Well posedness of fluid/solid mixture models for biofilm spread

TL;DR

This paper develops a mathematical framework to ensure well-posedness of biofilm spread models involving coupled fluid-solid interactions in moving domains, enabling more accurate and reliable simulations of biological processes.

Contribution

It introduces methods to characterize derivatives of quasi-stationary magnitudes and establishes conditions for well-posedness in moving boundary models of biofilm growth.

Findings

Constructed solutions for transport, diffusion, and elliptic submodels.

Provided conditions for well-posedness in moving domains.

Techniques applicable to similar biological and chemical engineering models.

Abstract

Two phase solid-fluid mixture models are ubiquitous in biological applications. For instance, models for growth of tissues and biofilms combine time dependent and quasi-stationary boundary value problems set in domains whose boundary moves in response to variations in the mechano-chemical variables. For a model of biofilm spread, we show how to obtain better posed models by characterizing the time derivatives of relevant quasi-stationary magnitudes in terms of additional boundary value problems. We also give conditions for well posedness of time dependent submodels set in moving domains depending on the motion of the boundary. After constructing solutions for transport, diffusion and elliptic submodels for volume fractions, displacements, velocities, pressures and concentrations with the required regularity, we are able to handle the full model of biofilm spread in moving domains assuming we know the dynamics of the boundary. These techniques are general and can be applied in models with a similar structure arising in biological and chemical engineering applications.