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December  2021, 16(4): 553-567. doi: 10.3934/nhm.2021017

Bi-Continuous semigroups for flows on infinite networks

Christian Budde 1, and  Marjeta Kramar Fijavž 2,,

1. 

North-West University, School of Mathematical and Statistical Sciences, Private Bag X6001-209, Potchefstroom 2520, South Africa
2. 

University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1000 Ljubljana, Slovenia, Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia

* Corresponding author: marjeta.kramar@fgg.uni-lj.si

Received  December 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: The authors are grateful to University of Wuppertal for the possibility of funding the stay of the first author at the University of Ljubljana within the Erasmus exchange program. The first author was supported by the DAAD-TKA Project 308019 "Coupled systems and innovative time integrators" and the second author by the Slovenian Research Agency, Grant No. P1-0222

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $ {\mathrm{L}}^{\infty} $-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.

Keywords: Transport equations, infinite metric graphs, bi-continuous operator semigroups, Bochner $ {\mathrm{L}}^{\infty} $-spaces, perturbations.
Mathematics Subject Classification: Primary: 35R02; Secondary: 35F46, 47D06, 46A70.
Citation: Christian Budde, Marjeta Kramar Fijavž. Bi-Continuous semigroups for flows on infinite networks. Networks & Heterogeneous Media, 2021, 16 (4) : 553-567. doi: 10.3934/nhm.2021017
References:
[1]

A. Albanese and F. Kühnemund, Trotter-Kato approximation theorems for locally equicontinuous semigroups, Riv. Mat. Univ. Parma (7), 1 (2002), 19-53.   Google Scholar

[2]

A. A. Albanese, L. Lorenzi and V. Manco, Mean ergodic theorems for bi-continuous semigroups, Semigroup Forum, 82 (2011), 141-171.  doi: 10.1007/s00233-010-9260-z.  Google Scholar

[3]

A. A. Albanese and E. Mangino, Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups, Journal of Mathematical Analysis and Applications, 289 (2004), 477-492.  doi: 10.1016/j.jmaa.2003.08.032.  Google Scholar

[4]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[5]

J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability, Appl. Math. Lett., 45 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.006.  Google Scholar

[6]

J. Banasiak and A. Falkiewicz, A singular limit for an age structured mutation problem, Math. Biosci. Eng., 14 (2017), 17-30.  doi: 10.3934/mbe.2017002.  Google Scholar

[7]

J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, Netw. Heterog. Media, 9 (2014), 197-216.  doi: 10.3934/nhm.2014.9.197.  Google Scholar

[8]

J. Banasiak and A. Puchalska, Generalized network transport and Euler-Hille formula, Discrete Contin. Dyn. Syst., Ser. B, 23 (2018), 1873-1893.  doi: 10.3934/dcdsb.2018185.  Google Scholar

[9]

A. Bátkai, M. Kramar Fijavž and A. Rhandi, Positive Operator Semigroups: From Finite to Infinite Dimensions, Operator Theory: Advances and Applications, Springer International Publishing, 257, 2017. doi: 10.1007/978-3-319-42813-0.  Google Scholar

[10]

F. Bayazit, B. Dorn and M. K. Fijavž, Asymptotic periodicity of flows in time-depending networks, Netw. Heterog. Media, 8 (2013), 843-855.  doi: 10.3934/nhm.2013.8.843.  Google Scholar

[11]

C. Budde and B. Farkas, Intermediate and extrapolated spaces for bi-continuous operator semigroups, J. Evol. Equ., 19 (2019), 321-359.  doi: 10.1007/s00028-018-0477-8.  Google Scholar

[12]

J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical surveys and monographs, American Mathematical Society, 1977. doi: 10.1090/surv/015.  Google Scholar

[13]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[14]

A. Dobrick, On the asymptotic behaviour of semigroups for flows in infinite networks, preprint. arXiv: 2011.07014. Google Scholar

[15]

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

[16]

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

[17]

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.  Google Scholar

[18]

K.-J. Engel, M. K. Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Networks & Heterogeneous Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.  Google Scholar

[19]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[20]

B. Farkas, Perturbations of Bi-Continuous Semigroups, PhD thesis, Eötvös Loránd University, 2003. Google Scholar

[21]

B. Farkas, Perturbations of bi-continuous semigroups, Studia Math., 161 (2004), 147-161.  doi: 10.4064/sm161-2-3.  Google Scholar

[22]

B. Farkas, Perturbations of bi-continuous semigroups with applications to transition semigroups on $C_b(H)$, Semigroup Forum, 68 (2004), 87-107.  doi: 10.1007/s00233-002-0024-2.  Google Scholar

[23]

B. Farkas, Adjoint bi-continuous semigroups and semigroups on the space of measures, Czechoslovak Mathematical Journal, 61 (2011), 309-322.  doi: 10.1007/s10587-011-0076-0.  Google Scholar

[24]

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

[25]

