模拟颗粒生物膜的球形自由边界问题分析
Analysis of a spherical free boundary problem modelling granular biofilms
摘要：
提出了一个与多物种颗粒生物膜形成有关的自由边界值问题。颗粒状生物膜被模拟为具有径向对称性的球形自由边界域。该模型是设想在连续介质力学的框架下，由：非线性双曲偏微分方程模型的平流运输和附着的物种，构成颗粒状生物膜基质的生长;半线性椭圆偏微分方程管理的扩散运输和转换的营养物质;和半线性椭圆偏微分方程描述的入侵现象和转换的浮游细胞悬浮在周围环境中。自由边界的演化由解释微生物生长、附着和分离的常微分方程控制。利用特征线法将构成颗粒生物膜模型的方程组转化为一个等效积分系统。利用不动点定理讨论并证明了该模型解的存在唯一性。
英文摘要：
A free boundary value problem related to the genesis of multispecies granular biofilms is presented. The granular biofilm is modelled as a spherical free boundary domain with radial symmetry. The proposed model is conceived in the framework of continuum mechanics and consists of: nonlinear hyperbolic PDEs which model the advective transport and growth of attached species that constitute the granular biofilm matrix; semilinear elliptic PDEs which govern the diffusive transport and conversion of nutrients; and semilinear elliptic PDEs describing the invasion phenomena and conversion of planktonic cells suspended in the surrounding environment. The evolution of the free boundary is governed by an ODE accounting for microbial growth, attachment, and detachment. By using the method of characteristics, the system of equations constituting the granular biofilm model is converted into an equivalent integral system. Existence and uniqueness of solutions are discussed and proved for the attachment regime using the fixed point theorem.
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