A non-local scalar conservation law describing navigation processes

Paulo Amorim (1) , Florent Berthelin (2) , Thierry Goudon (3)

1 UFRJ - Universidade Federal do Rio de Janeiro [Brasil] = Federal University of Rio de Janeiro [Brazil] = Université fédérale de Rio de Janeiro [Brésil]
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 COFFEE - COmplex Flows For Energy and Environment

In this work, we consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker-Planck equation. This Fokker-Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.

Domaines: Equations aux dérivées partielles [math.AP]

Paulo Amorim, Florent Berthelin, Thierry Goudon. A non-local scalar conservation law describing navigation processes. Journal of Hyperbolic Differential Equations, 2020, 17 (4). ⟨hal-02419349⟩