Minimizers of the Interaction Energy on a Space of Probability Measures

We consider a particular energy functional (or interaction energy) and we try to describe its minimizers following the approach of J. A. Carrillo et al.. The energy functional will be defined on a space of probability measures, since we would like to represent collective phenomena like swarm-like behaviour. The energy is characterised by a specific interaction potential W. Some specific properties of the local minima can be analysed in depth, especially in the case of strong and mild repulsive potential. The main example is characterised by the choice of a power-law potential. The most informative element is the Laplacian of our potential, whose behaviour near 0 identifies different types of repulsion. The minimization process can be translated into a partial differential equation (in the distributional sense), which is characterised by non-locality and non-linearity properties. Actually, the equation is obtained as a gradient flow for the interaction energy with respect to a non-standard Riemann-like structure and we give a sketch of its rigorous derivation in a metric space. The main references for the mathematical structures we need and for gradient flows on metric or probability spaces are the works of Ambrosio, Gigli and Savaré. In addition, working in Wesserstein spaces allows us to translate a wide range of propositions in terms of optimal transport problems. This correspondence has been widely treated in recent literature and we only give some hints about it. Furthermore, following another work by J. A. Carrillo et al., we are able to expose some results on the stability or instability of spherically symmetric steady states for certain classes of potentials. Finally, we discuss some basic results of existence and uniqueness which have proven to be really useful in literature.