Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension

TL;DR

This paper introduces a two-species tumour growth model coupled through Brinkman's law, establishes solution existence, and analyzes the incompressible limit using a novel BV-estimate approach in one dimension.

Contribution

It presents a new coupling mechanism via Brinkman's law in a two-species tumour model and a novel BV-estimate method for the incompressible limit in one dimension.

Findings

Existence of solutions to the coupled tumour model.

Successful analysis of the incompressible limit as pressure stiffness increases.

Introduction of a BV-estimate based approach differing from previous kinetic reformulations.

Abstract

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.