Local and global well-posedness of the stochastic KdV-Burgers equation

TL;DR

This paper proves local and global well-posedness of the stochastic KdV-Burgers equation with white noise initial data, advancing understanding of its invariance and solution behavior under noise smoothing.

Contribution

It establishes the first rigorous proof of well-posedness and invariance of white noise for the stochastic KdV-Burgers equation, including under noise smoothing.

Findings

Almost sure local well-posedness with noise smoothing

Global well-posedness under additional noise smoothing

Invariance of white noise for the stochastic KdV-Burgers equation

Abstract

The stochastic PDE known as the Kardar-Parisi-Zhang equation (KPZ) has been proposed as a model for a randomly growing interface. This equation can be reformulated as a stochastic Burgers equation. We study a stochastic KdV-Burgers equation as a toy model for this stochastic Burgers equation. Both of these equations formally preserve spatial white noise. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This paper establishes a result in this direction. After smoothing the additive noise (by less than one spatial derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.