Volume 27, 2021
|Number of page(s)||46|
|Published online||03 March 2021|
Linear hyperbolic systems on networks: well-posedness and qualitative properties*
University of Ljubljana, Faculty of Civil and Geodetic Engineering,
2 Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia.
3 Lehrgebiet Analysis, Fakultät Mathematik und Informatik, FernUniversität in Hagen, 58084 Hagen, Germany.
4 Université Polytechnique Hauts-de-France, LAMAV, FR CNRS 2956, 59313 Valenciennes Cedex 9, France.
** Corresponding author: firstname.lastname@example.org
Accepted: 17 December 2020
We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order reduction. We study forward and backward well-posedness; furthermore, we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first-order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied.
Mathematics Subject Classification: 47D06 / 35L40 / 35R02 / 81Q35
Key words: Hyperbolic systems / operator semigroups / PDEs on networks / invariance properties / Saint-Venant system / second sound
The work of M.K.F. was partially supported by the Slovenian Research Agency, Grant No. P1-0222, and the work of D.M. by the Deutsche Forschungsgemeinschaft (Grant 397230547). This article is based upon work from COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu.
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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