Global solutions for chemotaxis-Navier-Stokes system with Robin boundary conditions

TL;DR

This paper establishes the existence of global solutions for a chemotaxis-Navier-Stokes system with Robin boundary conditions, modeling cellular swimming with oxygen exchange, in both two and three dimensions.

Contribution

It proves global classical solutions in 2D and global weak solutions in 3D without requiring domain convexity, using boundary energy methods.

Findings

Unique global classical solutions in 2D.

Global weak solutions in 3D with bounded energy.

Removal of convexity assumption on the domain.

Abstract

We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account. This phenomenon results in an inhomogeneous Robin-type boundary condition. Moreover, the system is studied without the logistic growth of the bacteria population. We prove that in two dimensions, the system has a unique global classical solution, while the existence of a global weak solution is shown in three dimensions. In the latter case, we show that the energy is bounded uniformly in time. A key idea is to utilise a boundary energy to derive suitable {\it a priori} estimates. Moreover, we are able to remove the convexity assumption on the domain.