Segregation effects and gap formation in cross-diffusion models

Burger, Martin; Carrillo, José A.; Pietschmann, Jan-Frederik; Schmidtchen, Markus

doi:10.4171/IFB/438

Interfaces and Free Boundaries

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Volume 22, Issue 2, 2020, pp. 175–203

DOI: 10.4171/IFB/438

Published online: 2020-07-06

Segregation effects and gap formation in cross-diffusion models

Martin Burger^[1], José A. Carrillo^[2], Jan-Frederik Pietschmann^[3] and Markus Schmidtchen^[4]

(1) Universität Erlangen-Nürnberg, Germany
(2) University of Oxford, UK
(3) Technische Universität Chemnitz, Germany
(4) Imperial College London, UK

In this paper, we extend the results of [8] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.

Keywords: Nonlinear cross-diffusion, degenerate parabolic equations, segregated solutions, energy minimisation, pattern formation

Burger Martin, Carrillo José, Pietschmann Jan-Frederik, Schmidtchen Markus: Segregation effects and gap formation in cross-diffusion models. Interfaces Free Bound. 22 (2020), 175-203. doi: 10.4171/IFB/438