Existence and Uniqueness of Solution of a Continuous Flow Bioreactor Model with Two Species

TL;DR

This paper rigorously analyzes a coupled reaction-diffusion-advection model for a continuous flow bioreactor, establishing existence, uniqueness, and properties of solutions for both linear and nonlinear reactions, relevant for water treatment applications.

Contribution

It provides the first mathematical proof of existence and uniqueness for a coupled bioreactor model with both linear and nonlinear reactions, including solution properties.

Findings

Existence and uniqueness of solutions for linear reactions.

Existence and uniqueness for nonlinear reactions via Schauder Fixed Point.

Solutions are shown to be nonnegative and bounded.

Abstract

In this work, we study the mathematical analysis of a coupled system of two reaction-diffusion-advection equations and Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacterias) and a diluted substrate (e.g., nitrate) in a continuous flow bioreactor. This type of bioreactor can be used, for instance, for water treatment. First, we prove the existence and uniqueness of solution, under the hypothesis of linear reaction by using classical results for linear parabolic boundary value problems. Next, we prove the existence and uniqueness of solution for some nonlinear reactions by applying \textit{Schauder Fixed Point Theorem} and the theorem obtained for the linear case. Results about the nonnegativeness and boundedness of the solution are also proved here.