A result on the existence and uniqueness of stationary solutions for a bioconvective flow model

TL;DR

This paper proves the existence and uniqueness of weak solutions for a stationary bioconvective flow model, providing mathematical validation for the model's well-posedness.

Contribution

It establishes the first rigorous proof of existence and uniqueness for solutions to the stationary bioconvective flow boundary value problem.

Findings

Existence of weak solutions is proven using a priori estimates and Gossez theorem.

Uniqueness of solutions is demonstrated through comparison methods.

The results ensure the mathematical soundness of the bioconvective flow model.

Abstract

In this note we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem introduced by Tuval et. al. (2005, {\it PNAS} 102, 2277--2282). We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.