Since this seminal paper the technique of gradient flows in the Wasserstein space has been widely adopted as a method in approximating solutions to a variety of PDEs (from Fokker-Planck to the porus-medium).
In short, the solution to a PDE is approximated by the interpolation of the following "JKO scheme": given $\rho^0$ we find \begin{equation} \rho^{n}_{\tau}=\text{argmin}_{\rho\in \mathcal{P}^r_2(\mathbb{R}^d)} \Big\{ \frac{1}{2\tau}W^2_2(\rho^{n-1}_{\tau},\rho)+F(\rho) \Big\} \end{equation} as $\tau \to 0$. Here, $W_2$ is the Wasserstein metric, $\mathcal{P}^r_2(\mathbb{R}^d)$ is space of densities on $\mathbb{R}^d$ with finite second moments, and $F$ is some energy function.
It can be desirable to regularise the entropy, i.e. replace $W_2$ by $W_{2,\epsilon}$, where $$ W_{2,\epsilon}(\mu,\nu)=\inf_{\pi}\Big( \int_{\mathbb{R}^{2d}}|x-y|^2d\pi +\epsilon\,\int \pi \log \pi \Big)^{1/2}. $$
My question is: does anyone have any examples where $\rho^n_{\tau}$ has been explicitly calculated, i.e given a $\rho^0$ and a fixed $\tau$ has anyone calculated an explicit form of the sequence of minimisers $\rho^n_\tau$?
If I understand correctly, this is equivalent to asking for examples of closed form solutions of optimal transport problems. One typical example where these exist are gaussian distributions. See also this question.