American Institute of Mathematical Sciences

Advanced
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

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 & 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, (1991).  doi: 10.1007/978-1-4612-3038-0.  Google Scholar

[2]

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

[3]

J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications,, Springer Verlag, (2006).   Google Scholar

[4]

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

[5]

J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications,, 2nd ed., (2009).  doi: 10.1007/978-1-84800-998-1.  Google Scholar

[6]

N. Deo, Graph Theory with Applications to Engineering and Computer Science,, Prentice Hall, (1974).   Google Scholar

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Aproach,, Ph.D thesis, (2008).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations,, Springer Verlag, (2000).   Google Scholar

[12]

F. R. Gantmacher, Applications of the Theory of Matrices,, Interscience Publishers, (1959).   Google Scholar

[13]

Ch. Godsil and G. Royle, Algebraic Graph Theory,, Springer Verlag, (2001).  doi: 10.1007/978-1-4613-0163-9.  Google Scholar

[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.  doi: 10.1007/s00245-008-9057-6.  Google Scholar

[15]

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

[16]

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

[17]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar

[18]

H. Minc, Nonnegative Matrices,, John Wiley & Sons, (1988).   Google Scholar

[19]

R. Nagel, ed., One-parameter Semigroups of Positive Operators,, Springer Verlag, (1986).   Google Scholar

[20]

P. Namayanja, Transport on Network Structures,, Ph.D thesis, (2012).   Google Scholar

[21]

E. Seneta, Nonnegative Matrices and Markov Chains,, Springer Verlag, (1981).  doi: 10.1007/0-387-32792-4.  Google Scholar

[22]

E. Sikolya, Semigroups for Flows in Networks,, Ph.D dissertation, (2004).   Google Scholar

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications,, Springer Verlag, (2006).   Google Scholar

[4]

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

[5]

J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications,, 2nd ed., (2009).  doi: 10.1007/978-1-84800-998-1.  Google Scholar

[6]

N. Deo, Graph Theory with Applications to Engineering and Computer Science,, Prentice Hall, (1974).   Google Scholar

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Aproach,, Ph.D thesis, (2008).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations,, Springer Verlag, (2000).   Google Scholar

[12]

F. R. Gantmacher, Applications of the Theory of Matrices,, Interscience Publishers, (1959).   Google Scholar

[13]

Ch. Godsil and G. Royle, Algebraic Graph Theory,, Springer Verlag, (2001).  doi: 10.1007/978-1-4613-0163-9.  Google Scholar

[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.  doi: 10.1007/s00245-008-9057-6.  Google Scholar

[15]

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

[16]

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

[17]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar

[18]

H. Minc, Nonnegative Matrices,, John Wiley & Sons, (1988).   Google Scholar

[19]

R. Nagel, ed., One-parameter Semigroups of Positive Operators,, Springer Verlag, (1986).   Google Scholar

[20]

P. Namayanja, Transport on Network Structures,, Ph.D thesis, (2012).   Google Scholar

[21]

E. Seneta, Nonnegative Matrices and Markov Chains,, Springer Verlag, (1981).  doi: 10.1007/0-387-32792-4.  Google Scholar

[22]

E. Sikolya, Semigroups for Flows in Networks,, Ph.D dissertation, (2004).   Google Scholar

[23]

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

[1]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & 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 & 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 & Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469
[9]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[10]

George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207
[11]

Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167
[12]

Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
[13]

Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems & Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020
[14]

Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
[15]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008
[16]

Jean-Bernard Baillon, Patrick L. Combettes, Roberto Cominetti. Asymptotic behavior of compositions of under-relaxed nonexpansive operators. Journal of Dynamics & Games, 2014, 1 (3) : 331-346. doi: 10.3934/jdg.2014.1.331
[17]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
[18]

Mostafa Adimy, Laurent Pujo-Menjouet. Asymptotic behavior of a singular transport equation modelling cell division. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 439-456. doi: 10.3934/dcdsb.2003.3.439
[19]

Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[20]

Kenji Nakanishi. Modified wave operators for the Hartree equation with data, image and convergence in the same space. Communications on Pure & Applied Analysis, 2002, 1 (2) : 237-252. doi: 10.3934/cpaa.2002.1.237

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences