Analysis of Differential Equations Modelling the Reactive Flow through a Deformable System of Cells |
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Authors: | Willi Jäger Andro Mikeli? Maria Neuss-Radu |
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Institution: | 1. Institut für Angewandte Mathematik, Universit?t Heidelberg, Im Neuenheimer Feld 294, 69121, Heidelberg, Germany 2. Université de Lyon, 69003, Lyon Cedex 07, France 3. Université Lyon 1, Institut Camille Jordan, Site de Gerland, Bat. A, 50, avenue Tony Garnier, 69367, Lyon Cedex 07, France
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Abstract: | 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. |
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