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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.

A non-local scalar conservation law describing navigation processes