\begin{equation}\label{eqn:jmmgfdeltalater}
    \partial_t \rho^\delta_t = \nabla \cdot (\rho_t^\delta \nabla \Psi) +\tau \Delta \rho_t^\delta, \, \textrm{ with }\Psi = -K^\delta*f + K^\delta*K^\delta*\rho_t^\delta,
\end{equation} which is the Wasserstein gradient flow of \eqref{eqn:jmmfunctionallater}. Heuristically, as $\delta \rightarrow 0$, the solution $\rho^\delta_t$ converges to the solution $\rho_t$ of the viscous porous-medium equation:
