Reference request: analysis of a nonlinear Fokker-Planck type equation

Question

It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\right) \label{1}\tag{1}$$ where $\rho_\infty$ (say for instance of the form $\rho_\infty(x) \propto \mathrm{e}^{-V(x)}$ for some smooth and convex potential $V(x)$ growing sufficiently fast at infinity) is the unique equilibrium distribution to which the solution of (\ref{1}) converges, admits a family of Lyapunov functionals of the form (where $\phi$ is some convex function fulfilling certain properties) $$\mathrm{H}_\phi[\rho] = \int_{\mathbb R} \phi\left(\frac{\rho}{\rho_\infty}\right) \rho_\infty \,\mathrm{d} x \label{2}\tag{2}$$ for the study of the convergence to equilibrium problem, see for instance this monograph. However, I am wondering if there are references for investigation/study of the large-time convergence behavior of the following nonlinear Fokker-Planck type equation $$\partial_t \rho = \partial_x \left(\mathcal{F}[\rho]\, \partial_x\left(\frac{\rho}{\mathcal{F}[\rho]}\right)\right) \label{3}\tag{3}$$ where $\mathcal{F} \colon \rho \in \mathcal{P}(\mathbb R) \to \mathcal{F}[\rho] \in \mathcal{P}(\mathbb R)$ is a sort of "quasi-stationary distribution" for the PDE (\ref{3}) with $\mathcal{F}[\rho_\infty] = \rho_\infty$. Here quasi-stationarity rough means (loosely speaking) that $\mathcal{F}[\rho]$ "takes the same form" as the true equilibrium $\rho_\infty$ (for example, they are both Gaussian but with different variance or they are both exponential distributions with different mean values). I am wondering if there some recent or classical reference for the investigation of the large time behavior of such type of nonlinear Fokker-Planck equation, especially the construction of Lyapunov functionals. I have to admit that my question is a not very clear as the analysis of (\ref{3}) will depend on the specific choice of the map/nonlinearity $\mathcal{F}[\cdot]$, but I am hoping that some references pointing to analysis of Fokker-Planck type equations with the very specific structure as indicated in (\ref{3}) can be found.

Answer 1

Let me rewrite the equation (3) as $$ \partial_t \rho = \partial_x \left(\rho\, \partial_x\log\left(\frac{\rho}{\mathcal{F}[\rho]}\right)\right) \label{3'}\tag{3'}. $$ Then, this equation is a gradient flow with respect to the metric tensor after Otto inducing the Wasserstein distance iff there exists a driving free energy $\mathcal{E} : \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}$ such that its variational derivative satisfies $$ \mathcal{E}'(\rho) = \log\left(\frac{\rho}{\mathcal{F}[\rho]}\right) + C, $$ where the constant $C$ does not matter and could even depend on $\rho$.

In the first case with $\mathcal{F}[\rho]\equiv \rho_\infty$, one gets $$ \mathcal{E}(\rho) = H_{\phi}[\rho] \quad\text{with}\quad \phi(r) = r \log r -r + 1 . $$ A more interesting case is for some potential energy $V:\mathbb{R}^d \to \mathbb{R}$ and symmetric interaction energy $W:\mathbb{R^d}\times\mathbb{R^d} \to \mathbb{R}$ the map $$ \mathcal{F}[\rho](x) = Z[\rho]^{-1} \exp\left( -V(x)- \int W(x,y) \rho(y) dy \right) \quad\text{with}\quad Z[\rho] := \int \exp\left(- V(x)- \int W(x,y) \rho(y) dy \right) dx .$$ Then, upto a $\rho$-dependent constant, the free energy is given by $$ \mathcal{E}(\rho) = \int \rho \log \rho\, dx + \int V(x) \rho(x) \, dx + \frac{1}{2} \int \int W(x,y) \rho(x)\rho(y)\, dx\, dy . $$ This is the classic McKean-Vlasov model and the free energy above consisting of entropy, potential energy and interaction energy is studied a lot in different areas (gradient flows, density functional theory, statistical mechanics).

Depending on your map $\mathcal{F}$ there might be other free energies. The above one is the most common in the literature of mean-field limits for interacting particle systems, where it is usually even assumed that $W(x,y)=w(x-y)$ for some even $w:\mathbb{R}^d \to \mathbb{R}$.