On the regularity of the solution map of the porous media equation

Abstract: In this paper we consider the incompressible porous media equation in the Sobolev spaces $H^s(\R^2), s > 2$. We prove that for $T > 0$ the time $T$ solution map $\rho_0 \mapsto \rho(T)$ is nowhere locally uniformly continuous. On the other hand we show that the particle trajectories are analytic curves in $\R^2$.