PreprintCC BY 4.0arXiv
Persistence of C2 regularity for aggregation patches
Abstract
We prove persistence of C2 boundary regularity for patch solutions of the two-dimensional aggregation transport equation, in contrast with the known C2 ill-posedness of Euler vortex patches. More generally, we consider transport equations with kernels of the form L(∇ N), where N is the fundamental solution of the Laplacian and L is a linear map. We show that C2 regularity persists for every C2 initial patch if and only if L is a scalar multiple of the identity. Our approach builds on the curvature formulation introduced by Kiselev and Luo.
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BibTeX
@misc{cantero2026persistence,
title = {{Persistence of $C^2$ regularity for aggregation patches}},
author = {Juan Carlos Cantero and Marc Magaña},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03258},
eprint = {2611.03258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03258},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Persistence of C^2 regularity for aggregation patches
AU - Juan Carlos Cantero
AU - Marc Magaña
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03258
UR - https://arxiv.org/abs/2611.03258
AB - We prove persistence of C^2 boundary regularity for patch solutions of the two-dimensional aggregation transport equation, in contrast with the known C^2 ill-posedness of Euler vortex patches. More generally, we consider transport equations with kernels of the form L(∇ N), where N is the fundamental solution of the Laplacian and L is a linear map. We show that C^2 regularity persists for every C^2 initial patch if and only if L is a scalar multiple of the identity. Our approach builds on the curvature formulation introduced by Kiselev and Luo.
ER -