We will discus a priori estimates for the uniform norm of solutions of a class of viscous quasilinear stochastic partial differential equations. The estimates are obtained independently of the viscosity parameter. Hence, we can pass to the limit and deduce similar estimates for solutions of degenerate equations. In particular, we show that for any initial condition in $H^-1$, the solution $u(t)$ of the corresponding stochastic porous medium equation is a bounded function, for any positive time $t>0$.
This is a joint work with Benjamin Gess.