Well-posedness and singularity formation for inviscid Keller-Segel-fluid system of consumption type

TL;DR

This paper studies a coupled Keller-Segel-fluid system, proving local well-posedness and finite-time blow-up of oxygen density, revealing complex dynamics in consumption-driven chemotaxis-fluid interactions.

Contribution

It establishes local well-posedness for the inviscid Keller-Segel-fluid system and demonstrates finite-time blow-up of oxygen density under certain conditions.

Findings

Local well-posedness in Sobolev spaces for the system

Finite time blow-up of oxygen density's $C^{2}$-norm

Additional initial data assumptions needed for zero-density cases

Abstract

We consider the Keller-Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its $C^{2}$--norm with certain initial data.