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arXiv · 2302.07765

Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth

Christoph Helmer·Ansgar Jüngel
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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.

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Christoph Helmer, Ansgar Jüngel. 2023-02-15. Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth. https://arxiv.org/abs/2302.07765

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