American Institute of Mathematical Sciences

Advanced
December  2010, 5(4): 675-690. doi: 10.3934/nhm.2010.5.675

Coupling conditions for the $3\times 3$ Euler system

Rinaldo M. Colombo 1, and  Francesca Marcellini 2,

1. 

Dipartimento di Matematica, Università degli Studi di Brescia, Via Branze 38, 25123 Brescia, Italy
2. 

Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, Via Cozzi 53, 20126 Milano, Italy

Received  November 2009 Revised  May 2010 Published  November 2010

This paper is devoted to the extension to the full $3\times3$ 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.
Keywords: Conservation Laws at Junctions, Coupling Conditions at Junctions..
Mathematics Subject Classification: Primary: 35L65; secondary: 76N1.
Citation: Rinaldo M. Colombo, Francesca Marcellini. Coupling conditions for the $3\times 3$ Euler system. Networks and Heterogeneous Media, 2010, 5 (4) : 675-690. doi: 10.3934/nhm.2010.5.675
References:
[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 $2\times2$ 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.

show all references

References:
[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 $2\times2$ 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.

[1]

Jens Lang, Pascal Mindt. Entropy-preserving coupling conditions for one-dimensional Euler systems at junctions. Networks and Heterogeneous Media, 2018, 13 (1) : 177-190. doi: 10.3934/nhm.2018008
[2]

Laurent Bourgeois, Jean-François Fritsch, Arnaud Recoquillay. Imaging junctions of waveguides. Inverse Problems and Imaging, 2021, 15 (2) : 285-314. doi: 10.3934/ipi.2020065
[3]

Luis A. Caffarelli, Fang Hua Lin. Analysis on the junctions of domain walls. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 915-929. doi: 10.3934/dcds.2010.28.915
[4]

Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644
[5]

Pranay Goel, James Sneyd. Gap junctions and excitation patterns in continuum models of islets. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1969-1990. doi: 10.3934/dcdsb.2012.17.1969
[6]

Adnane Azzouzi, Yves Coudière, Rodolphe Turpault, Nejib Zemzemi. A mathematical model of the Purkinje-Muscle Junctions. Mathematical Biosciences & Engineering, 2011, 8 (4) : 915-930. doi: 10.3934/mbe.2011.8.915
[7]

Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191
[8]

Bertrand Haut, Georges Bastin. A second order model of road junctions in fluid models of traffic networks. Networks and Heterogeneous Media, 2007, 2 (2) : 227-253. doi: 10.3934/nhm.2007.2.227
[9]

Paola Goatin, Elena Rossi. Comparative study of macroscopic traffic flow models at road junctions. Networks and Heterogeneous Media, 2020, 15 (2) : 261-279. doi: 10.3934/nhm.2020012
[10]

Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[11]

Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[12]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[13]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[14]

Cyril Imbert, Régis Monneau. Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6405-6435. doi: 10.3934/dcds.2017278
[15]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[16]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[17]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure and Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[18]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[19]

Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks and Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393
[20]

Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673

2021 Impact Factor: 1.41

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy