# Mathematical Modeling of Biofilm Development

## TL;DR

This paper develops a complex mathematical model for biofilm growth involving coupled nonlinear equations, including fluid flow with obstacle constraints, advancing the theoretical understanding of biofilm development.

## Contribution

It introduces a novel coupled system of nonlinear equations with variational inequalities to model biofilm development, addressing solvability challenges.

## Key findings

- Model captures biomass growth and fluid flow interactions
- Addresses mathematical challenges of degenerate and singular diffusion
- Provides a framework for future analytical and numerical studies

## Abstract

We perform mathematical anaysis of the biofilm development process. A model describing biomass growth is proposed: It arises from coupling three parabolic nonlinear equations: a biomass equation with degenerate and singular diffusion, a nutrient tranport equation with a biomass-density dependent diffusion, and an equation of the Navier-Stokes type, describing the fluid flow in which the biofilm develops. This flow is subject to a biomass--density dependent obstacle. The model is treated as a system of three inclusions, or variational inequalities; the third one causes major difficulties for the system's solvability. Our approach is based on the recent development of the theory on Navier-Stokes variational inequalities.