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

Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions