On the Incompressible Limit for a Tumour Growth Model incorporating Convective Effects

TL;DR

This paper analyzes a tumor growth model with convective effects, establishing its incompressible limit to connect density-based models with free-boundary problems, and proves the uniqueness of the limit.

Contribution

It extends a previous tumor growth model by incorporating advective effects and rigorously derives the incompressible limit, linking it to free-boundary problems.

Findings

Established the incompressible limit of the model.

Proved the uniqueness of the limiting free-boundary problem.

Bridged the gap between density-based models and geometric free-boundary models.

Abstract

In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quir\'os, and V\'azquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.