Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions

2025/05/22 by Sun, Linlin, Zhu, Xiaobao
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.16414

Abstract

Let (M,g) be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begincases -Δu = ρ₁(\frach₁e^u∫_Mh₁e^udv_g-1) - ρ₂(\frach₂e^-u∫_Mh₂e^-udv_g-1),
∫_Mudv_g=0, \endcases where ρ₁=8π and ρ₂∈(0,8π] are parameters, and h₁, h₂ are smooth functions on M that are positive somewhere. By employing a refined Brezis-Merle type analysis, we establish sufficient conditions of Ding-Jost-Li-Wang type for the existence of solutions to this equation in critical cases, particularly when h₁ and h₂ may change signs. Our results extend Zhou's existence theorems (Nonlinear Anal. 69 (2008), no.~8, 2541--2552) for the case h₁=h₂≡ 1.

Citations

• Existence results for Toda systems with sign-changing prescribed functions: Part II
• Existence results for Toda systems with sign-changing prescribed functions: Part I
• Another remark on a result of Ding-Jost-Li-Wang
• The Prescribed Q-Curvature Flow for Arbitrary Even Dimension in a Critical Case
• A remark on a result of Ding-Jost-Li-Wang
• A general existence result for the Toda system on compact surfaces
• Analytic Aspects of the Toda System: II. Bubbling behavior and existence of solutions
• Solutions for Toda systems on Riemann surfaces
• Analytic aspects of the Toda system: I. A Moser-Trudinger inequality

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