Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network

Boris P. Andreianov; Giuseppe Maria Coclite; Carlotta Donadello

doi:10.3934/dcds.2017257

Article Contents

2017, Volume 37, Issue 11: 5913-5942. Doi: 10.3934/dcds.2017257

This issue Previous Article Visco-Energetic solutions to one-dimensional rate-independent problems Next Article Asymptotic large time behavior of singular solutions of the fast diffusion equation

Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network

1.
Laboratoire de Mathématiques et Physique Théorique CNRS UMR7350, Université de Tours, Parc de Grandmont, 37200 Tours, France
2.
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy
3.
Laboratoire de Mathématiques CNRS UMR6623, Université de Bourgogne-Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

* Corresponding author: G. M. Coclite
* Corresponding author: G. M. Coclite

Received: October 31, 2016

Revised: May 31, 2017

Published: October 2017

The second author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The work on this paper was supported by the French ANR-11-JS01-0006 project CoToCoLa

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Abstract

We provide a complete study of the model investigated in [Coclite, Garavello, SIAM J. Math. Anal., 2010]. We prove well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem for scalar conservation laws $ ρ_{h, t} + f_h(ρ_h)_x = 0$, for $h∈ \{1, ..., m+n\}$, on a junction where $m$ incoming and $n$ outgoing edges meet. Our analysis and the definition of the admissible solution rely upon the complete description of the set of edge-wise constant solutions and its properties, which is of some interest on its own. The Riemann solver at the junction is characterized. In order to prove uniqueness, we introduce a family of Kruzhkov-type adapted entropies at the junction. Existence is justified both by the vanishing viscosity method and via the proof of convergence of a monotone well-balanced finite volume discretization. Beyond the classical vanishing viscosity framework, the numerical procedure and the uniqueness argument can be applied to general junction solvers enjoying the crucial order-preservation property.

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Mathematics Subject Classification: Primary: 90B20; Secondary: 35L65.

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Figure 1. A junction consisting of $m$ incoming and $n$ outgoing roads.

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