A heat flow for the mean field equation on a finite graph

Abstract

Inspired by works of Castéras (Pac J Math 276:321–345, 2015), Li and Zhu (Calc Var Partial Differ Equ 58:1–18, 2019), Sun and Zhu (Calc Var Partial Differ Equ 60:1–26, 2021), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t\phi (u)=\Delta u-Q+\rho \frac{e^u}{\int _Ve^ud\mu }\\ u(\cdot ,0)=u_0, \end{array}\right. \end{aligned}$$

where $\Delta$ is the standard graph Laplacian, $\rho$ is a real number, $Q:V\rightarrow {\mathbb {R}}$ is a function satisfying $\int _VQd\mu =\rho$, and $\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ is one of certain smooth functions including $\phi (s)=e^s$. We prove that for any initial data $u_0$ and any $\rho \in {\mathbb {R}}$, there exists a unique solution $u:V\times [0,+\infty )\rightarrow {\mathbb {R}}$ of the above heat flow; moreover, u(x, t) converges to some function $u_\infty :V\rightarrow {\mathbb {R}}$ uniformly in $x\in V$ as $t\rightarrow +\infty$, and $u_\infty$ is a solution of the mean field equation

$$\begin{aligned} \Delta u_\infty -Q+\rho \frac{e^{u_\infty }}{\int _Ve^{u_\infty }d\mu }=0. \end{aligned}$$

Though G is a finite graph, this result is still unexpected, even in the special case $Q\equiv 0$. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz–Simon type inequality and use it to conclude the convergence of the heat flow.