  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.
