Analysis of a Degenerate and Singular Volume-Filling Cross-Diffusion System Modeling Biofilm Growth
Analysis of a Degenerate and Singular Volume-Filling Cross-Diffusion System Modeling Biofilm Growth
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模拟生物膜生长的简并单一体积填充交叉扩散系统分析
DOI：
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发表时间：
2018
影响因子：
2
通讯作者：
Nicola Zamponi
中科院分区：
文献类型：
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作者：
Esther S. Daus;Pina Milisic;Nicola Zamponi
我们分析了一个多物种生物膜交叉扩散模型的数学性质，以及有界域上的非常一般的反应项和混合Dirichlet-Neumann边界条件。该模型属于体积填充型交叉扩散系统，当总生物量消失时表现为多孔媒质型退化，当生物量达到最大细胞容量时表现为超扩散型奇异性，这使得分析具有极大的挑战性。这些方程还承认了一种非常有趣的非标准熵结构。我们证明了全局时间弱解的存在性，研究了解的渐近性和唯一性，并通过数值模拟补充了分析，说明了理论上得到的结果。
We analyze the mathematical properties of a multi-species biofilm cross-diffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volume-filling type cross-diffusion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdiffusion-type singularity when the biomass reaches its maximum cell capacity, which make the analysis extremely challenging. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.