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

Engin Başakoğlu, Faruk Temur, Oğuz Yılmaz

TL;DR

This work analyzes the defocusing stochastic generalized Korteweg–de Vries equation with additive noise for even $k\ge 4$, establishing local well-posedness at the scaling-critical regularity and global well-posedness with scattering in $L^2_x$ for the mass-critical case $k=4$ (small data) and in $H^1_x$ for the mass-supercritical case $k>4$ (large data). A key technical contribution is sharp oscillatory integral bounds for general dispersion relations and precise tail estimates for the stochastic convolution $z_*$, which enable almost surely controlled scattering via a Da Prato–Debussche decomposition and stochastic Gronwall-type arguments. The analysis combines Airy-propagator Strichartz/Kato-type estimates, BDG and martingale tools, and perturbative fixed-point arguments to bridge stochastic and deterministic dynamics, culminating in almost-sure scattering results under a decay assumption $|g(t)|=o(t^{-\\gamma})$ with $\gamma>\frac{2}{3}$. These results extend the stochastic gKdV theory to higher even nonlinearities and introduce oscillatory integral techniques that may apply to other dispersive PDEs with fractional or 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.

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

TL;DR

This work analyzes the defocusing stochastic generalized Korteweg–de Vries equation with additive noise for even , establishing local well-posedness at the scaling-critical regularity and global well-posedness with scattering in for the mass-critical case (small data) and in for the mass-supercritical case (large data). A key technical contribution is sharp oscillatory integral bounds for general dispersion relations and precise tail estimates for the stochastic convolution , which enable almost surely controlled scattering via a Da Prato–Debussche decomposition and stochastic Gronwall-type arguments. The analysis combines Airy-propagator Strichartz/Kato-type estimates, BDG and martingale tools, and perturbative fixed-point arguments to bridge stochastic and deterministic dynamics, culminating in almost-sure scattering results under a decay assumption with . These results extend the stochastic gKdV theory to higher even nonlinearities and introduce oscillatory integral techniques that may apply to other dispersive PDEs with fractional or 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 , we establish local well-posedness almost surely up to scaling critical regularity. We also prove global well-posedness and scattering in for the mass-critical equation with small initial data; also in 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.

Paper Structure

This paper contains 15 sections, 18 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. The tail of the stochastic convolution and the scattering results
  3. Notations, function spaces and useful lemmas
  4. Proof of Theorem \ref{['thm:well-posedness']}
  5. Local well-posedness
  6. A priori estimates
  7. Fixed point argument
  8. Global existence of solutions
  9. Estimates on the tail of the stochastic convolution
  10. Oscillatory integral estimates
  11. The fractional operator
  12. Proof of Proposition \ref{['mainprop']}
  13. Scattering in $L^{2}_{x}(\mathbb{R})$ and $H_x^1(\mathbb{R})$
  14. Proof of Theorem \ref{['thm:scat_in_L2']}
  15. Proof of Theorem \ref{['thm:scat_in_H1']}

Key Result

Theorem 1.1

For any initial data $u_{0}\in L^{2}_{x}(\mathbb{R})$ with $\left\lVert u_{0}\right\rVert_{L^{2}_{x}}<\delta$ for some $\delta>0$, the solution $y$ to eq:deterministic_gKdV_article with k=4 exists globally and satisfies

Theorems & Definitions (23)

  • Theorem 1.1: Theorem 2.8 in Kenig_gKdV_1993
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1: Van der Corput, Linares_book
  • Lemma 2.2: Burkholder-Davis-Gundy inequality, Hypothesis 6.4 in Da_Prato_book
  • Lemma 2.3: Stochastic Gronwall's Inequality, Xicheng_Zhang_Gronwall
  • Lemma 2.4
  • ...and 13 more