Global Well-posedness of the Chemotaxis-Navier-Stokes Equations in two dimensions

TL;DR

This paper proves the global existence and decay of solutions for the two-dimensional chemotaxis-Navier-Stokes equations with large initial data, extending understanding of these coupled PDEs in fluid dynamics and biological modeling.

Contribution

It establishes the absence of finite-time blow-up and decay properties for solutions with large initial data in 2D chemotaxis-Navier-Stokes equations, under specific assumptions.

Findings

No finite-time blow-up for large initial data

Solutions decay over time

Results extend previous local well-posedness to global in 2D

Abstract

We consider two dimensional Keller-Segel equations coupled with the Navier-Stokes equations modelled by Tuval et al.[32]. Assuming that the chemotactic sensitivity and oxygen consumption rate are nondecreasing and differentiable, we prove that there is no blow-up in a finite time for solutions with large initial data to chemotaxis-Navier-Stokes equations in two dimensions. In addition, temporal decays of solutions are shown, as time tends to infinity.