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Hmidi, Taoufik; Li, Dong

Dynamics of one-fold symmetric patches for the aggregation equation and collapse to singular measure. (English) Zbl 1433.35422

Anal. PDE 12, No. 8, 2003-2065 (2019).
Summary: 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.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B44 Blow-up in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
92D25 Population dynamics (general)
76B47 Vortex flows for incompressible inviscid fluids

Keywords:

aggregation equations; concentration; vortex patches
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\textit{T. Hmidi} and \textit{D. Li}, Anal. PDE 12, No. 8, 2003--2065 (2019; Zbl 1433.35422)
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