ANALYSIS OF TWO CLASSES OF CROSS DIFFUSION SYSTEMS

Abstract

A mathematical and numerical analysis has been carried out for two cross diffusion systems arising in applied mathematics. The first system appears in modelling the movement of two interacting cell populations whose kinetics are of competition type. The second system models axial segregation of a mixture of two different granular materials in a long rotating drum. 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 the fully discrete solutions is shown. By using entropy-type inequalities and compactness arguments, the convergence of the approximation of each system is proved and hence existence of a global weak solution is obtained. Providing further regularity of the solution of the axial segregation model, some uniqueness results and error estimates are established. The long time behaviour of both systems is investigated and estimates between the weak solutions and the mean integrals of the corresponding initial data are derived. 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.