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Well posedness of fluid/solid mixture models for biofilm spread

Ana Carpio, Gema Duro

TL;DR

This work develops a rigorous framework for well-posedness of moving-domain, two-phase biofilm models that couple quasi-stationary elliptic subproblems with time-dependent transport and boundary dynamics. By introducing domain-derivative techniques to express time derivatives as solutions of auxiliary boundary-value problems, and by reformulating diffusion in a fixed reference frame, the authors derive existence, uniqueness, and regularity results for the submodels and the full coupled system given known boundary motion. They establish precise elliptic and parabolic well-posedness results (including Faedo-Galerkin constructions and $H^k$-regularity) and provide conditions under which volume fractions remain nonnegative and bounded. The results lay a solid mathematical foundation for stable numerical schemes and can be extended to similar moving-domain multiphysics models in biology and chemical engineering.

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.

Well posedness of fluid/solid mixture models for biofilm spread

TL;DR

This work develops a rigorous framework for well-posedness of moving-domain, two-phase biofilm models that couple quasi-stationary elliptic subproblems with time-dependent transport and boundary dynamics. By introducing domain-derivative techniques to express time derivatives as solutions of auxiliary boundary-value problems, and by reformulating diffusion in a fixed reference frame, the authors derive existence, uniqueness, and regularity results for the submodels and the full coupled system given known boundary motion. They establish precise elliptic and parabolic well-posedness results (including Faedo-Galerkin constructions and -regularity) and provide conditions under which volume fractions remain nonnegative and bounded. The results lay a solid mathematical foundation for stable numerical schemes and can be extended to similar moving-domain multiphysics models in biology and chemical engineering.

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.
Paper Structure (10 sections, 96 equations, 2 figures)

This paper contains 10 sections, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Differentiation of quasi-stationary problems
  3. Study of diffusion problems in time dependent domains
  4. Variational formulation in the undeformed configuration
  5. Construction of stable solutions
  6. Well posedness results for the quasi-stationary submodels
  7. Elliptic problems for displacements, velocities and concentrations
  8. Conservation law for volume fractions
  9. Well posedness results for the full model with a known boundary dynamics
  10. Discussion and conclusions

Figures (2)

  • Figure 1: (a) Schematic view of biofilm microscopic structure. Cells are embedded in a network of polymeric threads forming the extracellular matrix (ECM), while a liquid solution containing nutrients and chemicals flows through the network. (b) Schematic view of a biofilm spreading on a surface.
  • Figure 2: Schematic representation of a biofilm slice $\Omega^t$ spreading on a surface (a) occupying a finite region and ending at triple contact points, (b) spreading over precursor layers. The upper boundary $\Gamma^t_+$ represents the biofilm/air interface. The lower boundary $\Gamma^t_-$ represents the biofilm/agar interface, which provides nutrients and resources necessary for biofilm growth in our framework.