Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth
Abstract
The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reaction-diffusion equation for the substrate concentration and a fourth-order Cahn-Hilliard-type equation for the volume fraction of the biomass, considered in a bounded domain with no-flux boundary conditions. The main difficulties are coming from the degenerate diffusivity and mobility, the singular potential arising from a logarithmic free energy, and the nonlinear reaction rates. These issues are overcome by a truncation technique and a Browder-Minty trick to identify the weak limits of the reaction terms. The qualitative behavior of the solutions is illustrated by numerical experiments in one space dimension, using a BDF2 (second-order backward Differentiation Formula) finite-volume scheme.
Keywords
Cite
@article{arxiv.2302.07765,
title = {Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth},
author = {Christoph Helmer and Ansgar Jüngel},
journal= {arXiv preprint arXiv:2302.07765},
year = {2023}
}
Comments
We corrected the definition of the approximate function $f_{1,\delta}$ in Section 2.1 to ensure that its derivative is bounded for all arguments