PreprintarXiv licencearXiv
Global weak solutions for a two-species Riesz cross-attraction system with quadratic diffusion
Abstract
We establish global existence for a two-species cross-attraction system on ℝ3, with quadratic diffusion and the common interaction operator (−Δ)−s, where 1/2<s<1. Nonnegative initial data in L1∩ L∞ with finite second moments require no smallness condition on the masses. A joint energy and second-moment estimate and a coupled Lp iteration give uniform a priori estimates for the regularized solutions. Compactness then yields mass-conserving global weak solutions, bounded on every finite time interval and strongly continuous in L1∩ L2. The energy inequality holds at every pair of time endpoints. Uniqueness and bounds uniform for all positive times are not asserted.
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BibTeX
@misc{liu2026global,
title = {{Global weak solutions for a two-species Riesz cross-attraction system with quadratic diffusion}},
author = {Zhenxing Liu and Shen Bian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25818},
eprint = {2610.25818},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25818},
}
RIS
TY - UNPB
TI - Global weak solutions for a two-species Riesz cross-attraction system with quadratic diffusion
AU - Zhenxing Liu
AU - Shen Bian
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25818
UR - https://arxiv.org/abs/2610.25818
AB - We establish global existence for a two-species cross-attraction system on ℝ^3, with quadratic diffusion and the common interaction operator (−Δ)^−s, where 1/2<s<1. Nonnegative initial data in L^1∩ L^∞ with finite second moments require no smallness condition on the masses. A joint energy and second-moment estimate and a coupled L^p iteration give uniform a priori estimates for the regularized solutions. Compactness then yields mass-conserving global weak solutions, bounded on every finite time interval and strongly continuous in L^1∩ L^2. The energy inequality holds at every pair of time endpoints. Uniqueness and bounds uniform for all positive times are not asserted.
ER -