Starting from the variational formulation of the Fokker-Planck equation provided by Jordan, Kinderlehrer and Otto in 1998, many evolution partial differential equations in the framework of probability spaces have been interpreted as gradient flow of a suitable free energy functional with respect to the Wasserstein distance. In particular, this is the case of aggregation-diffusion equations.

Generally speaking, the gradient flow structure is extremely useful not only to show well-posedness of certain PDEs, but it also helps to prove some properties such as stability results, trend to equilibrium, and characterisation of stationary states and energy minimisers. In the specific case of aggregation-diffusion phenomena, e.g. in the modelling of chemotaxis or in the context of Mean Field Games, it is interesting to see when a balance between repulsion and attraction occurs. These two effects should determine stationary states for the corresponding aggregation-diffusion equation including the stable solutions possibly given by local (global) minimisers of the free energy functional associated.

In the article “Global minimisers for anisotropic attractive–repulsive interactions’’ from the European Journal of Applied Mathematics, Gunnar Kaib, Kyungkeun Kang and Angela Stevens prove the existence of global minimisers for a class of attractive-repulsive interaction potentials that are in general not radially symmetric, which is indeed the main novelty of their paper. In fact, in most cases the interaction potential is assumed to be symmetric, and sometimes purely attractive.

In the aforementioned work from EJAM, the authors consider either the case with diffusion or the one with solely interactions. Roughly speaking, their strategy consists in “localising” the problem in a ball and then showing the result in the whole space, by also giving information on the support of the minimisers. Moreover, they conclude the paper with a formal argument supporting that for nonsymmetric potentials global minimisers may be neither radially symmetric nor unique.