Laboratoire de Physique Théorique
Toulouse - UMR 5152

Onsager derived a mean field equation in the high-energy limit of point vortex system where we can observe a recursive hierarchy of self-interaction controlled by a Hamiltonian. First we describe the situation and then notice the quantized blowup mechanism. Having examined variational and scaling properties, we formulate the Smoluchowki-Poisson equation by the Toland duality. We shall show the story of the proof of mass quantization in two-space dimension, and then talk about related models such as degenerate parabolic equatios. Similarity and differences between harmonic maps are emphasized.