Mathematical Biology Tumor Growth
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Abstract
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.
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@article{He2026Hele,
title = {Hele–Shaw limit of chemotaxis-Navier–Stokes flows},
author = {Qingyou He and Ling-Yun Shou and Leyun Wu},
journal = {Mathematical Models and Methods in Applied Sciences},
year = {2026},
doi = {10.1142/s021820252650051x},
url = {https://doi.org/10.1142/s021820252650051x}
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