Global existence of suitable weak solutions to the 3D chemotaxis-Navier-Stokes equations

TL;DR

This paper proves the global existence of suitable weak solutions for the 3D chemotaxis-Navier-Stokes equations, advancing understanding of fluid dynamics coupled with chemotactic behavior.

Contribution

It establishes the existence of suitable weak solutions for the 3D chemotaxis-Navier-Stokes system using partial regularity theory and local energy inequalities.

Findings

Existence of suitable weak solutions is proven.

Addresses singularity properties of vortices in chemotaxis-fluid models.

Advances mathematical understanding of coupled chemotaxis and fluid flow.

Abstract

In 2004, Dombrowski et al. showed that suspensions of aerobic bacteria often develop flows from the interplay of chemotaxis and buoyancy, which is described as the chemotaxis-Navier-Stokes model, and they observed self-concentration occurs as a turbulence by exhibiting transient, reconstituting, high-speed jets, which entrains nearby fluid to produce paired, oppositely signed vortices. In order to investigate the properties of these vortices (singular points), one approach is to follow the partial regularity theory of Caffarelli-Kohn-Nirenberg to study the singularity properties of suitable weak solutions. In this paper, we established the existence of suitable weak solution for the three dimensional chemotaxis-Navier-Stokes equations, where the main difficulty is to establish appropriate local energy inequalities.