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Singular limits in mechanical models of tissue growth
Abstract
Based on the mechanical viewpoint that living tissues exhibit a fluid-like behavior, PDE models inspired by fluid dynamics are now 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 (viscoelastic models) or Darcy’s law (porous medium equations, PME). Furthermore, the stiffness of the pressure law plays a crucial role in distinguishing density-based (compressible) models from free-boundary (incompressible) problems, in which the density is saturated. In this talk, I will show how to connect different mechanical models of living tissues through singular limits. In particular, I will discuss the inviscid limit toward the PME, the incompressible limit of the PME leading to free-boundary problems of the Hele-Shaw type, and finally the joint limit.
