Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type

TL;DR

This paper proves the global existence of smooth solutions for a 2D Keller-Segel-Euler system modeling bacteria movement in fluids, extending previous results from Navier-Stokes to Euler equations under small initial oxygen density.

Contribution

It establishes the global well-posedness of the coupled Keller-Segel-Euler system in 2D for small initial oxygen density, improving upon prior results with Navier-Stokes coupling.

Findings

Global smooth solutions exist if initial oxygen density is small.

Local existence of solutions is valid for arbitrary smooth initial data.

The proof uses $W^{1,q}$-energy estimates inspired by Boussinesq systems.

Abstract

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in $\mathbb{R}^2$. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the $W^{1,q}$-energy estimate $(q>2)$ motivated by the partially inviscid two-dimensional Boussinesq system in \cite{C06}. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.