2021/08/03 by Yong Lin, Lin, Yong, Yunyan Yang +1 · 1 citation
Computer Science · Mathematics · #34B45 #35R02 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2108.01416
openalex publication_date 2021/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \∂tϕ(u)=Δu-Q+ρ(eu)/(∫Veudμ)
u(⋅,0)=u0,. where Δ is the standard graph Laplacian, ρ is a real number, Q:V→ℝ is a function satisfying ∫VQdμ=ρ, and ϕ:ℝ→ℝ is one of certain smooth functions including ϕ(s)=es. We prove that for any initial data u0 and any ρ∈ℝ, there exists a unique solution u:V×[0,+∞)→ℝ of the above heat flow; moreover, u(x,t) converges to some function u_∞:V→ℝ uniformly in x∈ V as t→+∞, and u_∞ is a solution of the mean field equation Δu_∞-Q+ρ\fraceu_∞∫Veu_∞dμ=0. Though G is a finite graph, this result is still unexpected, even in the special case Q≡ 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.