Regularization by noise and stochastic Burgers equations

We study a generalized 1d periodic SPDE of Burgers type: ∂tu=−Aθu+∂xu2+Aθ/2ξ where θ>1/2 , −A is the 1d Laplacian, ξ is a space–time white noise and the initial condition u0 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 θ>1/2 . When θ>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