Non degeneracy, Mean Field Equations and the Onsager theory of 2D turbulence

TL;DR

This paper investigates the mathematical properties of solutions to the mean field equations related to 2D turbulence, establishing non-degeneracy and convexity results crucial for understanding large energy states in Onsager's theory.

Contribution

It proves the non-degeneracy of bubbling solutions under certain conditions, leading to smoothness and convexity of the Onsager entropy in high energy regimes.

Findings

Bubbling solutions are non-degenerate under specified assumptions.

Onsager entropy is shown to be smooth and strictly convex at high energies.

Results contribute to the mathematical understanding of 2D turbulence models.

Abstract

The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated $m$-vortex Hamiltonian, the $m$-point bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.