Hele-Shaw limit of chemotaxis-Navier-Stokes flows

This paper investigates the connection between the chemotaxis–Navier–Stokes system with porous medium type nonlinear diffusion and the Hele–Shaw problem in ${\mathbb{R}}^d$ ($d\ge 2$). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier-Stokes system with the general initial data, uniformly in the diffusion range $m\in [3,\mathrm{\infty })$. Then, we rigorously justify the Hele–Shaw limit for this system as $m\to \mathrm{\infty }$, showing the convergence to a free boundary problem of Hele–Shaw type, where the bacterium (cell) diffusion is governed by the stiff pressure law. Moreover, the complementarity relation characterizing the limiting bacterium (cell) pressure via a degenerate elliptic equation is verified by a novel application of the Hele–Shaw framework.