Hele-Shaw limit of chemotaxis-Navier-Stokes flows

TL;DR

This paper establishes the global existence of solutions for a chemotaxis-Navier-Stokes system with nonlinear diffusion and rigorously derives its limit as the diffusion becomes infinitely stiff, leading to a Hele-Shaw free boundary problem.

Contribution

It proves global solutions for the chemotaxis-Navier-Stokes system with general initial data and justifies the Hele-Shaw limit as the diffusion parameter tends to infinity.

Findings

Existence of global weak solutions for the system.

Convergence to Hele-Shaw free boundary problem as diffusion becomes stiff.

Verification of the complementarity relation in the limit.

Abstract

This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). 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 $m\in [3,\infty)$. Then, we rigorously justify the Hele--Shaw limit for this system as $m\rightarrow\infty$, 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.