Title: A heat flow for the mean field equation on a finite graph Show Chinese title

Title: 一个有限图上的平均场方程的热流

Abstract: Inspired by works of Cast\'eras (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 $$ \left\{\begin{array}{lll} \partial_t\phi(u)=\Delta u-Q+\rho \frac{e^u}{\int_Ve^ud\mu}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$ 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 $$\Delta u_\infty-Q+\rho\frac{e^{u_\infty}}{\int_Ve^{u_\infty}d\mu}=0.$$ 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. Show Chinese abstract

Abstract: 受Castéras（Pacific J. Math., 2015）、Li-Zhu（Calc. Var., 2019）和Sun-Zhu（Calc. Var., 2020）工作的启发，我们提出了一个在连通有限图$G=(V,E)$上的平均场方程的热流。 即$$ \left\{\begin{array}{lll} \partial_t\phi(u)=\Delta u-Q+\rho \frac{e^u}{\int_Ve^ud\mu}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$其中 $\Delta$是标准图拉普拉斯算子，$\rho$是一个实数， $Q:V\rightarrow\mathbb{R}$是满足$\int_VQd\mu=\rho$的函数，且 $\phi:\mathbb{R}\rightarrow\mathbb{R}$是包括$\phi(s)=e^s$在内的某些光滑函数之一。 我们证明了对于任何初始数据$u_0$和任何$\rho\in\mathbb{R}$，上述热流存在唯一的解$u:V\times[0,+\infty)\rightarrow\mathbb{R}$；此外，$u(x,t)$在$x\in V$上一致收敛到某个函数$u_\infty:V\rightarrow\mathbb{R}$，当$t\rightarrow+\infty$时，$u_\infty$是平均场方程$$\Delta u_\infty-Q+\rho\frac{e^{u_\infty}}{\int_Ve^{u_\infty}d\mu}=0.$$的解。尽管$G$是一个有限图，这个结果仍然是出乎意料的，即使在特殊情形$Q\equiv 0$中也是如此。 我们的方法如下：热流的短时间存在性由常微分方程理论得出；各种积分估计给出了其长时间存在性；此外我们建立了洛萨列夫-西蒙类型不等式，并用它来得出热流的收敛性。