发表机构
TU Wien（维也纳工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带体积填充效应的Keller--Segel系统，该研究分析三种渐近 regime 的收敛性，分别通过能量-熵估计、动力学约化、BV紧性得到所需的密度强紧性，明确了不同条件下的极限方程。
AI 中文摘要
我们研究一类带多孔介质扩散和体积填充灵敏度函数$u(1-u)$的抛物-椭圆型Keller--Segel系统的三种渐近 regime。首先，当$D=\epsilon^2\to0$、$\delta=1$、$1\le m<2$且趋化灵敏度系数低于依赖于$m$的显式抛物性阈值时，我们证明其强收敛到一个标量非线性扩散方程。其次，当$\delta=\epsilon\to0$且$D=1$时，我们通过约化动力学方法得到双曲-椭圆型Keller--Segel极限。第三，在标度$\chi=\epsilon^{-1}$、$D=\epsilon^2$、$\delta=1$且具有有限周长的特征初值条件下，我们证明在固定时间区间上收敛到$BV(\Omega;\{0,1\})$中的不随时间变化的初始斑块。由于灵敏度函数关于密度非线性，仅弱收敛不足以确定趋化通量，因此三种 regime 均需要密度的强紧性，分别通过能量-熵估计、动力学约化和$BV$紧性获得。
英文摘要
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 $D=ε^2\to0$, $δ=1$, $1\le m<2$, 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 $δ=ε\to0$ 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.
