Analysis of three Classes of Cross Diffusion Systems

Abstract

A mathematical and numerical analysis has been undertaken for three cross diffusion systems which arise in the modelling of biological systems. The first system appears in modelling the movement of multiple interacting cell populations whose kinetics are of competition type. The second model is the mechanical tumor-growth model of Jackson and Byrne that consists of nonlinear parabolic cross-diffusion equations in one space dimension for the volume fractions of tumor cells and an extracellular matrix (ECM), and describes tumor encapsulation influenced by a cell-induced pressure coefficient. The third system is the Keller-Segel model in multiple-space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal.


A fully practical piecewise linear finite element approximation for each system is proposed and studied. With the aid of a fixed point theorem, existence of fully discrete solution is shown. By using entropy type inequalities and compactness arguments, the convergence of each approximation is proved and hence existence of a global weak solution is obtained. In the case of the Keller-Segel model, we were able to obtain additional regularity to provide an improved weak formulation. Further, for the Keller-Segel model we established uniqueness results and error estimates. Finally, a practical algorithm for computing the numerical solutions of each system is described and some numerical experiments are performed to illustrate and verify the theoretical results.