Application of Tikhonov fixed point theorem to analyze an inverse problem for a bioconvective flow model

In this paper, we study the inverse problem of determining the density function modeling the vector external source for the linear momentum of particles, in a mathematical model for the bioconvective flow problem. The model consists of three equations: linear momentum of particles, a conservation law for the microorganisms, and the incompressibility condition. We analyze the direct problem obtaining results for the well posedness. We prove the existence of weak solutions under general assumptions and the uniqueness of weak solutions for a particular class of density functions. To solve the inverse problem, we assume that an integral overspecification condition is given. Then, we prove the local uniqueness of the inverse problem. The proof is based on the characterization of the inverse problem solutions using an operator equation of second kind, the introduction of several a priori estimates, and the application of the Tikhonov fixed point theorem.