Calculus of Variations and Geometric Measure Theory

G. Bellettini - A. Betti - M. Paolini

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

created by belletti on 24 Jun 2022


Submitted Paper

Inserted: 24 jun 2022
Last Updated: 24 jun 2022

Year: 2022


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.