发表机构
Institute of Natural Sciences, Shanghai Jiaotong University; School of Physics and Astronomy, Shanghai Jiaotong University; Yau Mathematical Sciences Center, Tsinghua University; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications; School of Mathematics, Shanghai Jiaotong University; School of Mathematics and Computational Science, Xiangtan University; Department of Mathematical Sciences, Tsinghua University; Department of Mathematics, City University of Hong Kong（上海交通大学自然科学研究院; 上海交通大学物理与天文学院; 清华大学丘成桐数学科学中心; 北京雁栖湖应用数学研究院; 上海交通大学数学学院; 湘潭大学数学与计算科学学院; 清华大学数学科学系; 香港城市大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究增广Keller-Segel模型中时空依赖扩散和对流系数的逆问题，建立条件Lipschitz稳定性、有限时间分离及松弛速率唯一识别，并提出两阶段梯度重建策略，数值实验验证其可行性。
AI 中文摘要
我们研究了一个增广Keller-Segel模型中的逆系数问题，该模型源于对趋光性和趋化性种群动力学的描述。种群密度的时空演化由对流-扩散方程控制，其中扩散系数和对流系数均依赖于空间和时间。由于潜在的细胞内适应机制，即使在时间无关的外部环境中，扩散系数\(D(x,t)\)和标量对流系数\(K(x,t)\)仍然保持时间依赖性，并分别指数松弛至其各自的稳态分布。重要的是，它们的松弛由相同的速率控制，该速率由细胞内适应动力学决定。根据密度的内部测量，我们建立了条件Lipschitz稳定性，用于在一个系数已知时恢复另一个系数，并推导了交叉敏感性估计，量化了产生相同密度数据的扩散和对流扰动之间的补偿。我们进一步确定了瞬态和稳态之间的有限时间分离：在后期，完整动力学允许指数精确的冻结系数近似。当稳态系数已知时，我们还证明了从解的长时间行为中唯一识别松弛速率。最后，我们推导了正演映射的显式线性化，并受上述时间分离的启发，开发了一种两阶段基于梯度的重建策略。数值实验证明了所提出方法的可行性。
英文摘要
We investigate an inverse coefficient problem for an augmented Keller--Segel model arising in the description of phototactic and chemotactic population dynamics. The spatiotemporal evolution of the population density is governed by an advection-diffusion equation in which both the diffusion coefficient and the advection coefficient depend on space and time. Due to the underlying intracellular adaptation mechanism, even in a time-independent external environment, the diffusion coefficient \(D(x,t)\) and the scalar advection coefficient \(K(x,t)\) remain time-dependent and relax exponentially toward their respective steady-state profiles. Importantly, their relaxation is governed by the same rate, determined by the intracellular adaptation dynamics. From internal measurements of the density, we establish conditional Lipschitz stability for recovering either coefficient when the other is known, and derive cross-sensitivity estimates quantifying the compensation between diffusion and advection perturbations that produce the same density data. We further identify a finite-time separation between the transient and steady regimes: the full dynamics admit an exponentially accurate frozen-coefficient approximation at late times. When the steady-state coefficients are known, we also prove unique identification of the relaxation rate from the long-time behavior of the solution. Finally, we derive an explicit linearization of the forward map and, motivated by the temporal separation above, develop a two-stage gradient-based reconstruction strategy. Numerical experiments demonstrate the feasibility of the proposed method.
