Both compressible and incompressible porous medium models are used in the literature to describe the mechanical properties of living tissues. These two classes of models can be related using a stiff pressure law. In the incompressible limit, the compressible model generates a free boundary problem of Hele-Shaw type where incompressibility holds in the saturated phase.

Here we consider the case with a nutrient. Then, a badly coupled system of equations describes the cell density number and the nutrient concentration. For that reason, the derivation of the free boundary (incompressible) limit was an open problem, in particular a difficulty is to establish the so-called complementarity relation which allows to recover the pressure using an elliptic equation. To establish the limit, we use two new ideas. The first idea, also used recently for related problems, is to extend the usual Aronson-Bénilan estimate in $L^{\infty }$ to an $L^2$ setting. The second idea is to derive a sharp uniform $L^4$ estimate on the pressure gradient, independently of the space dimension.

文献中使用可压缩和不可压缩多孔介质模型来描述活组织的机械特性。这两类模型可以使用刚性压力定律相关联。在不可压缩极限下，可压缩模型生成 Hele-Shaw 类型的自由边界问题，其中不可压缩性在饱和相中成立。

在这里，我们考虑具有营养素的情况。然后，一个严重耦合的方程组描述了细胞密度数和营养浓度。出于这个原因，自由边界（不可压缩）极限的推导是一个开放的问题，特别是难点是建立所谓的互补关系，允许使用椭圆方程恢复压力。为了建立极限，我们使用了两个新想法。最近也用于相关问题的第一个想法是扩展通常的 Aronson-Bénilan 估计${升}^{\infty }$ 到 ${升}^2$环境。第二个想法是推导出锐利的制服${升}^4$ 估计压力梯度，独立于空间维度。