Vol. 12, No. 8, 2019

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Dynamics of one-fold symmetric patches for the aggregation equation and collapse to singular measure

Taoufik Hmidi and Dong Li

Vol. 12 (2019), No. 8, 2003–2065
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.

Keywords
aggregation equations, concentration, vortex patches
Mathematical Subject Classification 2010
Primary: 35B44, 35A07, 35Q92
Milestones
Received: 3 April 2018
Revised: 26 October 2018
Accepted: 30 November 2018
Published: 28 October 2019
Authors
Taoufik Hmidi
Université de Rennes 1
CNRS
IRMAR - UMR 6625
Rennes
France
Dong Li
Department of Mathematics
The Hong Kong University of Science and Technology
Kowloon
Hong Kong