Stability estimates for a family of cross-diffusion systems

Luca Alasio, Maria Bruna, Yves Capdeboscq

Published 2018-01-19Version 1

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 the appropriate Sobolev space. Under additional assumptions we show that these estimates are time independent. These results apply to several problems from mathematical biology; they allow the comparison of the behaviour of the solutions of the 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.