Fernando Betancourt, Raimund Bürger, Kenneth H. KARLSEN:
A strongly degenerate parabolic aggregation equation
This paper is concerned with a strongly degenerate convection-diffusion equation in one space dimension whose convective flux involves a non-linear function of the total mass to one side of the given position. This equation can be understood as a model of aggregation of the individuals of a population with the solution representing their local density. The aggregation mechanism is balanced by a degenerate diffusion term accounting for dispersal. In the strongly degenerate case, solutions of the non-local problem are usually discontinuous and need to be defined as weak solutions satisfying an entropy condition. A finite difference scheme for the non-local problem is formulated and its convergence to the unique entropy solution is proved. The scheme emerges from taking divided differences of a monotone scheme for the local PDE for the primitive. Numerical examples illustrate the behaviour of entropy solutions of the non-local problem, in particular the aggregation phenomenon.
This preprint gave rise to the following definitive publication(s):
Fernando BETANCOURT, Raimund BüRGER, Kenneth H. KARLSEN: A strongly degenerate parabolic aggregation equation. Communications in Mathematical Sciences, vol. 9, 3, pp. 711-742, (2011).