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

Felisia Angela Chiarello, Simone Fagioli, Massimiliano Daniele Rosini

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.

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

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.

Paper Structure

This paper contains 2 sections, 1 theorem, 11 equations.

Table of Contents

  1. Introduction
  2. The system and main notation

Key Result

Lemma 2.2

Fix $w_2\leqslant0$. The function $F_{w_2}(\rho) \doteq \rho \, V\mathopen{}\mathclose{\left( w_2 - p(\rho) \right)$ given in e:F is well defined, strictly decreasing, strictly concave down, $F_{w_2}\mathopen{}\mathclose{\left(p^{-1}( w_2 )\right) = 0$, and Lipschitz continuous with

Theorems & Definitions (3)

  • Example 2.1
  • Lemma 2.2
  • proof