Let $n\in\mathbb N$, let $T\in(0,\infty)$, let $U\in C^2((0,\infty);\mathbb R)$, let $V\in C^1(\mathbb R^n;\mathbb R)$, and let $W\in C^1(\mathbb R^n;\mathbb R)$ be even, meaning $W(z)=W(-z)$ for every $z\in\mathbb R^n$. For each positive probability density $\rho:\mathbb R^n\to(0,\infty)$ for which the following integrals are finite, define the aggregation-diffusion energy by
be a smooth curve of probability densities, and write $\rho_t:\mathbb R^n\to(0,\infty)$ for the map $x\mapsto \rho(t,x)$. Assume that $\mathcal E[\rho_t]$ is finite for every $t\in(0,T)$, that $U'\circ\rho_t$, $V$, and $W*\rho_t$ are $C^1$ in the spatial variable, where
and that differentiation under the convolution integral, the first-variation computations below, and the integrations by parts on $\mathbb R^n$ are justified by the assumed smoothness and decay.
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Then the formal $2$-Wasserstein gradient flow of $\mathcal E$ along $(\rho_t)_{t\in(0,T)}$ is the continuity equation