Vanishing viscosity on a star-shaped graph under general transmission conditions at the node

Giuseppe Maria Coclite; Carlotta Donadello

doi:10.3934/nhm.2020009

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2020, Volume 15, Issue 2: 197-213. Doi: 10.3934/nhm.2020009

This issue Previous Article Convexity and starshapedness of feasible sets in stationary flow networks Next Article A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity

Vanishing viscosity on a star-shaped graph under general transmission conditions at the node

Giuseppe Maria Coclite^1, , and
Carlotta Donadello^2,

1.
Department of Mechanics, Mathematics and Management, Polytechnic University of Bari, Via E. Orabona 4, 70125 Bari, Italy
2.
Laboratoire de Mathématiques CNRS UMR6623, Université de Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

^* Corresponding author: Giuseppe Maria Coclite
^* Corresponding author: Giuseppe Maria Coclite

Received: April 30, 2019

Revised: October 31, 2019

Early access: May 2020

Published: June 2020

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Abstract

In this paper we consider a family of scalar conservation laws defined on an oriented star shaped graph and we study their vanishing viscosity approximations subject to general matching conditions at the node. In particular, we prove the existence of converging subsequence and we show that the limit is a weak solution of the original problem.

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Mathematics Subject Classification: Primary: 35L65; Secondary: 35R02.

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

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