Existence result for a 2 x 2 system of conservation laws with discontinuous flux and applications

TL;DR

This paper proves the existence of entropy solutions for 2x2 conservation law systems with discontinuous flux, using wave-front tracking and adapted Riemann invariants, with applications to traffic flow models on inhomogeneous roads.

Contribution

Introduces a Kruzhkov-type entropy condition and adapted Riemann invariants to handle discontinuous flux in conservation laws, establishing global existence results.

Findings

Proved global existence of entropy solutions for large data.

Developed a wave-front tracking approximation method.

Applied results to traffic flow models with abrupt road changes.

Abstract

This paper is concerned with one-dimensional 2 x 2 systems of conservation laws with a flux f=f(x, U) that is discontinuous with respect to the spatial variable. No monotonicity assumption is imposed on the mapping x \to f(x,U). We introduce a Kruzhkov-type entropy condition and establish the global existence of entropy solutions for large data. The proof relies on a wave-front tracking approximation. The main technical novelty consists in the introduction of adapted Riemann invariant coordinates, specifically designed to account for the discontinuities of the flux, which yield a uniform-in-time bound on the total variation of the approximate solutions U^n(t). We also outline several alternative approaches that may lead to existence results under possibly weaker assumptions. As an application, we propose second-order vehicular traffic models on inhomogeneous roads featuring abrupt ''collective'' changes in the speed law or road capacity.