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.
Citation
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
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