## Mathematics Faculty Works

#### Document Type

Article - post-print

2016

#### 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 $C^{1+\gamma}$ ,$0<\gamma<1$. 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 $L^1$ sense to a self-similar expanding ball solution. In this work, we prove $C^{1+\gamma}$ regularity of the domain's boundary on the time interval on which the solution exists as an $L^\infty$ patch, up to the collapse time going forward in time and for all finite times going backward in time.

#### Citation / Publisher Attribution

This is an author-manuscript of an article accepted for publication in SIAM Journal of Math. Anal. following peer review. The version of record: A. Bertozzi, J. Garnett, T. Laurent, and J. Verdera. "The Regularity of the Boundary of a Multidimensional Aggregation Patch." SIAM J. Math. Anal., 48(6), 3789–3819. is available online at: DOI:10.1137/15M1033125.

#### Publisher Statement

© 2016, Society for Industrial and Applied Mathematics

#### Recommended Citation

A. Bertozzi, J. Garnett, T. Laurent, and J. Verdera. "The Regularity of the Boundary of a Multidimensional Aggregation Patch." SIAM J. Math. Anal., 48(6), 3789–3819. DOI:10.1137/15M1033125.

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