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2019 Dynamics of one-fold symmetric patches for the aggregation equation and collapse to singular measure
Taoufik Hmidi, Dong Li
Anal. PDE 12(8): 2003-2065 (2019). DOI: 10.2140/apde.2019.12.2003

Abstract

We are concerned with the dynamics of one-fold symmetric patches for the two-dimensional aggregation equation associated to 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 us to analyze the concentration phenomenon of the aggregation patches near the blow-up 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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Taoufik Hmidi. Dong Li. "Dynamics of one-fold symmetric patches for the aggregation equation and collapse to singular measure." Anal. PDE 12 (8) 2003 - 2065, 2019. https://doi.org/10.2140/apde.2019.12.2003

Information

Received: 3 April 2018; Revised: 26 October 2018; Accepted: 30 November 2018; Published: 2019
First available in Project Euclid: 14 December 2019

zbMATH: 07143409
MathSciNet: MR4023974
Digital Object Identifier: 10.2140/apde.2019.12.2003

Subjects:
Primary: 35A07, 35B44, 35Q92

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 8 • 2019
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