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 nonlinear 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 describing the effect of dispersal. In the strongly degenerate case, solutions of the nonlocal problem are usually discontinuous and need to be defined as weak solutions. A finite difference scheme for the nonlocal problem is formulated and its convergence to the unique weak solution is proved. This scheme emerges from taking divided differences of a monotone scheme for the local PDE for the primitive. Some numerical examples illustrate the behaviour of solutions of the nonlocal problem, in particular the aggregation phenomenon.

Keywords

Aggregation, strongly degenerate convection-diffusion equation, nonlocal flux, well-posedness, finite difference scheme

2010 Mathematics Subject Classification

35K65, 65N06, 92Cxx

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Published 11 March 2011