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

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.

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 $$w:\mathbb{R}^d \to \mathbb{R}$$.
• Hello Andre, first thanks for your answer and post. However, the map $\mathcal{F}$ I have in mind is not the classical one that you wrote (and was well-known to be honest). I am wondering if there exists a general guidance for the construction of the Lyapunov-type functional (for general nonlinearity $\mathcal{F}[\cdot]$), or one has to deal with the specific $\mathcal{F}$ "case by case". What I have in my project is actually a discrete version of equation (3) where the state space is $\mathbb N$... Jul 31, 2023 at 22:10
• In this generality, the best you can try is to find a free energy functional that satisfies the unnumbered equation above $\mathcal{E}'(\rho) = \dots$. Something similar is possible for discrete models. Together with Joe Conlon, I once studied a class of models on $[0,\infty)$ and also $\mathbb{N}$ that have similar properties to McKean-Vlasov: arXiv:1711.00782, see in particular Section 6 about the gradient flow structures and Remark 1.4 about its relation to McKean-Vlasov. Otherwise, I don't know of any general theory beyond what I wrote above. Aug 1, 2023 at 20:32