A Keller-Segel-fluid system with singular sensitivity: Generalized solutions

TL;DR

This paper establishes the global existence of generalized solutions for a Keller-Segel-fluid system with singular sensitivity in two and three dimensions, preventing blow-up into Dirac-type singularities under certain conditions.

Contribution

It extends the analysis of Keller-Segel-fluid systems by proving global solutions with singular sensitivity, avoiding blow-up phenomena in bounded domains.

Findings

Global solutions exist for hi<0; in 2D, for all hi; in 3D, for hi<5/3.

Blow-up into Dirac-type singularities is excluded for the solutions.

The results apply to bounded smooth domains in  and  dimensions.

Abstract

In bounded smooth domains $\Omega\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} 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 &= \Delta u + \nabla P + n\nabla \phi, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if \[ \chi<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases} \] These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.