AI 中文总结
本文首次严格研究受限活性物质中跑 tumble 模型的非线性边界层，证明了其消失扩散率极限的定量收敛速率，揭示边界质量的动态边界条件耦合机制，数值上呈现边界层的相变与滞后行为。
AI 中文摘要
受限活性物质系统的一个显著特征是运动粒子倾向于在固体边界附近积累。在各种无通量边界条件的线性模型中，这种积累通过在粒子扩散率κ较小时形成尖锐边界层实现。本文首次对受限活性物质背景下的非线性边界层进行严格研究，具体考虑半直线ℝ₊上具有非线性平流和翻滚的一维跑 tumble 模型族，严格证明了消失扩散率极限的定量收敛速率。在极限系统中，边界质量作为新变量求解非线性常微分方程（ODE），并通过动态边界条件与偏微分方程（PDE）耦合。有趣的是，κ=0时的边界非线性性需参考κ≪1时的边界层分析才能得到。数值上，这些模型呈现丰富行为，包括边界层中的相变与滞后现象。
英文摘要
A notable feature of confined active matter systems is the tendency for motile particles to accumulate near solid boundaries. In various linear models with no-flux boundary conditions, this accumulation is realized through the development of sharp boundary layers at small particle diffusivity $κ$. In this paper, we present the first rigorous investigation of nonlinear boundary layers in the context of confined active matter. Specifically, we consider a family of 1D run-and-tumble models with nonlinear advection and tumbling on the half-line $\mathbb{R}_+$. We rigorously prove the vanishing diffusivity limit with quantitative convergence rates. In the limiting system, the boundary mass enters as a new variable which solves a nonlinear ODE, coupled to the PDE through a dynamic boundary condition. Interestingly, the nonlinearity on the boundary at $κ= 0$ cannot be obtained without reference to the boundary layer analysis at $κ\ll 1$. Numerically, these models exhibit rich behavior, including phase transition and hysteresis in the boundary layer.
Comments57 pages, 3 figures
