Segregation and gap formation in cross-diffusion models

Journal article

In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion perturbations. This criticality is showcased by proving that adding an extra cross-diffusion term that breaks the symmetry of the population pressure in the system leads to completely different behaviours, namely segregation or mixing, depending on the sign of the perturbation. We show these results at the level of the minimisers of the associated free energy functionals. We also analyse certain implications for the gradient flow systems of the associated PDEs and present a numerical exploration of the time evolution of these phenomena.

Authors: Burger, M; Carrillo, JA; Pietschmann, J-F; Schmidtchen, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: European Mathematical Society
• Journal: Interfaces and Free Boundaries
• Volume: 22
• Issue: 2
• Pages: 175–203
• Publication date: 2020-07-06
• Acceptance date: 2020-05-01
• DOI: 10.4171/IFB/438
• EISSN: 1463-9971
• ISSN: 1463-9963
• Language: English
• Keywords: nonlinear cross-diffusion; pattern formation.; energy minimisation; degenerate parabolic equations; segregated solutions; FFR