We discuss the analysis and stability of a family of cross-diffusion boundary value problems with nonlinear diffusion and drift terms. We assume that these systems are close, in a suitable sense, to a set of decoupled and linear problems. We focus on stability estimates, that is, continuous dependence of solutions with respect to the nonlinearities in the diffusion and in the drift terms. We establish well-posedness and stability estimates in an appropriate Banach space. Under additional assumptions we show that these estimates are time independent. These results apply to several problems from mathematical biology; they allow comparisons between the solutions of different models a priori. For specific cell motility models from the literature, we illustrate the limit of the stability estimates we have derived numerically, and we document the behaviour of the solutions for extremal values of the parameters.
Mots clés : Cross diffusion, continuous dependence, quasilinear parabolic systems.
@article{M2AN_2018__52_3_1109_0,
author = {Alasio, Luca and Bruna, Maria and Capdeboscq, Yves},
title = {Stability estimates for systems with small cross-diffusion\protect\textsuperscript{,}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1109--1135},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2018036},
zbl = {1403.35126},
mrnumber = {3865560},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018036/}
}
TY - JOUR AU - Alasio, Luca AU - Bruna, Maria AU - Capdeboscq, Yves TI - Stability estimates for systems with small cross-diffusion, JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1109 EP - 1135 VL - 52 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018036/ DO - 10.1051/m2an/2018036 LA - en ID - M2AN_2018__52_3_1109_0 ER -
%0 Journal Article %A Alasio, Luca %A Bruna, Maria %A Capdeboscq, Yves %T Stability estimates for systems with small cross-diffusion, %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1109-1135 %V 52 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018036/ %R 10.1051/m2an/2018036 %G en %F M2AN_2018__52_3_1109_0
Alasio, Luca; Bruna, Maria; Capdeboscq, Yves. Stability estimates for systems with small cross-diffusion,. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1109-1135. doi : 10.1051/m2an/2018036. http://www.numdam.org/articles/10.1051/m2an/2018036/
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