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On some properties of mean fields of equilibrium vortices described by the Hamiltonian

According to the scenario of Onsager to explain the reason why large-scale long-lived coherent structures are often observed in two-dimensional flows, the mean fields for the equilibrium of a large number of vortices are considered. In this paper we are concerned with our recent results on the variational structure of the blow-up solutions to the Gelʼfand problem, which would give insights on the relation between mean fields of equilibrium vortices and the Hamiltonian of vortices.