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

Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise