Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion

TL;DR

This paper proves the global existence of weak solutions for a coupled Keller-Segel and fluid dynamics system, modeling bacteria movement in fluid with oxygen consumption, in three dimensions.

Contribution

It establishes the first global-in-time existence results for weak solutions to this complex coupled system, including the case with the Stokes approximation.

Findings

Global weak solutions exist in 3D for the coupled system.

Bounded weak solutions exist when using the Stokes system.

Results extend understanding of chemotaxis-fluid interaction models.

Abstract

We consider a coupled system consisting of the Navier-Stokes equations and a porous medium type of Keller-Segel system that model the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global-in-time existence of weak solutions for the Cauchy problem of the system in dimension three. In addition, if the Stokes system, instead Navier-Stokes system, is considered for the fluid equation, we prove that bounded weak solutions exist globally in time.