Singular sensitivity in a Keller-Segel-fluid system

TL;DR

This paper proves the global existence of classical solutions for a chemotaxis-fluid system with singular sensitivity in bounded domains, under specific conditions on parameters and fluid types, extending understanding of such coupled PDE models.

Contribution

It establishes the first global existence results for a Keller-Segel-fluid system with singular sensitivity in bounded domains for certain dimensions and fluid regimes.

Findings

Global classical solutions exist under specified conditions.

Singular sensitivity does not prevent solution existence.

Results apply to both Stokes and Navier-Stokes fluids in 2D and 3D.

Abstract

In bounded smooth domains $\Omega\subset\mathbb{R}^N$, $N\in\{2,3\}$, considering the chemotaxis--fluid system \[ \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= \Delta n - \chi \nabla \cdot(\frac{n}{c}\nabla c) &\\ & c_t + u\cdot \nabla c &= \Delta c - c + n &\\ & u_t + \kappa (u\cdot \nabla) u &= \Delta u + \nabla P + n\nabla \Phi & \end{split}\end{cases} \] with singular sensitivity, we prove global existence of classical solutions for given $\Phi\in C^2(\bar{\Omega})$, for $\kappa=0$ (Stokes-fluid) if $N=3$ and $\kappa\in\{0,1\}$ (Stokes- or Navier--Stokes fluid) if $N=2$ and under the condition that \[ 0<\chi<\sqrt{\frac{2}{N}}. \]