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

Tristan Robert

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 Schrödinger equation and the Wick-ordered fractional cubic nonlinear Schrödinger equation, all with periodic boundary conditions.

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

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 Schrödinger equation and the Wick-ordered fractional cubic nonlinear Schrödinger equation, all with periodic boundary conditions.

Paper Structure

This paper contains 21 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Regularization by noise
  3. Assumptions
  4. On the irregularity of $W_t$
  5. Assumptions on the dispersion and the nonlinearity
  6. Main results
  7. The strongly non-resonant case
  8. Examples for Theorem \ref{['THM:LWP-NR']}
  9. Strongly non-resonant/completely resonant case
  10. Examples of completely resonant perturbations
  11. Sketch of the argument
  12. Preliminaries
  13. Notations
  14. Properties associated with the noise
  15. Function spaces
  16. ...and 6 more sections

Key Result

Theorem 1.1

If the PDE EQ:deterministic satisfies a strong non-resonant condition, then the modulated equation EQ is well-posed in $H^s(\mathcal{M})$ for any$s\in\mathbb{R}$ provided that $W_t$ is irregular enough depending on $s$.

Theorems & Definitions (43)

  • Theorem 1.1: Informal version
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 33 more