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Abstract:
Tissue growth, as in solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics.
We will focus on the links between two types of mathematical models. The 'compressible' description describes the cell population density using systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form.
The mathematical tool to perform the incompressible limit is the Aronson-Benilan estimate and we will show why a $L^2$ version is needed. We will also show that a $L^4$ estimate on the pressure gradient can be derived.