AI 中文总结
该研究针对带漂移的两物种Brinkman模型，在任意空间维度及仅可积初始数据下建立不可压缩极限，通过新的L²紧性理论等方法完成分析，得到极限系统的相关性质与正则化效应。
AI 中文摘要
我们研究用于组织生长的两物种模型，其中两种群均受外部漂移和由Brinkman定律确定的速度势输运，压力由依赖于总密度的刚性本构关系产生。我们的主要结果是：当刚性指数趋于无穷大时，在任意空间维度且初始数据仅可积的情况下，建立不可压缩极限；极限系统包含两个平衡律与Brinkman方程耦合、硬阻塞约束0≤n_∞≤1、图关系p_∞(1−n_∞)=0以及相应的互补关系。分析的关键是一种新的基于L²的紧性理论，该理论避免了对压力的一致L^∞界以及早期方法中使用的动力学重述。我们首先为有界数据构造弱解，然后借助受Bresch-Jabin启发的加权紧性论证消除有界性假设；还证明了基于振荡控制的Aubin-Lions-Simon型引理，得到所构造解的时间连续性。最后，对Bresch-Jabin紧性泛函的精细耗散估计给出刚性极限中压力的强紧性，并蕴含正则化效应：即使近似压力仅可积，极限压力也是有界的。
英文摘要
We study a two-species model for tissue growth in which both populations are transported by an external drift and by a velocity potential determined through Brinkman's law. The pressure is generated by a stiff constitutive relation depending on the total density. Our main result establishes the incompressible limit as the stiffness exponent tends to infinity, in arbitrary space dimension and for merely integrable initial data. The limit system consists of the two balance laws coupled to Brinkman's equation, the hard-congestion constraint $0\leq n_\infty\leq 1$, the graph relation $p_\infty(1-n_\infty)=0$, and the corresponding complementarity relation. A key point of the analysis is a new $L^2$-based compactness theory that avoids both uniform $L^\infty$-bounds on the pressure and the kinetic reformulation used in earlier approaches. We first construct weak solutions for bounded data and then remove the boundedness assumption by means of a weighted compactness argument inspired by Bresch--Jabin. We also prove an Aubin--Lions--Simon type lemma based on oscillation control, yielding time continuity of the constructed solutions. Finally, a refined dissipation estimate for the Bresch--Jabin compactness functional gives strong compactness of the pressure in the stiff limit and implies a regularising effect: the limiting pressure is bounded even when the approximating pressures are only integrable.
