  We prove the pathwise well-posedness of stochastic porous media and fast
diffusion equations driven by nonlinear, conservative noise. As a consequence,
the generation of a random dynamical system is obtained. This extends results
of the second author and Souganidis, who considered analogous spatially
homogeneous and first-order equations, and earlier works of Lions, Perthame,
and Souganidis.
