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.
Highlights
In recent years there has been an increasing interest in multi-phase models applied to tumour growth
Tumour growth was modelled using a single equation describing the evolution of the abnormal cell density
Schmidtchen where n(i) represents the normal cells, for i = 1, 2, and k ∈ N is a given constant modelling the stiffness of the total population pressure, pk, which is generated by both species, i.e., pk n(k1) + n(k2)
Summary
In recent years there has been an increasing interest in multi-phase models applied to tumour growth. Abnormal) cells, for i = 1, 2, and k ∈ N is a given constant modelling the stiffness of the total population pressure, pk, which is generated by both species, i.e., pk. An easy application of the chain rule in conjunction with Eq (1) leads to The change to these new variables was first introduced in [2,3,4] in the context of a twospecies system where the two species avoid overcrowding. In a way, their works paved the way for more modern approaches to tumour models linked through Darcy’s law, cf [5, 6, 8, 13, 18].
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