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

In this paper we consider the incompressible porous media equation in the Sobolev spaces $H^s({\mathbb{R}}^2),s>2$. 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 ${\mathbb{R}}^2$.