Volume 52, Number 3, May–June 2018
|Page(s)||1109 - 1135|
|Published online||13 September 2018|
Stability estimates for systems with small cross-diffusion★,★★
Mathematical Institute, University of Oxford,
OX2 6GG, UK.
* Corresponding author: firstname.lastname@example.org
Accepted: 24 May 2018
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.
Mathematics Subject Classification: 35K55 / 35B30 / 35Q92 / 65M15
Key words: Cross diffusion / continuous dependence / quasilinear parabolic systems.
© EDP Sciences, SMAI 2018
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