On the inviscid limit connecting Brinkman's and Darcy's models of tissue growth with nonlinear pressure

TL;DR

This paper extends the connection between Brinkman's and Darcy's tissue growth models from linear to nonlinear pressure laws, demonstrating an inviscid limit using pressure-potential relations.

Contribution

It proves the inviscid limit connection for nonlinear, power-law pressure models, advancing understanding of multi-phase tissue growth models.

Findings

Established the inviscid limit for nonlinear pressure models

Used pressure-potential relations to deduce compactness

Extended previous linear pressure results to nonlinear case

Abstract

Several recent papers have addressed modelling of the tissue growth by the multi-phase models where the velocity is related to the pressure by one of the physical laws (Stoke's, Brinkman's or Darcy's). While each of these models has been extensively studied, not so much is known about the connection between them. In the recent paper (arXiv:2303.10620), assuming the linear form of the pressure, the Authors connected two multi-phase models by an inviscid limit: the viscoelastic one (of Brinkman's type) and the inviscid one (of Darcy's type). Here, we prove that the same is true for a nonlinear, power-law pressure. The new ingredient is that we use relation between the pressure $p$ and the Brinkman potential $W$ to deduce compactness in space of $p$ from the compactness in space of $W$.