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

TL;DR

This paper investigates Neri's mean field equation in 2D turbulence, establishing mass quantization for blow-up sequences and constructing minimax solutions, thus extending classical results to a stochastic vortex circulation setting.

Contribution

It demonstrates that Neri's equation can be seen as a perturbation of the standard mean field equation and proves mass quantization under certain conditions, which was not previously known.

Findings

Mass quantization for blow-up sequences in Neri's equation.

Construction of minimax solutions on bounded domains and compact 2-manifolds.

Neri's equation as a perturbation of the standard mean field equation.

Abstract

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 $\mathcal P\in\mathcal M([-1,1])$ 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 $\mathcal P=\delta_1$. In particular, in the physically relevant case where $\mathcal P$ is non-negatively supported and $\mathcal P(\{1\})>0$, we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in $\mathbb R^2$ and on compact 2-manifolds without boundary.