One-dimensional aggregation equation after blow up: existence, uniqueness and numerical simulation

TL;DR

This paper studies the one-dimensional aggregation equation, establishing existence and uniqueness of measure solutions after blow-up, and introduces a convergent numerical scheme to simulate post-blow-up dynamics.

Contribution

It provides a rigorous framework for measure solutions post-blow-up and proves convergence of a new numerical scheme for this nonlinear nonlocal PDE.

Findings

Existence and uniqueness of measure solutions after blow-up

Development of a convergent upwind finite volume scheme

Numerical simulations illustrating solution dynamics post-blow-up

Abstract

The nonlocal nonlinear aggregation equation in one space dimension is investigated. In the so-called attractive case smooth solutions blow up in finite time, so that weak measure solutions are introduced. The velocity involved in the equation becomes discontinuous, and a particular care has to be paid to its definition as well as the formulation of the corresponding flux. When this is done, the notion of duality solutions allows to obtain global in time existence and uniqueness for measure solutions. An upwind finite volume scheme is also analyzed, and the convergence towards the unique solution is proved. Numerical examples show the dynamics of the solutions after the blow up time.