Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

Abstract

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result.Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-Bénilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L 1 version in place of the standard upper bound.

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

“…In stark contrast, Figure 1(d) shows an almost smooth transition of the pressure indicating a much higher regularity. This is in perfect alignment with the findings of [6], as the case ν = 0 yields, at least formally, the system studied in the latter. As a matter of fact, the pressure gradient was shown to be square-integrable in the Darcy case, i.e., ν = 0.…”

“…Even in the case ν = 0 corresponding to the inviscid case, the system nature of the problem gives rise to a whole range of difficulties, cf. [6,8,13]. At first glance, the pressure gains in regularity, however, it gains just enough regularity to obtain compactness of its gradient, requiring a minute derivation of suitable estimates.…”

“…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].…”

Global existence theorem for a model governing the motion of two cell populations

“…We also refer the reader to [16] and the references therein for numerical results that deal with how the mobilities change the morphology of the interfaces between the two cell populations and analytical study of traveling wave solutions with composite shapes and discontinuities for cell-density models of avascular tumor growth. The limit of (u 1 , u 2 ) as γ → ∞ is very interesting from the modeling perspective ans has been investigated in [3].…”

“…where w p is given as in (H1). We define an operator M from L 2 (Ω T ) 3 into itself as follows: 3 . We first consider the initial boundary value…”