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

Christoph Helmer; Ansgar Jüngel

doi:10.3934/dcds.2023069

Article Contents

2023, Volume 43, Issue 10: 3839-3861. Doi: 10.3934/dcds.2023069

This issue Previous Article Box dimension of the graphs of the generalized Weierstrass-type functions Next Article

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

Christoph Helmer^1, and
Ansgar Jüngel^1, ,

1.
Institute of Analysis and Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria

^*Corresponding author: Ansgar Jüngel
^*Corresponding author: Ansgar Jüngel

Received: February 2023

Revised: May 2023

Early access: June 2023

Published: October 2023

Abstract / Introduction Full Text(HTML) Figure(4) / Table(1) Related Papers Cited by

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:

Mathematics Subject Classification: Primary: 35K35, 35K65, 35K67, 35Q92; Secondary: 92C17.

Citation:

Full Text(HTML)

Figure 1. Biomass fraction $ u $ (top) and substrate concentration $ v $ (bottom) in test case $ 1 $ for our system (left) and the system of [26] (right)

Download: Full-size image PowerPoint slide

Figure 2. Biomass $ u $ in test case $ 2 $ for our system (left) and the system of [26] (right)

Download: Full-size image PowerPoint slide

Figure 3. Biomass $ u $ in test case $ 3 $ for our system (left) and the system of [26] (right)

Download: Full-size image PowerPoint slide

Figure 4. Convergence in space (left) and convergence in time (right) at time $ T = 1 $

