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Aggregation Equation and Collapse to Singular Measure

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Research in PDEs and Related Fields

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Abstract

We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments, and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.

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Author information

Authors and Affiliations

  1. IRMAR, Université de Rennes 1, Rennes, France

    Taoufik Hmidi

  2. Department of Mathematics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong

    Dong Li

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  1. Taoufik Hmidi
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  2. Dong Li
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Correspondence to Taoufik Hmidi .

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  1. Department of Mathematics, University of Monastir, Monastir, Tunisia

    Kaïs Ammari

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Hmidi, T., Li, D. (2022). Aggregation Equation and Collapse to Singular Measure. In: Ammari, K. (eds) Research in PDEs and Related Fields. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-14268-0_4

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