Analysis of Differential Equations Modelling the Reactive Flow through a Deformable System of Cells

A system of model equations coupling fluid flow, deformation of solid structure and chemical reactions is formulated starting from processes in biological tissue. The main aim of this paper is to analyse this non-standard system, where the elasticity modules are functionals of a concentration and the diffusion coefficients of the chemical substances are functions of their concentrations. A new approach and new methods are required and adapted to these nonlinearities and the transmission conditions on the interface solid–fluid. Strong solutions for the initial and boundary value problem are constructed under suitable regularity assumptions on the data, and stability estimates of the solutions with respect to the initial and boundary values are proved. These estimates imply uniqueness directly. The approach of the paper can be used in more general problems modeling reactive flow and transport and its interaction with elastic cell structures. In a forthcoming paper the approach of this paper is used for getting the upscaled system modeling reactive flow through biological tissue on the macroscopic scale, starting from a system on the cell level.