The Regularity of the Boundary of a Multidimensional Aggregation Patch
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The Regularity of the Boundary of a Multidimensional Aggregation Patch

  • Author(s): Bertozzi, A
  • Garnett, J
  • Laurent, T
  • Verdera, J
  • et al.
Abstract

We consider solutions to the aggregation equation with Newtonian potential where the initial data are the characteristic function of a domain with boundary of class C1+γ ,0 < γ < 1. Such initial data are known to yield a solution that, going forward in time, retains a patch-like structure with a constant time-dependent density inside an evolving region, which collapses on itself in a finite time, and which, going backward in time, converges in an L1 sense to a self-similar expanding ball solution. In this work, we prove C1+γ regularity of the domain’s boundary on the time interval on which the solution exists as an L∞ patch, up to the collapse time going forward in time and for all finite times going backward in time.

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