Multidimensional stochastic Burgers equation

TL;DR

This paper proves the existence and uniqueness of strong global solutions for the multidimensional stochastic Burgers equation on both the torus and the whole space, including uniform estimates and a zero-viscosity limit analysis.

Contribution

It establishes the first rigorous results on strong solutions and uniform estimates for the multidimensional stochastic Burgers equation in both bounded and unbounded domains.

Findings

Existence and uniqueness of strong solutions for positive viscosity.

Uniform a priori estimates independent of viscosity on the torus.

Analysis of the zero-viscosity limit under certain conditions.

Abstract

We consider multidimensional stochastic Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. In both cases we show that for positive viscosity $\nu>0$ there exists a unique strong global solution in $L^p$ for $p>d$. In the case of torus we also establish a uniform in $\nu$ a priori estimate and consider a limit $\nu\todown 0$ for potential solutions. In the case of $\Rd$ uniform with respect to $\nu$ a priori estimate established if a Beale-Kato-Majda type condition is satisfied.