Seminarium:
Równania różniczkowe
Osoba referująca:
Piotr Gwiazda (UW, IMPAN)
Data:
poniedziałek, 14. Styczeń 2019 - 15:15
Sala:
603
Opis:
A two species hyperbolic-parabolic model of tissue growth.
Models of tissue growth are now well established, in particular in
relation to their applications to cancer. They describe the dynamics
of cells subject to motion resulting from a pressure gradient
generated by the death and birth of cells, itself controlled primarily
by pressure through contact inhibition. In the compressible regime we
consider, when pressure results from the cell densities and when two
different populations of cells are considered, a specific difficulty
arises from the hyperbolic character of the equation for each cell
density, and to the parabolic aspect of the equation for the total
cell density. For that reason, few a priori estimates are available
and discontinuities may occur. Therefore the existence of solutions is
a difficult problem. Here, we establish the existence of weak
solutions to the model with two cell populations which react similarly
to the pressure in terms of their motion but undergo different
growth/death rates. In opposition to the method used in the recent
paper of J. A. Carrillo, S. Fagioli, F. Santambrogio, and
M.Schmidtchen, Splitting schemes & segregation in
reaction-(cross-)diffusion systems, our strategy is to ignore
compactness on the cell densities and to prove strong compactness on
the pressure gradient. We improve known results in two directions; we
obtain new estimates, we treat higher dimension than 1 and we deal
with singularities resulting from vacuum.