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

Abstract

Inspired by works of Castéras (Pacific J. Math., 2015), Li-Zhu (Calc. Var., 2019) and Sun-Zhu (Calc. Var., 2020), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely $$\{\begin{array}{l}{\partial }_t\varphi (u)=\Delta u-Q+\rho \frac{e^u}{{\int }_V{e}^{u}d\mu }\\ u(\cdot ,0)=u_0,\end{array}$$ where $\Delta$ is the standard graph Laplacian, $\rho$ is a real number, $Q:V\to \mathbb{R}$ is a function satisfying ${\int }_{V}Qd\mu =\rho$, and $\varphi :\mathbb{R}\to \mathbb{R}$ is one of certain smooth functions including $\varphi (s)=e^s$. We prove that for any initial data $u_0$ and any $\rho \in \mathbb{R}$, there exists a unique solution $u:V\times [0,+\infty )\to \mathbb{R}$ of the above heat flow; moreover, $u(x,t)$ converges to some function $u_{\infty }:V\to \mathbb{R}$ uniformly in $x\in V$ as $t\to +\infty$, and $u_{\infty }$ is a solution of the mean field equation $$\Delta u_{\infty }-Q+\rho \frac{e^{u_{\infty }}}{{\int }_V{e}^{u_{\infty }}d\mu }=0.$$ 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.

15 pages

References in corpus (1)

• Kazdan-Warner equation on infinite graphs