Document Type
Article - post-print
Publication Date
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.
Original Publication 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.
Publisher Statement
© 2016, Society for Industrial and Applied Mathematics
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.
Digital Commons @ LMU & LLS Citation
Bertozzi, Andrea L.; Garnett, John B.; Laurent, Thomas; and Verdera, Joan, "The regularity of the boundary of a multidimensional aggregation patch" (2016). Mathematics, Statistics and Data Science Faculty Works. 93.
https://digitalcommons.lmu.edu/math_fac/93
