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 [Formula: see text] ([Formula: see text]). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier–Stokes system with general initial data, uniformly in the diffusion range [Formula: see text]. In particular, this existence result does not require additional structural assumptions on the chemotactic sensitivity [Formula: see text] and the oxygen consumption rate [Formula: see text], and remains valid in arbitrary dimensions. Then, we rigorously justify the Hele–Shaw limit for this system as [Formula: see text], 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.