Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a phase-segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson–Bénilan estimates cannot be established in our context. We are led, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an $$L^1$$ L 1 version in place of the standard upper bound.

中文翻译：

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活组织的两个反应（交叉）扩散方程系统的 Hele-Shaw 极限

多相机械模型现在通常用于描述活组织，包括肿瘤生长。我们在这里研究的具体模型由两个混合抛物线型和双曲线型方程组成，它们扩展了标准的可压缩多孔介质方程，包括交叉反应项。我们研究了不可压缩极限，当压力变得僵硬时，就会产生一个自由边界问题。我们建立了互补关系和相分离结果。在两个物种的情况下出现了几个主要的数学难题。首先，系统结构使比较原则失效。其次，隔离层和内部层将某些数量的可用规律性限制为 BV。第三，不能在我们的上下文中建立 Aronson-Bénilan 估计。我们被引导，因为它是经典的，添加校正项。此过程需要基于仅在一个空间维度中有效的 BV 估计值的技术操作。另一个新颖之处是建立一个 $$L^1$$ L 1 版本来代替标准上限。