具有退化迁移率和奇异势的反应扩散cahn - hilliard型系统模拟生物膜生长的存在性分析

IF 1.1 3区数学 Q1 MATHEMATICS

Discrete and Continuous Dynamical Systems Pub Date : 2023-02-15 DOI:10.3934/dcds.2023069

Christoph Helmer, A. Jungel

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引用次数: 0

摘要

证明了模拟生物膜生长的扩散系统有界弱解的整体存在性。该方程由底物浓度的反应-扩散方程和生物质体积分数的四阶cahn - hilliard型方程组成，在无通量边界条件下考虑有界域。主要的困难来自于简并扩散率和迁移率，由对数自由能引起的奇异势，以及非线性反应速率。这些问题通过截断技术和布劳德-明蒂技巧来克服，以确定反应项的弱极限。采用二阶后向微分方程有限体积格式，在一维空间上进行了数值实验，说明了解的定性行为。

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

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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来源期刊

Discrete and Continuous Dynamical Systems 数学-数学

CiteScore

2.50

自引率

0.00%

发文量

175

审稿时长

6 months

期刊介绍： DCDS, series A includes peer-reviewed original papers and invited expository papers on the theory and methods of analysis, differential equations and dynamical systems. This journal is committed to recording important new results in its field and maintains the highest standards of innovation and quality. To be published in this journal, an original paper must be correct, new, nontrivial and of interest to a substantial number of readers.