Pathwise well-posedness of a stochastic porous medium equation with nonlinear, conservative noise

In this talk, which is based on joint work with Benjamin Gess, I will present a pathwise well-posedness theory for porous medium equations driven by conservative, nonlinear noise. The solution theory is based upon the kinetic formulation of the equation. On this level, the noise enters linearly, and the corresponding system of stochastic characteristics may be understood as rough paths. This yields a well-defined, pathwise notion of solution, for which test functions are transported along the inverse system of characteristics. I will discuss the existence and uniqueness of such solutions, and I will mention some estimates proving that, locally in time, pathwise solutions possess the basic spacial regularity of solutions to the deterministic porous medium equation.