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arXiv 2108.01416math.APmath.CO
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有限图上平均场方程的热流

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

Yong Lin, Yunyan Yang

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15 pages
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摘要：

受Castéras（Pacific J. Math.，2015），Li-Zhu（Calc.变量，2019）和孙朱（Calc.变量，2020），我们提出了一个热流的平均场方程上的连接有限图$G=(V,E)$。即$$ \left\{\begin{array}{lll} \partial_tϕ(u)=Δu-Q+ρ\frac{e^u}{\int_Ve^udμ}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$，其中$Δ$是标准图拉普拉斯算子，$ρ$是实数，$Q:V\rightarrow\mathbb{R}$是满足$\int_VQdμ=ρ$的函数，而$ϕ:\mathbb{R}\rightarrow\mathbb{R}$是包括$ϕ(s)=e^s$的某些光滑函数之一。证明了对于任意的初始数据$u_0$和$ρ\in\mathbb{R}$，上述热流存在唯一解$u:V\times[0,+\infty)\rightarrow\mathbb{R}$，并且$u(x,t)$在$x\in V$中一致收敛于某个函数$u_\infty:V\rightarrow\mathbb{R}$，$t\rightarrow+\infty$，$u_\infty$是平均场方程$Q\equiv 0$的解。我们的做法如下：根据常微分方程理论，给出了热流的短时存在性，各种积分估计给出了热流的长时存在性，并建立了Lojasiewicz-Simon型不等式，利用该不等式得到了热流的收敛性.

英文摘要：

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 $$ \left\{\begin{array}{lll} \partial_tϕ(u)=Δu-Q+ρ\frac{e^u}{\int_Ve^udμ}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$ where $Δ$ is the standard graph Laplacian, $ρ$ is a real number, $Q:V\rightarrow\mathbb{R}$ is a function satisfying $\int_VQdμ=ρ$, and $ϕ:\mathbb{R}\rightarrow\mathbb{R}$ is one of certain smooth functions including $ϕ(s)=e^s$. We prove that for any initial data $u_0$ and any $ρ\in\mathbb{R}$, 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 $$Δu_\infty-Q+ρ\frac{e^{u_\infty}}{\int_Ve^{u_\infty}dμ}=0.$$ 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.

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