# American Institute of Mathematical Sciences

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

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

 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.
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.
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