Free boundary limit of a tumor growth model with nutrient

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-Benilan estimate in L-infinity 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. (C) 2021 Elsevier Masson SAS. All rights reserved.

Automated summary

The text discusses the application of compressible and incompressible porous medium models in describing the mechanical properties of living tissues, and how these two models can be related using a stiff pressure law. It also highlights the challenges of deriving the free boundary limit and establishing the complementarity relation to recover pressure, with the introduction of two new ideas to address the issues.