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We propose and analyze a one-dimensional multi-species cross-diffusion system with non-zero-flux boundary conditions on a moving domain, motivated by the modeling of a Physical Vapor Deposition process. Using the boundedness by entropy method introduced and developped in [5, 16], we prove the existence of a global weak solution to the obtained system. In addition, existence of a solution to an optimization problem defined on the fluxes is established under the assumption that the solution to the considered cross-diffusion system is unique. Lastly, we prove that in the case when the imposed external fluxes are constant and positive and the entropy density is defined as a classical logarithmic entropy, the concentrations of the different species converge in the long-time limit to constant profiles at a rate inversely proportional to time. These theoretical results are illustrated by numerical tests.
Bakhta, Athmane 1 ; Ehrlacher, Virginie 2
@article{M2AN_2018__52_4_1385_0,
author = {Bakhta, Athmane and Ehrlacher, Virginie},
title = {Cross-diffusion systems with non-zero flux and moving boundary conditions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1385--1415},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017053},
mrnumber = {3875290},
zbl = {1408.65051},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017053/}
}
TY - JOUR AU - Bakhta, Athmane AU - Ehrlacher, Virginie TI - Cross-diffusion systems with non-zero flux and moving boundary conditions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1385 EP - 1415 VL - 52 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017053/ DO - 10.1051/m2an/2017053 LA - en ID - M2AN_2018__52_4_1385_0 ER -
%0 Journal Article %A Bakhta, Athmane %A Ehrlacher, Virginie %T Cross-diffusion systems with non-zero flux and moving boundary conditions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1385-1415 %V 52 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017053/ %R 10.1051/m2an/2017053 %G en %F M2AN_2018__52_4_1385_0
Bakhta, Athmane; Ehrlacher, Virginie. Cross-diffusion systems with non-zero flux and moving boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1385-1415. doi: 10.1051/m2an/2017053
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