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Analysis of differential equations modelling the reactive flow through a deformable system of cells. (English) Zbl 1160.74034

Summary: 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 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 solid-fluid interface. 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 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.

MSC:

74L15 Biomechanical solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
76S05 Flows in porous media; filtration; seepage
76V05 Reaction effects in flows
92C10 Biomechanics
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