Vanishing cross-diffusion limit in a Keller–Segel system with additional cross-diffusion. (March 2020)

• Record Type: Journal Article
• Authors: Jüngel, Ansgar; Leingang, Oliver; Wang, Shu
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 35B40 -- 35K51 -- 35Q92 -- 92C17; Keller–Segel model -- Asymptotic analysis -- Vanishing cross-diffusion limit -- Entropy method -- Higher-order estimates -- Numerical simulations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111698
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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 cross-diffusion 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 ( Ω ) 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.