We present a two-species model with applications in tumour modelling. The main novelty is the coupling of both species through the so-called Brinkman law which is typically used in the context of visco-elastic media, where the velocity field is linked to the total population pressure via an elliptic equation. The same model for only one species has been studied by Perthame and Vauchelet in the past. The first part of this paper is dedicated to establishing existence of solutions to the problem, while the second part deals with the incompressible limit as the stiffness of the pressure law tends to infinity. Here we present a novel approach in one spatial dimension that differs from the kinetic reformulation used in the aforementioned study and, instead, relies on uniform BV-estimates.

中文翻译：

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一维通过布林克曼定律耦合的两种肿瘤模型的不可压缩极限

我们提出了两种模型在肿瘤建模中的应用。主要的新颖性是通过所谓的布林克曼定律（Brinkman law）将两个物种耦合在一起，该定律通常用于粘弹性介质中，其中，速度场通过椭圆方程与总人口压力相关联。过去，Perthame和Vauchelet研究了仅针对一种物种的相同模型。本文的第一部分致力于建立解决问题的方法，而第二部分则处理不可压缩极限，因为压力定律的刚度趋于无穷大。在这里，我们提出一种新颖的方法，它在一个空间维度上不同于前述研究中使用的动力学重构，而是依靠统一的BV估计。