2023/02/15 by Christoph Helmer, Ansgar Jüngel, Helmer, Christoph +1
Earth and Planetary Sciences · Engineering · Materials Science · #35K35 #35K65 #35K67 #35Q92 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Solidification and crystal growth phenomena #nanoparticles nucleation surface interactions
paper · pdf · doi:10.48550/arxiv.2302.07765
openalex publication_date 2023/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.