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.

在本文中，我们研究了在生物对流问题的数学模型中确定密度函数的逆问题，该函数对粒子线性动量的矢量外部源进行建模。该模型由三个方程组成：粒子的线性动量、微生物守恒定律和不可压缩条件。我们分析了获得适定性结果的直接问题。我们证明了一般假设下弱解的存在性以及特定类别密度函数弱解的唯一性。为了解决反问题，我们假设给出了积分超规格条件。然后，我们证明了反问题的局部唯一性。该证明基于使用第二类算子方程对反演问题解的表征，