24 may 2017 -- 17:00   [open in google calendar]

Scuola Normale Superiore, Aula Russo

Abstract.

We discuss the solution of two long standing open problems closely related to the mean field Liouville-type equation (P_\lm). On one side, we find the global behaviour of the entropy for the mean field Microcanonical Variational Principle ((MVP) for short), as it arises in the Onsager description of two-dimensional turbulence on strictly starshaped domains of second kind. Among other things we find a region of strict convexity of the entropy. On the other side, to achieve this goal, we have to catch the global bifurcation diagram of solutions of the mean field equation (P_\lm), emanating from \lm = 0 and crossing \lm = 8\pi. The (MVP) suggests the right variable (which is the energy) to be used to obtain a global parametrization of solutions of (P_\lm). In particular a crucial spectral simplification is obtained by using the fact that, by definition, solutions of the (MVP) maximize the entropy at fixed energy and total vorticity.