  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.
