Contributions to Books:
A. Jüngel, O. Leingang, S. Wang:
"Vanishing cross-diffusion limit in a Keller-Segel system with additional cross-diffusion";
in: "ASC Report 19/2019",
issued by: Institute for Analysis and Scientific Computing;
Vienna University of Technology,
Keller-Segel systems in two and three space dimensions with an additional cross-diffusion term in the equation for the chemical concentration are analyzed. The cross-diffusion term has a stabilizing effect and leads to the global-in-time existence of weak solutions. The limit of vanishing cross-diffusion parameter is proved rigorously
in the parabolic-elliptic and parabolic-parabolic cases. When the signal production is sublinear, the existence of global-in-time weak solutions as well as the convergence of the solutions to those of the classical parabolic-elliptic Keller-Segel equations are proved.
The proof is based on a reformulation of the equations eliminating the additional crossdiffusion term but making the equation for the cell density quasilinear. For superlinear signal production terms, convergence rates in the cross-diffusion parameter are proved for
local-in-time smooth solutions (since finite-time blow up is possible). The proof is based on careful H^s(Omega) estimates and a variant of the Gronwall lemma. Numerical experiments in two space dimensions illustrate the theoretical results and quantify the shape of the cell aggregation bumps as a function of the cross-diffusion parameter.
Keller-Segel model, asymptotic analysis, vanishing cross-diffusion limit, entropy method, higher-order estimates, numerical simulations.
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.