Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials
Abstract
In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $\lambda-$Wasserstein metric topology with $1\le \lambda<\infty$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\infty$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2019
- DOI:
- 10.48550/arXiv.1907.07004
- arXiv:
- arXiv:1907.07004
- Bibcode:
- 2019arXiv190707004K
- Keywords:
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- Mathematics - Probability;
- Mathematics - Analysis of PDEs;
- Mathematics - Optimization and Control
- E-Print:
- 21 pages. Minor corrections in Theorem 1 and Lemma 4 were made in v3