Three asymptotic regimes for a Keller--Segel system with volume-filling effect
Abstract
We study three asymptotic regimes for a parabolic--elliptic Keller--Segel system with porous-medium diffusion and the volume-filling sensitivity function u(1−u). First, when D=ε2→0, δ=1, 1≤ m<2, and the chemotactic sensitivity coefficient is below an explicit m-dependent parabolicity threshold, we prove strong convergence to a scalar nonlinear diffusion equation. Second, as δ=ε→0 with D=1, we obtain a hyperbolic--elliptic Keller--Segel limit by means of a kinetic reduction argument. Third, under the scaling χ=ε−1, D=ε2, δ=1, and for characteristic initial data of finite perimeter, we prove convergence on fixed time intervals to the time-independent initial patch in BV(Ω;{0,1}). Because the sensitivity function is nonlinear in the density, weak convergence alone is insufficient to identify the chemotactic flux. Strong compactness of the density is therefore required in all three regimes and is obtained, respectively, through energy--entropy estimates, kinetic reduction, and BV compactness.
Related
- CAT(0) spaces without linear filling at the asymptotic rank
- A posteriori error estimates for a finite volume approximation of a local sensing Keller-Segel model
- Global Well-posedness and Asymptotic Extinction for Keller-Segel Systems with Superlinear Noise
- Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions
- Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces
BibTeX
@misc{zhang2026three,
title = {{Three asymptotic regimes for a Keller--Segel system with volume-filling effect}},
author = {Mingyue Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04599},
eprint = {2609.04599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04599},
}
RIS
TY - UNPB
TI - Three asymptotic regimes for a Keller--Segel system with volume-filling effect
AU - Mingyue Zhang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04599
UR - https://arxiv.org/abs/2609.04599
AB - We study three asymptotic regimes for a parabolic--elliptic Keller--Segel system with porous-medium diffusion and the volume-filling sensitivity function u(1−u). First, when D=ε^2→0, δ=1, 1≤ m<2, and the chemotactic sensitivity coefficient is below an explicit m-dependent parabolicity threshold, we prove strong convergence to a scalar nonlinear diffusion equation. Second, as δ=ε→0 with D=1, we obtain a hyperbolic--elliptic Keller--Segel limit by means of a kinetic reduction argument. Third, under the scaling χ=ε^−1, D=ε^2, δ=1, and for characteristic initial data of finite perimeter, we prove convergence on fixed time intervals to the time-independent initial patch in BV(Ω;{0,1}). Because the sensitivity function is nonlinear in the density, weak convergence alone is insufficient to identify the chemotactic flux. Strong compactness of the density is therefore required in all three regimes and is obtained, respectively, through energy--entropy estimates, kinetic reduction, and BV compactness.
ER -