AI 中文总结
针对阿尔茨海默病相关的小胶质细胞-淀粉样蛋白趋化模型，引入适配测度边界源的弱解概念，构造有限体积格式并证明其离散解收敛到连续问题的弱解。
AI 中文摘要
本研究针对受阿尔茨海默病中小胶质细胞向β-淀粉样蛋白相关信号募集启发的抛物-抛物趋化系统展开研究。该信号在边界的血管部分受到非负Radon测度值Neumann流入，而小胶质细胞对信号的非局部空间平均产生响应。对于固定感知长度σ>0，趋化速度为b_σ[v]=∇K_σ[v]；对于每个固定σ>0，非局部算子将有限信号质量映射为有界空间Lipschitz速度场。我们引入适配边界测度诱导的低正则性的弱解概念，并构造全隐式迎风格式有限体积近似，其中边界源通过其在每个边界面-时间单元上的精确质量进行离散。我们证明了离散解的存在性与正性，以及一致质量、能量、离散梯度和紧性估计，最终证明离散解的子序列收敛到连续问题的非负弱解。
英文摘要
We study a parabolic-parabolic chemotaxis system motivated by microglial recruitment toward an amyloid-$β$-associated signal in Alzheimer's disease. The signal is subject to a nonnegative Radon measure-valued Neumann influx on a vascular portion of the boundary, while microglial cells respond to a nonlocal spatial average of the signal. For a fixed sensing length $σ>0$, the chemotactic velocity is $b_σ[v]=\nabla K_σ[v]$. For every fixed $σ>0$, the nonlocal operator maps finite signal mass into a bounded spatially Lipschitz velocity field. We introduce a weak-solution concept adapted to the low regularity induced by the boundary measure and construct a fully implicit upwind finite-volume approximation in which the boundary source is discretized through its exact mass on each boundary face-time cell. We establish existence and positivity of discrete solutions, together with uniform mass, energy, discrete-gradient, and compactness estimates. Finally, we prove subsequential convergence of the discrete solutions toward a nonnegative weak solution of the continuous problem.
Comments29 pages, 2 tables, 1 four-panel figure
