June  2014, 9(2): 197-216. doi: 10.3934/nhm.2014.9.197

Asymptotic behaviour of flows on reducible networks

1. 

School of Mathematical Sciences, UKZN, Durban

2. 

School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa

Received  March 2013 Revised  December 2013 Published  July 2014

In this paper we extend some of the previous results for a system of transport equations on a closed network. We consider the Cauchy problem for a flow on a reducible network; that is, a network represented by a diagraph which is not strongly connected. In particular, such a network can contain sources and sinks. We prove well-posedness of the problem with generalized Kirchhoff's conditions, which allow for amplification and/or reduction of the flow at the nodes, on such reducible networks with sources but show that the problem becomes ill-posed if the network has a sink. Furthermore, we extend the existing results on the asymptotic periodicity of the flow to such networks. In particular, in contrast to previous papers, we consider networks with acyclic parts and we prove that such parts of the network become depleted in a finite time, an estimate of which is also provided. Finally, we show how to apply these results to open networks where a portion of the flowing material is allowed to leave the network.
Citation: Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks and Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
References:
[1]

W. J. Anderson, Continuous-Time Markov Chains. An Application Oriented Approach, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0.

[2]

W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349. doi: 10.1112/plms/s3-54.2.321.

[3]

J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications, Springer Verlag, London, 2006.

[4]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001.

[5]

J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1.

[6]

N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall, Inc., Englewood Cliffs, 1974.

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Aproach, Ph.D thesis, University of Tübingen, 2008.

[8]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.

[9]

B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.

[10]

B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x.

[11]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 2000.

[12]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, Inc. New York, 1959.

[13]

Ch. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York, 2001. doi: 10.1007/978-1-4613-0163-9.

[14]

F.M. Hante, G. Leugering and T. I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. & Optimization, 59 (2009), 275-292. doi: 10.1007/s00245-008-9057-6.

[15]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.

[16]

T. Matrai and E. Sikolya, Asymptotic behaviour of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.

[17]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.

[18]

H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.

[19]

R. Nagel, ed., One-parameter Semigroups of Positive Operators, Springer Verlag, Berlin, 1986.

[20]

P. Namayanja, Transport on Network Structures, Ph.D thesis, UKZN, 2012.

[21]

E. Seneta, Nonnegative Matrices and Markov Chains, Springer Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.

[22]

E. Sikolya, Semigroups for Flows in Networks, Ph.D dissertation, University of Tübingen, 2004.

[23]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.

show all references

References:
[1]

W. J. Anderson, Continuous-Time Markov Chains. An Application Oriented Approach, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0.

[2]

W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349. doi: 10.1112/plms/s3-54.2.321.

[3]

J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications, Springer Verlag, London, 2006.

[4]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001.

[5]

J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1.

[6]

N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall, Inc., Englewood Cliffs, 1974.

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Aproach, Ph.D thesis, University of Tübingen, 2008.

[8]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.

[9]

B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.

[10]

B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x.

[11]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 2000.

[12]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, Inc. New York, 1959.

[13]

Ch. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York, 2001. doi: 10.1007/978-1-4613-0163-9.

[14]

F.M. Hante, G. Leugering and T. I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. & Optimization, 59 (2009), 275-292. doi: 10.1007/s00245-008-9057-6.

[15]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.

[16]

T. Matrai and E. Sikolya, Asymptotic behaviour of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.

[17]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.

[18]

H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.

[19]

R. Nagel, ed., One-parameter Semigroups of Positive Operators, Springer Verlag, Berlin, 1986.

[20]

P. Namayanja, Transport on Network Structures, Ph.D thesis, UKZN, 2012.

[21]

E. Seneta, Nonnegative Matrices and Markov Chains, Springer Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.

[22]

E. Sikolya, Semigroups for Flows in Networks, Ph.D dissertation, University of Tübingen, 2004.

[23]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.

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