Regularization by noise and stochastic Burgers equations

TL;DR

This paper investigates a stochastic Burgers equation with noise-induced regularization, establishing existence, uniqueness, and estimates for solutions depending on the parameter , and explores applications to related models.

Contribution

It introduces a weak solution framework for a generalized stochastic Burgers equation with noise regularization effects and extends the approach to other approximations and models.

Findings

Noise provides a regularizing effect enabling existence proofs for  > 1/2.

Pathwise uniqueness is achieved for  > 5/4.

Method applies to stationary 2D stochastic Navier-Stokes models.

Abstract

We study a generalized 1d periodic SPDE of Burgers type: $$ \partial_t u =- A^\theta u + \partial_x u^2 + A^{\theta/2} \xi $$ where $\theta > 1/2$, $-A$ is the 1d Laplacian, $\xi$ is a space-time white noise and the initial condition $u_0$ is taken to be (space) white noise. We introduce a notion of weak solution for this equation in the stationary setting. For these solutions we point out how the noise provide a regularizing effect allowing to prove existence and suitable estimates when $\theta>1/2$. When $\theta>5/4$ we obtain pathwise uniqueness. We discuss the use of the same method to study different approximations of the same equation and for a model of stationary 2d stochastic Navier-Stokes evolution.