Download: Full-size image PowerPoint slide

Table 1. Parameters used in the numerical simulations

Symbol	Parameter	Value	Unit
$ D $	Diffusivity	$ 10^{-10} $	m$ ^2 $ s$ ^{-1} $
$ M' $	Mobility	$ 2.5\cdot 10^{-8} $	s
$ R_c $	Consumption rate	$ 10^{-2} $	s$ ^{-1} $
$ R_p $	Production rate	$ 10^{-2} $	kg m$ ^{-3} $ s$ ^{-1} $
$ K_v $	Half-saturation constant	$ 10^{-4} $	kg m$ ^{-3} $
$ \Gamma_1 $	Distortional energy	$ 4\cdot 10^{-15} $	m$ ^4 $ s$ ^{-2} $
$ \Gamma_2 $	Mixing free energy	$ 4\cdot 10^{-6} $	m$ ^{2} $ s$ ^{-2} $
$ N $	Polymerization parameter	$ 10^{3} $
$ \lambda $	Flory–Huggins parameter	0.55
$ x_0 $	Characteristic length	$ 10^{-4} $	m
$ t_0 $	Characteristic time	$ 10^2 $	s
$ v_0 $	Characteristic concentration	$ 10^{-3} $	kg m$ ^{-3} $
$ k_BT $	Thermal energy at $ T=300 $ K	$ 4\cdot 10^{-21} $	kg m$ ^2 $ s$ ^{-2} $
$ \widetilde{K} $	Half-saturation constant for model of [26]	$ 5\cdot 10^{-4} $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	D. Antonopoulou, G. Karali and G. T. Kossioris, Asymptotics for a generalized Cahn–Hilliard equation with forcing terms, Discrete Cont. Dyn. Sys., 30 (2011), 1037-1054. doi: 10.3934/dcds.2011.30.1037.
[2]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.
[3]	L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn–Hilliard equation with biological applications, Discrete Cont. Dyn. Sys. B, 19 (2014), 2013-2026. doi: 10.3934/dcdsb.2014.19.2013.
[4]	N. G. Cogan and J. P. Keener, The role of the biofilm matrix in structural development, Math. Med. Biol., 21 (2004), 147-166. doi: 10.1093/imammb/21.2.147.
[5]	B. D'Acunto and L. Frunzo, Qualitative analysis and simulations of a free boundary problem for multispecies biofilm models, Math. Computer Model., 53 (2011), 1596-1606. doi: 10.1016/j.mcm.2010.12.024.
[6]	B. D'Acunto, L. Frunzo, V. Luongo and M. R. Mattei, Free boundary approach for the attachment in the initial phase of multispecies biofilm growth, Z. Angew. Math. Phys., 70 (2019), no. 91, 16 pp. doi: 10.1007/s00033-019-1134-y.
[7]	E. S. Daus, A. Jüngel and A. Zurek, Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms, IMA J. Numer. Anal., 41 (2021), 935-973. doi: 10.1093/imanum/draa040.
[8]	M. Ebenbeck, Cahn–Hilliard–Brinkmann Models for Tumour Growth: Modelling, Analysis and Optimal Control, Dissertation, University of Regensburg, Germany, 2020. doi: 10.5283/epub.43376.
[9]	M. Ebenbeck, H. Garcke and R. Nürnberg, Cahn–Hilliard–Brinkmann systems for tumour growth, Discrete Cont. Dyn. Sys. S, 14 (2021), 3989-4033. doi: 10.3934/dcdss.2021034.
[10]	H. G. Eberl, M. A. Efendiev, D. Wrzosek and A. Zhigun, Analysis of a degenerate biofilm model with a nutrient taxis term, Discrete Cont. Dyn. Sys., 34 (2014), 99-119. doi: 10.3934/dcds.2014.34.99.
[11]	H. J. Eberl, D. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Medicine, 3 (2001), 161-175. doi: 10.1080/10273660108833072.
[12]	M. A. Efendiev, H. J. Eberl and S. V. Zelik, Existence and longtime behavior of solutions of the nonlinear reaction-diffusion system arising in the modelling of biofilms, In: Nonlinear Diffusive Systems and Related Topics, Surikaisekikenkyusho Kokyuroku, 1258 (2002), 49-71.
[13]	B. O. Emerenini, S. Sonner and H. J. Eberl, Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects, Math. Biosci. Eng., 14 (2017), 625-653. doi: 10.3934/mbe.2017036.
[14]	C. M. Elliott and H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662.
[15]	P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys., 10 (1942), 51-61. doi: 10.1063/1.1723621.
[16]	A. Friedman, B. Hu and C. Xue, On a multiphase multicomponent model of biofilm growth, Arch. Ration. Mech. Anal., 211 (2014), 257-300. doi: 10.1007/s00205-013-0665-1.
[17]	S. Frigeri, On a nonlocal Cahn–Hilliard/Navier–Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities, Ann. Inst. H. Poincaré – Anal. Non Lin., 38 (2021), 647-687. doi: 10.1016/j.anihpc.2020.08.005.
[18]	H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. doi: 10.3934/Math.2016.3.318.
[19]	C. Helmer, A. Jüngel and A. Zurek, Analysis of a finite-volume scheme for a single-species biofilm model, Appl. Numer. Math., 185 (2023), 386-405. doi: 10.1016/j.apnum.2022.12.002.
[20]	M. L. Huggins, Solutions of long chain compounds, J. Chem. Phys., 9 (1941), 440. doi: 10.1063/1.1750930.
[21]	A. Iuorio and S. Melchionna, Long-time behavior of a nonlocal Cahn–Hilliard equation with reaction, Discrete Cont. Dyn. Sys., 38 (2018), 3765-3788. doi: 10.3934/dcds.2018163.
[22]	M. Jurak, I. Radišić and A. Žgaljić Keko, Two-phase two-component flow in porous media in low solubility regime, SIAM J. Math. Anal., 51 (2019), 2019-2052. doi: 10.1137/18M1182206.
[23]	B. Perthame and A. Poulain, Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility, Europ. J. Appl. Math., 32 (2021), 89-112. doi: 10.1017/S0956792520000054.
[24]	J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Math. Pura. Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[25]	Q. Wang and T. Zhang, Review of mathematical models for biofilms, Solid State Commun., 150 (2010), 1009-1022. doi: 10.1016/j.ssc.2010.01.021.
[26]	Q. Wang and T. Zhang, Kinetic theories for biofilms, Discrete Cont. Dyn. Sys. B, 17 (2012), 1027-1059. doi: 10.3934/dcdsb.2012.17.1027.
[27]	O. Wanner and W. Gujer, A multispecies biofilm model, Biotechnol. Bioengin., 28 (1986), 314-328. doi: 10.1002/bit.260280304.
[28]	J.-X. Yin, On the existence of nonnegative continuous solutions of the Cahn–Hilliard equation, J. Differ. Eqs., 97 (1992), 310-327. doi: 10.1016/0022-0396(92)90075-X.
[29]	T. Zhang, Modeling of biocide action against biofilm, Bull. Math. Biol., 74 (2012), 1427-1447. doi: 10.1007/s11538-012-9719-z.
[30]	T. Zhang, N. G. Cogan and Q. Wang, Phase-field models for biofilms. Ⅰ. Theory and 1-D simulations, SIAM J. Appl. Math., 69 (2008), 641-669. doi: 10.1137/070691966.
[31]	T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms. Ⅱ. 2-D numerical simulations of biofilm-flow interaction, Commun. Comput. Phys., 4 (2008), 72-101.