Onsager's Mean Field Theory of Vortex Flows with Singular Sources: Blow-Up and Concentration without Quantization

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles.