk-Gubinelli-Li-Li-Oh to a more general context. We quantify the irregularity of the noise required to reach a given regularity index in terms of the regularity of its occupation measure in the sense of Catellier-Gubinelli. As examples, we discuss the cases of dispersive perturbations of the Burger's equation, including the dispersion-generalized Korteweg-de Vries and Benjamin-Ono equations, the intermediate long wave equation, the Wick-ordered modified dispersion-generalized Korteweg-de Vries equation, and the fifth-order Korteweg-de Vries equation. We also treat the completely non-resonant nonlinear Schr\"odinger equation and the Wick-ordered fractional cubic nonlinear Schr\"odinger equation, all with periodic boundary conditions.