On the singular limit of Brinkman's law to Darcy's law
Noemi David, Matt Jacobs, Inwon Kim
TL;DR
This work addresses the singular joint limit from a compressible Brinkman model to incompressible Darcy/Hele–Shaw dynamics in congestion-averse tissue growth by developing a general energy evolution framework. The authors introduce a flexible pair of convex energies $e_\nu$ and their pressure counterparts $z_\nu$ to derive a family of energy evolution equations (EEE) that yield uniform dissipation and strong compactness, enabling passage to the limit in nonlinear terms. They prove that, as $\nu\to 0$, the limiting density $\rho$ and pressure $p$ satisfy $\partial_t \rho - \nabla\cdot(\rho\nabla p)=\rho G(p)$ with $p\in\partial f_0(\rho)$, and they establish $m=\rho\nabla p$ and strong convergence $\nabla w_\nu \to \nabla p$, along with $R=\rho G(p)$. The results link the compressible Brinkman problem, compressible Darcy, and the incompressible Hele–Shaw limit, providing a robust framework for joint singular limits in congestion-penalized growth models and offering insight into energy-dissipation structures beyond this particular setting.
Abstract
In this paper we study singular limits of congestion-averse growth models, connecting different models describing the effect of congestion. These models arise in particular in the context of tissue growth. The main ingredient of our analysis is a family of energy evolution equations and their dissipation structures, which are novel and of independent interest. This strategy allows us to consider a larger family of pressure laws as well as proving the joint limit, from a compressible Brinkman's model to the incompressible Darcy's law, where the latter is a Hele-Shaw type free boundary problem.