Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.
Cite this article
Katy Craig, Ihsan Topaloglu, Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 2, pp. 239–279DOI 10.1016/J.ANIHPC.2019.10.003