Calculus of Variations and Geometric Measure Theory
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K. Craig - I. Topaloglu

Aggregation-Diffusion to Constrained Interaction: Minimizers & Gradient Flows in the Slow Diffusion Limit

created by topaloglu1 on 03 Aug 2019

[BibTeX]

preprint

Inserted: 3 aug 2019

Year: 2018

ArXiv: 1806.07415 PDF

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.

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