A class of germs arising from homogenization in traffic flow on junctions

Abstract

We consider traffic flows described by conservation laws. We mainly study a 2:1 junction (with two incoming roads and one outgoing road). At the mesoscopic level, the priority law at the junction is given by traffic lights, which are periodic in time; the traffic can also be slowed down by periodic in time flux-limiters. Looking at long-time behavior and on large space scale, we intuitively expect an effective junction condition to emerge, thus deriving a macroscopic model from the mesoscopic one. At the limit of the rescaling, we show rigorous homogenization of the problem and identify the effective junction condition, which belongs to a general class of germs (in the terminology of [B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of [Formula: see text]-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal. 201 (2011) 27–86; U. S. Fjordholm, M. Musch and N. H. Risebro, Well-posedness and convergence of a finite volume method for conservation laws on networks, SIAM J. Numer. Anal. 60(2) (2022) 606–630; M. Musch, U. S. Fjordholm and N. H. Risebro, Well-posedness theory for nonlinear scalar conservation laws on networks, Netw. Heterog. Media 17 (2022) 101–128]). The proof of homogenization requires two steps: our first key result is the identification of this germ and of a characteristic subgerm which determines the whole germ. The second key result is the construction of a family of correctors whose values at infinity are related to each element of the characteristic subgerm. This construction is indeed explicit at the level of some mixed Hamilton–Jacobi equations for concave Hamiltonians (i.e. fluxes). The solutions are found in the spirit of representation formulas for optimal control problems.