Asymptotic behaviour of flows on reducible networks

Previous Article
Variational evolution of one-dimensional Lennard-Jones systems
NHM Home
This Issue
Next Article

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

Asymptotic behaviour of flows on reducible networks

Jacek Banasiak ^1, and Proscovia Namayanja ^2,

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

Abstract
Related Papers

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.

Keywords: semigroups of operators, asymptotic periodicity., topological sorting algorithm, network, Transport equation, resolvent positive operators, diagraph.

Mathematics Subject Classification: Primary: 47D06, 35F45; Secondary: 05C90, 82C7.

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.

[1]	Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[2]	Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa. On the positive semigroups generated by Fleming-Viot type differential operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 323-340. doi: 10.3934/cpaa.2019017
[3]	Danilo Costarelli, Gianluca Vinti. Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators. Mathematical Foundations of Computing, 2020, 3 (1) : 41-50. doi: 10.3934/mfc.2020004
[4]	Daliang Zhao, Yansheng Liu, Xiaodi Li. Controllability for a class of semilinear fractional evolution systems via resolvent operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 455-478. doi: 10.3934/cpaa.2019023
[5]	Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[6]	V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure and Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481
[7]	Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
[8]	Bertrand Lods, Mustapha Mokhtar-Kharroubi, Mohammed Sbihi. Spectral properties of general advection operators and weighted translation semigroups. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469
[9]	Jorge J. Betancor, Alejandro J. Castro, Marta De León-Contreras. Variation operators for semigroups associated with Fourier-Bessel expansions. Communications on Pure and Applied Analysis, 2022, 21 (1) : 239-273. doi: 10.3934/cpaa.2021176
[10]	Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[11]	Ana-Maria Acu, Ioan Cristian Buscu, Ioan Rasa. Generalized Kantorovich modifications of positive linear operators. Mathematical Foundations of Computing, 2022 doi: 10.3934/mfc.2021042
[12]	George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207
[13]	Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167
[14]	Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
[15]	Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems and Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020
[16]	Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
[17]	Zheng Chen, Liu Liu, Lin Mu. Solving the linear transport equation by a deep neural network approach. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 669-686. doi: 10.3934/dcdss.2021070
[18]	Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008
[19]	Jean-Bernard Baillon, Patrick L. Combettes, Roberto Cominetti. Asymptotic behavior of compositions of under-relaxed nonexpansive operators. Journal of Dynamics and Games, 2014, 1 (3) : 331-346. doi: 10.3934/jdg.2014.1.331
[20]	Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic and Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032

2021 Impact Factor: 1.41

Tools

Metrics

Other articles
by authors

on AIMS
Jacek Banasiak Proscovia Namayanja
on Google Scholar
Jacek Banasiak Proscovia Namayanja

[Back to Top]