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

Francois James (1) , Nicolas Vauchelet (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 LJLL - Laboratoire Jacques-Louis Lions

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.

Mots clés: Aggregation equation; Weak measure solutions; Transport equation; Blow up; Finite volume scheme

Domaines: Equations aux dérivées partielles [math.AP]

Francois James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: existence, uniqueness and numerical simulation. CoToCoLa, Jan 2015, Besançon, France. ⟨hal-01234162⟩