Mass quantization and minimax solutions for Neri's mean field equation in 2D-turbulence

We study the mean field equation derived by Neri in the context of the statistical mechanics description of 2D-turbulence, under a “stochastic” assumption on the vortex circulations. The corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity, containing a probability measure P∈M([−1,1])P∈M([−1,1]) which describes the distribution of the vortex circulations. Unlike the more investigated “deterministic” version, we prove that Neri's equation may be viewed as a perturbation of the widely analyzed standard mean field equation, obtained by taking P=δ1P=δ1. In particular, in the physically relevant case where PP is non-negatively supported and P({1})>0P({1})>0, we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in R2R2 and on compact 2-manifolds without boundary.