Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

  • Katy Craig

    Department of Mathematics, University of California, Santa Barbara, CA, United States of America
  • Ihsan Topaloglu

    Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, United States of America
Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit cover
Download PDF

A subscription is required to access this article.

Abstract

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. 239–279

DOI 10.1016/J.ANIHPC.2019.10.003