Multidimensional stochastic Burgers equation

Zdzisław Brzeźniak, Ben Goldys and Misha Neklyudov

Abstract

We consider multidimensional stochastic Burgers equation on the torus ${\mathbb{T}}^d$ and the whole space ${\mathbb{R}}^d$. 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 $\nu \searrow 0$ for potential solutions. In the case of ${\mathbb{R}}^d$ uniform with respect to $\nu$ a priori estimate is established if a Beale-Kato-Majda type condition is satisfied.

Keywords: stochastic Burgers equation, stochastic integral, Beale-Kato-Majda condition, Maximum principle.

AMS Subject Classification: Primary 35K45; secondary 35K55, 35R60, 60H15.