Let $n\in\mathbb N$ and let $m>1$. Let $\mathcal P_2(\mathbb R^n)$ denote the set of Borel probability measures $\mu$ on $\mathbb R^n$ with finite second moment, meaning that $\int_{\mathbb R^n}|x|^2\,d\mu(x)<\infty$. For every smooth positive probability density $r:\mathbb R^n\to(0,\infty)$ with $r^m\in L^1(\mathbb R^n,\mathcal B(\mathbb R^n),\mathcal L^n)$, define

\begin{align*} \mathcal U_m[r]:=\frac{1}{m-1}\int_{\mathbb R^n}r(x)^m\,d\mathcal L^n(x). \end{align*}

Let $\rho\in C^\infty([0,\infty)\times\mathbb R^n;(0,\infty))$, and for each $t\ge 0$ let $\rho_t:\mathbb R^n\to(0,\infty)$ be the smooth density $x\mapsto \rho(t,x)$. Assume that, for every $t\ge 0$, the measure $\mu_t:=\rho_t\mathcal L^n$ belongs to $\mathcal P_2(\mathbb R^n)$, that $\rho_t^m\in L^1(\mathbb R^n,\mathcal B(\mathbb R^n),\mathcal L^n)$, and that $\rho$, $\partial_t\rho$, and all spatial derivatives of $\rho$ and $\rho^m$ that occur below have enough decay in the spatial variable, locally uniformly in $t$, to justify the stated differentiations under the integral sign, products with rapidly decreasing perturbations, and integrations by parts with no boundary terms.

For each $t>0$, interpret the first variation of $\mathcal U_m$ at $\rho_t$ modulo additive constants as

\begin{align*} \frac{\delta\mathcal U_m}{\delta\rho}(\rho_t):\mathbb R^n\to\mathbb R,\qquad x\mapsto \frac{m}{m-1}\rho_t(x)^{m-1}. \end{align*}

Then $\rho$ satisfies the formal Wasserstein gradient-flow equation for $\mathcal U_m$, namely there is a smooth velocity field $v:(0,\infty)\times\mathbb R^n\to\mathbb R^n$ such that, for each $t>0$, $v_t:\mathbb R^n\to\mathbb R^n$ is given by

\begin{align*} v_t=-\nabla\left(\frac{\delta\mathcal U_m}{\delta\rho}(\rho_t)\right) \end{align*}

and the continuity equation

\begin{align*} \partial_t\rho_t+\nabla\cdot(\rho_t v_t)=0 \end{align*}

holds on $(0,\infty)\times\mathbb R^n$, if and only if $\rho$ satisfies the porous medium equation

\begin{align*} \partial_t\rho_t=\Delta(\rho_t^m) \end{align*}

on $(0,\infty)\times\mathbb R^n$.