Regularization by noise and stochastic Burgers equations

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.

clarifications and small corrections

References in corpus (1)

• Universality of KPZ equation