Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence

Abstract

On strictly starshaped domains of second kind (see Definition 1.2) we find sufficient conditions which allow the solution of two long standing open problems closely related to the mean field equation ${({\mathbf {P}}_{\lambda })}$ below. On one side we describe the global behaviour of the Entropy for the mean field Microcanonical Variational Principle ((MVP) for short) arising in the Onsager description of two-dimensional turbulence. This is the completion of well known results first established in Caglioti et al. (Commun Math Phys 174:229–260, 1995). Among other things we find a full unbounded interval of strict convexity of the Entropy. On the other side, to achieve this goal, we have to provide a detailed qualitative description of the global branch of solutions of ${({\mathbf {P}}_{\lambda })}$ emanating from $\lambda =0$ and crossing $\lambda =8\pi$. This is the completion of well known results first established in Suzuki (Ann Inst H Poincaré Anal Non Linéaire 9(4):367–398,1992) and Chang et al. (in: Lecture on partial differential equations, International Press, Somerville, pp 61–93, 2003) for $\lambda \le 8\pi$, and it has an independent mathematical interest, since the shape of global branches of semilinear elliptic equations, with very few well known exceptions, are poorly understood. It turns out that the MVP suggests the right variable (which is the energy) to be used to obtain a global parametrization of solutions of ${({\mathbf {P}}_{\lambda })}$. 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.