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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, Wien, 2019, ISBN: 978-3-902627-12-4, 1 - 25.

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.

http://www.asc.tuwien.ac.at/preprint/2019/asc19x2019.pdf