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 \(\nu\searrow0\) 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.