Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Based on the mechanical viewpoint that living tissues present a fluid-like behaviour, Partial Differential Equations models inspired by fluid dynamics are nowadays well established as one of the main mathematical tools for the macroscopic description of tissue growth. Depending on the type of tissue, these models link the pressure to the velocity field using either Brinkman’s law (visco-elastic models) or Darcy’s law (porous-medium equations (PME)). Moreover, the stiffness of the pressure law plays a crucial role in distinguishing density-based (compressible) models from free boundary (incompressible) problems where saturation of the density holds. This mechanical description of cell movement is also similarly applied to models of crowd motion.
In this talk, I will present how we can relate different mechanical models of living tissues through singular limits. In particular, I will show how the inviscid limit of Brinkman's law leads to the PME, how the incompressible limit links the PME to a Hele-Shaw free boundary problem, as well as how to pass to the joint limit.