Homogenization of a microscopic pedestrians model on a convergent junction

Abstract

In this paper, we establish a rigorous connection between a microscopic and a macroscopic pedestrians model on a convergent junction. At the microscopic level, we consider a “follow the leader” model far from the junction point and we assume that a rule to enter the junction point is imposed. At the macroscopic level, we obtain the Hamilton-Jacobi equation with a flux limiter condition at x = 0 introduced in Imbert and Monneau [Ann. Sci. l’École Normale Supér. 50 (2017) 357-414], To obtain our result, we inject using the “cumulative distribution functions” the microscopic model into a non-local PDE. Then, we show that the viscosity solution of the non-local PDE converges locally uniformly towards the solution of the macroscopic one.

Regularizing effect for unbounded flux-limited viscosity solutions of a discontinuous Hamilton-Jacobi equation on junction

“…In this paper, we want to develop regularizing results for discontinuous Hamilton-Jacobi equations. The study of discontinuous (in space) Hamilton-Jacobi equations is recent (see [1,11]) but the basic results, like comparison principle, existence, stability, homogenization are now well understood (see [10,13,5,9,16,2,14,8]) and we are now able to attack more difficult problems like the regularity or the regularizing effect.…”