A free boundary singular transport equation as a formal limit of a discrete dynamical system

TL;DR

This paper analyzes a singular transport PDE derived as a limit of a discrete game, establishing existence, uniqueness, and properties of solutions, including examples and a Lyapunov functional.

Contribution

It introduces a novel PDE model for the continuous limit of open mancala, with a nonlocal free boundary formulation and analysis of solution properties.

Findings

Existence and uniqueness of solutions proved.

The PDE admits a Lyapunov functional.

Examples including the Riemann problem illustrate singularity formation.

Abstract

We study the continuous version of a hyperbolic rescaling of a discrete game, called open mancala. The resulting PDE turns out to be a singular transport equation, with a forcing term taking values in $\{0,1\}$, and discontinuous in the solution itself. We prove existence and uniqueness of a certain formulation of the problem, based on a nonlocal equation satisfied by the free boundary dividing the region where the forcing is one (active region) and the region where there is no forcing (tail region). Several examples, most notably the Riemann problem, are provided, related to singularity formation. Interestingly, the solution can be obtained by a suitable vertical rearrangement of a multi-function. Furthermore, the PDE admits a Lyapunov functional.