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

Craig, Katy; Topaloglu, Ihsan

doi:10.1016/j.anihpc.2019.10.003

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

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 239-279.

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Résumé

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
Mots-clés : Global minimizers, Pair potentials, Aggregation-diffusion equation, Gradient flow, Γ-convergence, Porous medium equation

Affiliations des auteurs :

Craig, Katy ¹ ; Topaloglu, Ihsan ²

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     author = {Craig, Katy and Topaloglu, Ihsan},
     title = {Aggregation-diffusion to constrained interaction: {Minimizers} & gradient flows in the slow diffusion limit},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {239--279},
     publisher = {Elsevier},
     volume = {37},
     number = {2},
     year = {2020},
     doi = {10.1016/j.anihpc.2019.10.003},
     mrnumber = {4072808},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.10.003/}
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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, Tome 37 (2020) no. 2, pp. 239-279. doi : 10.1016/j.anihpc.2019.10.003. http://www.numdam.org/articles/10.1016/j.anihpc.2019.10.003/

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