Mathematical models of living tissues and the Hele-Shaw limit

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 focuss 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.

This work is a collaboration with F. Quiros and J.-L. Vazquez (Universidad Autonoma Madrid), A. Mellet, M. Tang (SJTU), N. Vauchelet (Sorbonne-Paris-Nord) and N. David (LJLL).