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Finite- and infinite-time cluster formation for alignment dynamics on the real line

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Abstract

We show that the locations where finite- and infinite-time clustering occur for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated with a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated with a scalar balance law formulation of the system.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while TL was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the summer of 2023. CT acknowledges the support of NSF grants DMS-2108264 and DMS-2238219.

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Authors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Trevor M. Leslie

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Changhui Tan

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  1. Trevor M. Leslie
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  2. Changhui Tan
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Correspondence to Changhui Tan.

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Leslie, T.M., Tan, C. Finite- and infinite-time cluster formation for alignment dynamics on the real line. J. Evol. Equ. 24, 8 (2024). https://doi.org/10.1007/s00028-023-00939-2

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