Generalized KdV equation subject to a stochastic perturbation

TL;DR

This paper proves the global well-posedness of the subcritical generalized KdV equation under additive stochastic perturbations, extending previous results for the stochastic KdV to more general nonlinearities.

Contribution

It establishes global well-posedness for the gKdV with quartic nonlinearity under stochastic forcing, broadening the understanding of stochastic PDEs with nonlinear dispersive equations.

Findings

Solutions are globally well-posed in H^1 for stochastic gKdV with Hilbert-Schmidt covariance.

Extends previous results from stochastic KdV to generalized versions with higher nonlinearity.

Provides conditions on noise for well-posedness in Sobolev spaces.

Abstract

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on $L^2(\mathbb{R})$ and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with $H^1(\mathbb{R})$ data are globally well-posed in $H^1(\mathbb{R})$. This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation.