# On radial two-species Onsager vortices near the critical temperature

## TL;DR

This paper compares two mean field models of hydrodynamic turbulence near the critical temperature, revealing that deterministic models behave more like single vortex cases than stochastic ones, with new insights into critical temperature values.

## Contribution

It introduces a comparison between deterministic and stochastic mean field equations for vortex turbulence, highlighting differences in solution properties near the critical temperature.

## Key findings

- Deterministic models resemble single vortex intensity behavior.
- Stochastic models differ significantly in qualitative properties.
- New variational interpretations of critical temperature are provided.

## Abstract

We compare two mean field equations describing hydrodynamic turbulence in equilibrium, which are derived under a deterministic vs.\ stochastic assumption on the variable vortex intensity distribution. Mathematically, such equations correspond to non-local Liouville type problems, and the critical temperature corresponds to the optimal Moser-Trudinger constant. We consider the radial case and we assume that the inverse temperature is near its critical value. Under these assumptions we show that, unlike previously existing results, the qualitative properties of the solution set in the deterministic case is more similar to the single vortex intensity case than the stochastic case. Some new variational interpretations of the value explicit values of the critical temperature are also provided.