Existence results for mean field equations with turbulence

2007/05/11 by Cheikh Birahim Ndiaye, Ndiaye, Cheikh Birahim
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.0705.1687

arxiv created 2007/05/11 · openalex publication_date 2007/05/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the following form of the so-called Mean field equation arising from the statistical mechanics description of two dimensional turbulence - \D_g u = ρ₁ (\frace^u∫_\Sig e^u dV_g-1)-ρ₂ (\frace^-u∫_\Sig e^-u dV_g - 1) on a given closed orientable Riemannian surface (Σ, g) with volume 1, where ρ₁, ρ₂ are real parameters. Exploiting the variational structure of the problem and running a min-max scheme introduced by Djadli and Malchiodi, we prove that if k is a positive integer, ρ₁ and ρ₂ two real numbers such that ρ₁∈ (8kπ, 8(k+1)π) and ρ₂<4π then \eqrefeq:study is solvable.

Citations

• The existence of surfaces of constant mean curvature with free boundaries
• Solutions for Toda systems on Riemann surfaces

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