Laboratoire Jacques-Louis Lions, Sorbonne Université
Date(s) : 13/12/2022 iCal
11 h 00 min - 12 h 00 min
Both compressible and incompressible porous medium models have been used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems of Hele-Shaw type where saturation holds in the moving domain. In this talk, I will present the study of the incompressible limit for advection-porous medium equations motivated by tumor development. The derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3 -version of the celebrated Aronson-Bénilan estimate and a sharp uniform L4 -bound on the pressure gradient. Moreover, we provide an estimate of the convergence rate at the incompressible limit in a Sobolev negative norm.