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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 & 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.   Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41.   Google Scholar

[3]

A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications \textbf{20}, 20 (2000).   Google Scholar

[4]

R. M. Colombo and M. Garavello, On the $p$-system at a junction,, in, 426 (2007), 193.   Google Scholar

[5]

R. M. Colombo and M. Garavello, On the 1D modeling of fluid flowing through a junction,, preprint, (2009).   Google Scholar

[6]

R. M. Colombo and G. Guerra, On general balance laws with boundary,, J. Diff. Equations, 248 (2010), 1017.  doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[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.   Google Scholar

[8]

R. M. Colombo, M. Herty and V. Sachers, On $2\times2$ conservation laws at a junction,, SIAM J. Math. Anal., 40 (2008), 605.  doi: 10.1137/070690298.  Google Scholar

[9]

R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the p-system,, J. Math. Anal. Appl., 361 (2010), 440.  doi: 10.1016/j.jmaa.2009.07.022.  Google Scholar

[10]

R. M. Colombo and C. Mauri, Euler system at a junction,, Journal of Hyperbolic Differential Equations, 5 (2008), 547.  doi: 10.1142/S0219891608001593.  Google Scholar

[11]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics \textbf{1}, 1 (2006).   Google Scholar

[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.  doi: 10.1016/j.anihpc.2004.02.002.  Google Scholar

[13]

G. Guerra, F. Marcellini and V. Schleper, Balance laws with integrable unbounded source,, SIAM J. Math. Anal., 41 (2009), 1164.  doi: 10.1137/080735436.  Google Scholar

[14]

H. Holden and N. H. Risebro, Riemann problems with a kink,, SIAM J. Math. Anal., 30 (1999), 497.  doi: 10.1137/S0036141097327033.  Google Scholar

[15]

T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle,, Comm. Math. Phys., 83 (1982), 243.  doi: 10.1007/BF01976043.  Google Scholar

[16]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition, (1994).   Google Scholar

[17]

G. B. Whitham, "Linear and Nonlinear Waves,", John Wiley & Sons Inc., (1999).   Google Scholar

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.   Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41.   Google Scholar

[3]

A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications \textbf{20}, 20 (2000).   Google Scholar

[4]

R. M. Colombo and M. Garavello, On the $p$-system at a junction,, in, 426 (2007), 193.   Google Scholar

[5]

R. M. Colombo and M. Garavello, On the 1D modeling of fluid flowing through a junction,, preprint, (2009).   Google Scholar

[6]

R. M. Colombo and G. Guerra, On general balance laws with boundary,, J. Diff. Equations, 248 (2010), 1017.  doi: 10.1016/j.jde.2009.12.002.  Google Scholar

[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.   Google Scholar

[8]

R. M. Colombo, M. Herty and V. Sachers, On $2\times2$ conservation laws at a junction,, SIAM J. Math. Anal., 40 (2008), 605.  doi: 10.1137/070690298.  Google Scholar

[9]

R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the p-system,, J. Math. Anal. Appl., 361 (2010), 440.  doi: 10.1016/j.jmaa.2009.07.022.  Google Scholar

[10]

R. M. Colombo and C. Mauri, Euler system at a junction,, Journal of Hyperbolic Differential Equations, 5 (2008), 547.  doi: 10.1142/S0219891608001593.  Google Scholar

[11]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics \textbf{1}, 1 (2006).   Google Scholar

[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.  doi: 10.1016/j.anihpc.2004.02.002.  Google Scholar

[13]

G. Guerra, F. Marcellini and V. Schleper, Balance laws with integrable unbounded source,, SIAM J. Math. Anal., 41 (2009), 1164.  doi: 10.1137/080735436.  Google Scholar

[14]

H. Holden and N. H. Risebro, Riemann problems with a kink,, SIAM J. Math. Anal., 30 (1999), 497.  doi: 10.1137/S0036141097327033.  Google Scholar

[15]

T. P. Liu, Nonlinear stability and instability of transonic flows through a nozzle,, Comm. Math. Phys., 83 (1982), 243.  doi: 10.1007/BF01976043.  Google Scholar

[16]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition, (1994).   Google Scholar

[17]

G. B. Whitham, "Linear and Nonlinear Waves,", John Wiley & Sons Inc., (1999).   Google Scholar

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