A non-local scalar conservation law describing navigation processes

Abstract

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.