Incompressible Limit for a Two-Species Tumour Model with Coupling Through Brinkman’s Law in One Dimension

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.

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

“…The rigorous derivation is already known in one spatial dimension, cf. [14] and, for one species, in any dimension, cf. [25].…”

“…In both works the authors emphasised the possibility of jump discontinuities in the pressure which renders the problem of obtaining compactness rather challenging. An extension of the strategy of [14] to higher dimensions appears futile, as does the extension of [25] to two species due to the contribution of the individual species and their role in the identification of weak-⋆ limits in the kinetic reformulation. The subsequent sections are concerned with the proof of the main theorem.…”

“…The total population plays an important role in the analysis of these types of models that was exploited in many related results, cf. [11,14]. It is easily verified that the total population density, n k , satisfies ∂n k ∂t − ∇ • (n k ∇W k ) = n k r k G (1) (p k ) + (1 − r k )G (2) (p k ) , (1) where r k denotes the concentration…”

“…Two of the authors were able to establish the incompressible limit in the one dimensional case, cf. [14]. Their proof mainly relies on establishing uniform BV -bounds for the two species as well as the total population.…”

“…At first glance, this approach may seem rather absurd and the reader might wonder what could possibly be gained by incorporating the total population into the estimates that has not already been obtained from the two individual species. However, this strategy already proved most useful in [14] as certain cancellations can be obtained in doing so. Having established the strong compactness of the individual species, we address the compactness of the pressure in Section 5.…”