one formally arrives at the continuous energy $\mathcal E$ given in . The $\Gamma$-convergence of $( \mathcal E_N )_N$ to the lower-semicontinuous envelope of $\mathcal{E}$ with respect to the narrow topology has been shown in [@13-FornasierHuetter-VarProperties].

Results from the literature {#sec:well-posedness}
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As indicated above, equation results as the mean-field limit of particle dynamics. In [@FHS12] that limit process has been made rigorous under the well-known assumption $\psi_a,\psi_r\in C^{1,1}$. Under this classical smoothness condition, the general gradient flow theory in probability spaces [@AGS08] is applicable to , yielding well-posedness of the initial value problem and contraction/expansion estimates on the flow.

If instead the repulsive kernel $\psi_r$ is of lower regularity (note that we only have $\psi_r\in C^{1,q_r-1}$), then well-posedness and particle approximation are difficult to analyze in general. Many results have been proven recently for and similar equations with non-smooth kernels. The main focus has been on equations without external confinement ($\omega\equiv0$), but with more general “self-interaction” kernels $\psi_r=-K$, i.e., the equation $$\begin{aligned}
  \label{eq:K}
  \partial_t\mu = \nabla\cdot(\rho\,\nabla K\ast\rho)\end{aligned}$$ is considered. A typical choice for $K$ are combined repulsion-attraction kernels, that are repulsive at short distances but attractive at large distances. We refer to [@13-Carrillo-Choi-Hauray-MFL] for a recent survey on the 