no. 2
Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit
Craig, Katy1; Topaloglu, Ihsan2
1 Department of Mathematics, University of California, Santa Barbara, CA, United States of America
2 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, United States of America
Annales de l'I.H.P. Analyse non linéaire, Volume 37 (2020) no. 2, pp. 239-279

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.

DOI: 10.1016/j.anihpc.2019.10.003
Classification: 49J45, 82B21, 82B05, 35R09, 45K05
Keywords: Global minimizers, Pair potentials, Aggregation-diffusion equation, Gradient flow, Γ-convergence, Porous medium equation

Craig, Katy 1; Topaloglu, Ihsan 2

1 Department of Mathematics, University of California, Santa Barbara, CA, United States of America
2 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, United States of America 
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Craig, Katy; Topaloglu, Ihsan. Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit. Annales de l'I.H.P. Analyse non linéaire, Volume 37 (2020) no. 2, pp. 239-279. doi: 10.1016/j.anihpc.2019.10.003

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