Incompressible limit for a two-species Brinkman model with drift

Abstract

We study a two-species model for tissue growth in which both populations are transported by an external drift and by a velocity potential determined through Brinkman's law. The pressure is generated by a stiff constitutive relation depending on the total density. Our main result establishes the incompressible limit as the stiffness exponent tends to infinity, in arbitrary space dimension and for merely integrable initial data. The limit system consists of the two balance laws coupled to Brinkman's equation, the hard-congestion constraint $0\leq n_\infty\leq 1$, the graph relation $p_\infty(1-n_\infty)=0$, and the corresponding complementarity relation. A key point of the analysis is a new $L^2$-based compactness theory that avoids both uniform $L^\infty$-bounds on the pressure and the kinetic reformulation used in earlier approaches. We first construct weak solutions for bounded data and then remove the boundedness assumption by means of a weighted compactness argument inspired by Bresch--Jabin. We also prove an Aubin--Lions--Simon type lemma based on oscillation control, yielding time continuity of the constructed solutions. Finally, a refined dissipation estimate for the Bresch--Jabin compactness functional gives strong compactness of the pressure in the stiff limit and implies a regularising effect: the limiting pressure is bounded even when the approximating pressures are only integrable.