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Dynamics of one-fold
symmetric patches for the aggregation equation and collapse
to singular measure
Taoufik Hmidi and Dong Li |
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Vol. 12 (2019), No. 8, 2003–2065
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Abstract |
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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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Keywords
aggregation equations, concentration, vortex patches
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Mathematical Subject Classification 2010
Primary: 35B44, 35A07, 35Q92
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Milestones
Received: 3 April 2018
Revised: 26 October 2018
Accepted: 30 November 2018
Published: 28 October 2019
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Authors |
| Taoufik Hmidi | |
| Université de Rennes 1 CNRS IRMAR - UMR 6625 Rennes France |
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| Dong Li | |
| Department of Mathematics The Hong Kong University of Science and Technology Kowloon Hong Kong |
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