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Networks and Heterogeneous Media

2010, Volume 5, Issue 4: 675-690. doi: 10.3934/nhm.2010.5.675
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Coupling conditions for the 3×3 Euler system

  • 1.
    Dipartimento di Matematica, Università degli Studi di Brescia, Via Branze 38, 25123 Brescia
  • 2.
    Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, Via Cozzi 53, 20126 Milano
  • Received: 01 November 2009 Revised: 01 May 2010

  • Primary: 35L65; secondary: 76N10.

  • This paper is devoted to the extension to the full 3×3 Euler system of the basic analytical properties of the equations governing a fluid flowing in a duct with varying section. First, we consider the Cauchy problem for a pipeline consisting of 2 ducts joined at a junction. Then, this result is extended to more complex pipes. A key assumption in these theorems is the boundedness of the total variation of the pipe's section. We provide explicit examples to show that this bound is necessary.

    Citation: Rinaldo M. Colombo, Francesca Marcellini. Coupling conditions for the 3×3 Euler system[J]. Networks and Heterogeneous Media, 2010, 5(4): 675-690. doi: 10.3934/nhm.2010.5.675

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  • This paper is devoted to the extension to the full 3×3 Euler system of the basic analytical properties of the equations governing a fluid flowing in a duct with varying section. First, we consider the Cauchy problem for a pipeline consisting of 2 ducts joined at a junction. Then, this result is extended to more complex pipes. A key assumption in these theorems is the boundedness of the total variation of the pipe's section. We provide explicit examples to show that this bound is necessary.


    [1] M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314 (electronic).
    [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56 (electronic).
    [3] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications 20, Oxford University Press, Oxford, 2000.
    [4] R. M. Colombo and M. Garavello, On the p-system at a junction, in "Control Methods in Pde-Dynamical Systems," volume 426 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2007), 193-217.
    [5] R. M. Colombo and M. Garavello, On the 1D modeling of fluid flowing through a junction, preprint, (2009).
    [6] R. M. Colombo and G. Guerra, On general balance laws with boundary, J. Diff. Equations, 248 (2010), 1017-1043. doi: 10.1016/j.jde.2009.12.002
    [7] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Modeling and optimal control of networks of pipes and canals, SIAM J. Math. Anal., 48 (2009), 2032-2050.
    [8] R. M. Colombo, M. Herty and V. Sachers, On 2×2 conservation laws at a junction, SIAM J. Math. Anal., 40 (2008), 605-622. doi: 10.1137/070690298
    [9] R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the p-system, J. Math. Anal. Appl., 361 (2010), 440-456. doi: 10.1016/j.jmaa.2009.07.022
    [10] R. M. Colombo and C. Mauri, Euler system at a junction, Journal of Hyperbolic Differential Equations, 5 (2008), 547-568. doi: 10.1142/S0219891608001593
    [11] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.
    [12] P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902. doi: 10.1016/j.anihpc.2004.02.002
    [13] G. Guerra, F. Marcellini and V. Schleper, Balance laws with integrable unbounded source, SIAM J. Math. Anal., 41 (2009), 1164-1189. doi: 10.1137/080735436
    [14] H. Holden and N. H. Risebro, Riemann problems with a kink, SIAM J. Math. Anal., 30 (1999), 497-515 (electronic). doi: 10.1137/S0036141097327033
    [15] T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys., 83 (1982), 243-260. doi: 10.1007/BF01976043
    [16] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Springer-Verlag, New York, 1994.
    [17] G. B. Whitham, "Linear and Nonlinear Waves," John Wiley & Sons Inc., New York, 1999, reprint of the 1974 original, A Wiley-Interscience Publication.

  • This article has been cited by:

    1. Gunhild A. Reigstad, Existence and Uniqueness of Solutions to the Generalized Riemann Problem for Isentropic Flow, 2015, 75, 0036-1399, 679, 10.1137/140962759
    2. Martin Gugat, Michael Herty, 2022, 23, 9780323850599, 59, 10.1016/bs.hna.2021.12.002
    3. Rinaldo M. Colombo, Graziano Guerra, Yannick Holle, Non conservative products in fluid dynamics, 2022, 66, 14681218, 103539, 10.1016/j.nonrwa.2022.103539
    4. Mapundi K. Banda, Axel-Stefan Häck, Michael Herty, Numerical Discretization of Coupling Conditions by High-Order Schemes, 2016, 69, 0885-7474, 122, 10.1007/s10915-016-0185-x
    5. Edwige Godlewski, Pierre-Arnaud Raviart, 2021, Chapter 7, 978-1-0716-1342-9, 627, 10.1007/978-1-0716-1344-3_7
    6. Rinaldo M. Colombo, Graziano Guerra, Yannick Holle, 2024, Chapter 27, 978-3-031-55263-2, 313, 10.1007/978-3-031-55264-9_27
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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