A mean field type flow with sign-changing prescribed function on a symmetric Riemann surface

Abstract

Let (Σ,g) be a closed Riemann surface, and G={σ1,⋯,σN} be a finite isometric group acting on it. Denote a positive integer ℓ=minx∈Σ⁡I(x), where I(x) is the number of all distinct points of the set {σ1(x),⋯,σN(x)}. In this paper, we consider the following G-invariant mean field type flow{∂∂teu=Δgu+8πℓ(feu∫Σfeudvg−1|Σ|)u(⋅,0)=u0, where u0 belongs to C2+α(Σ) for some α∈(0,1), f is a sign-changing smooth function such that ∫Σfeu0dvg≠0, both u0 and f are G-invariant, and |Σ| denotes the area of (Σ,g). Such kind of flow was originally proposed by Castéras [6]. Through a priori estimates, we prove that the flow u(x,t) exists for all time t∈[0,∞). Moreover, by employing blow-up procedure, we obtain that under certain geometric conditions, u(x,t) converges to u(x) in H2(Σ) as t→∞, where u(x) is a solution of the mean field equation−Δgu=8πℓ(feu∫Σfeudvg−1|Σ|). This generalizes recent results of Li-Zhu [27] and Sun-Zhu [37].