Stability estimates for systems with small cross-diffusion

TL;DR

This paper analyzes the stability of cross-diffusion systems with nonlinear terms, providing estimates that show how solutions depend continuously on model parameters, with applications to biological models and numerical illustrations.

Contribution

It establishes well-posedness and stability estimates for nonlinear cross-diffusion systems close to linear problems, including time-independent bounds under certain conditions.

Findings

Stability estimates depend continuously on nonlinearities.

Solutions are well-posed in an appropriate Banach space.

Numerical illustrations show limits of stability for biological models.

Abstract

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.