IBVPs for Scalar Conservation Laws with Time Discontinuous Fluxes

TL;DR

This paper establishes well-posedness and stability for scalar conservation laws with time discontinuous fluxes, extending to traffic models with time-dependent speed limits and demonstrating the existence of optimal traffic management solutions.

Contribution

It proves well-posedness and stability for scalar conservation laws with time discontinuous fluxes, including applications to traffic flow models with minimal regularity assumptions.

Findings

Well-posedness of scalar conservation laws with time discontinuous fluxes.

Stability of solutions with respect to flux variations.

Existence of optimal solutions in traffic management problems.

Abstract

The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely $\mathbf{L}^{\infty}$ dependence on time. These results ensure, for instance, the well posedness of a class of vehicular traffic models with time dependent speed limits. A traffic management problem is then shown to admit an optimal solution.