Existence results for mean field equation with turbulence

on Σ; ( where ∆g is the Laplace-Beltrami, ρ a real parameter) which arises in the study of limit of point vortices of Euler flows, spherical Onsager vortex theory and condensates in some Chern-Simons-Higgs models, see for example the papers [3], [4], [8], [9], [10], [11], [14], [25], [40] and the references therein. An other example is what we refer as mean field equation with turbulence on a closed orientable Riemannian surface (Σ, g) (3) −∆gu = ρ1 ( e