论文标题
肿瘤生长模型的自由边界极限
Free boundary limit of tumor growth model with nutrient
论文作者
论文摘要
文献中都使用可压缩和不可压缩的多孔培养基模型来描述活组织的机械性能。这两类模型可以使用僵硬的压力定律相关。在不可压缩的极限中，可压缩模型会产生Hele-Shaw类型的自由边界问题，其中不可压缩性在饱和阶段。在这里，我们认为有营养的情况。然后，一个不良耦合的方程系统描述了细胞密度的数量和养分浓度。因此，自由边界（不可压缩）极限的推导是一个开放的问题，特别是一个困难是建立所谓的互补关系，该关系允许使用椭圆方程来恢复压力。为了建立限制，我们使用两个新想法。最近也用于相关问题的第一个想法是将通常的Aronson-Bénilan估计量扩展到$ l^\ infty $的$ l^2 $设置。第二个想法是在压力梯度上得出尖锐的均匀$ l^4 $估计，而与空间尺寸无关。
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 estimates 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 space dimension.
