Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions

We establish pathwise existence of solutions for porous media and fast diffusion equations with nonlinear gradient noise, in the full regime $m\in(0,\infty)$ and for any initial data in $L^2$. Moreover, if the initial data is positive, solutions are pathwise unique. In turn, the solution map of these equations is a continuous function of the driving noise and it generates an associated random dynamical system. Finally, in the regime $m\in\{1\}\cup(2,\infty)$, all the aforementioned results also hold for signed initial data.