Global solutions to the free boundary value problem of a chemotaxis-Navier-Stokes system

TL;DR

This paper proves the global existence and uniqueness of solutions for a chemotaxis-Navier-Stokes system on a three-dimensional moving domain with free boundary, addressing a previously unstudied problem in mathematical fluid dynamics and biological modeling.

Contribution

It provides the first analytical proof of well-posedness for the chemotaxis-Navier-Stokes system on a time-dependent domain with free boundary conditions.

Findings

Established global existence and uniqueness of solutions near a constant state.

Addressed boundary conditions matching experimental and numerical setups.

First analytical work on chemotaxis-Navier-Stokes on a moving domain.

Abstract

In this paper, we investigate the global solvability of the chemotaxis-Navier-Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with boundary conditions that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state $(0,\hat{c},\mathbf{0})$, where $\hat{c}$ is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-Navier-Stokes system on a time-dependent domain.