F. Kühnemund, Bi-Continuous Semigroups on Spaces with Two Topologies: Theory and Applications, PhD thesis, Eberhard-Karls-Universität Tübingen, 2001. Google Scholar

[26]

F. Kühnemund, A Hille-Yosida theorem for bi-continuous semigroups, Semigroup Forum, 67 (2003), 205-225.  doi: 10.1007/s00233-002-5000-3.  Google Scholar

[27]

H. P. Lotz, Uniform convergence of operators on $L^\infty$ and similar spaces, Math. Z., 190 (1985), 207-220.  doi: 10.1007/BF01160459.  Google Scholar

[28]

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

[29]

A. K. Scirrat, Evolution Semigroups for Well-Posed, NonAutonomous Evolution Families, PhD thesis, Louisiana State University and Agricultural and Mechanical College, 2016. Google Scholar

[30]

W. van Zuijlen, Integration of Functions with Values in a Riesz Space, Master's thesis, Radboud Universiteit Nijmegen, 2012. Google Scholar

[31]

J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks, Results Math., 47 (2005), 199-225.  doi: 10.1007/BF03323026.  Google Scholar

[32]

J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks under generalized node transition, Results Math., 54 (2009), 15-39.  doi: 10.1007/s00025-009-0376-y.  Google Scholar

show all references

References:
[1]

A. Albanese and F. Kühnemund, Trotter-Kato approximation theorems for locally equicontinuous semigroups, Riv. Mat. Univ. Parma (7), 1 (2002), 19-53.   Google Scholar

[2]

A. A. Albanese, L. Lorenzi and V. Manco, Mean ergodic theorems for bi-continuous semigroups, Semigroup Forum, 82 (2011), 141-171.  doi: 10.1007/s00233-010-9260-z.  Google Scholar

[3]

A. A. Albanese and E. Mangino, Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups, Journal of Mathematical Analysis and Applications, 289 (2004), 477-492.  doi: 10.1016/j.jmaa.2003.08.032.  Google Scholar

[4]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.  Google Scholar

[5]

J. Banasiak and A. Falkiewicz, Some transport and diffusion processes on networks and their graph realizability, Appl. Math. Lett., 45 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.006.  Google Scholar

[6]

J. Banasiak and A. Falkiewicz, A singular limit for an age structured mutation problem, Math. Biosci. Eng., 14 (2017), 17-30.  doi: 10.3934/mbe.2017002.  Google Scholar

[7]

J. Banasiak and P. Namayanja, Asymptotic behaviour of flows on reducible networks, Netw. Heterog. Media, 9 (2014), 197-216.  doi: 10.3934/nhm.2014.9.197.  Google Scholar

[8]

J. Banasiak and A. Puchalska, Generalized network transport and Euler-Hille formula, Discrete Contin. Dyn. Syst., Ser. B, 23 (2018), 1873-1893.  doi: 10.3934/dcdsb.2018185.  Google Scholar

[9]

A. Bátkai, M. Kramar Fijavž and A. Rhandi, Positive Operator Semigroups: From Finite to Infinite Dimensions, Operator Theory: Advances and Applications, Springer International Publishing, 257, 2017. doi: 10.1007/978-3-319-42813-0.  Google Scholar

[10]

F. Bayazit, B. Dorn and M. K. Fijavž, Asymptotic periodicity of flows in time-depending networks, Netw. Heterog. Media, 8 (2013), 843-855.  doi: 10.3934/nhm.2013.8.843.  Google Scholar

[11]

C. Budde and B. Farkas, Intermediate and extrapolated spaces for bi-continuous operator semigroups, J. Evol. Equ., 19 (2019), 321-359.  doi: 10.1007/s00028-018-0477-8.  Google Scholar

[12]

J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical surveys and monographs, American Mathematical Society, 1977. doi: 10.1090/surv/015.  Google Scholar

[13]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[14]

A. Dobrick, On the asymptotic behaviour of semigroups for flows in infinite networks, preprint. arXiv: 2011.07014. Google Scholar

[15]

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

[16]

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

[17]

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.  Google Scholar

[18]

K.-J. Engel, M. K. Fijavž, R. Nagel and E. Sikolya, Vertex control of flows in networks, Networks & Heterogeneous Media, 3 (2008), 709-722.  doi: 10.3934/nhm.2008.3.709.  Google Scholar

[19]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. doi: 10.1007/b97696.  Google Scholar

[20]

B. Farkas, Perturbations of Bi-Continuous Semigroups, PhD thesis, Eötvös Loránd University, 2003. Google Scholar

[21]

B. Farkas, Perturbations of bi-continuous semigroups, Studia Math., 161 (2004), 147-161.  doi: 10.4064/sm161-2-3.  Google Scholar

[22]

B. Farkas, Perturbations of bi-continuous semigroups with applications to transition semigroups on $C_b(H)$, Semigroup Forum, 68 (2004), 87-107.  doi: 10.1007/s00233-002-0024-2.  Google Scholar

[23]

B. Farkas, Adjoint bi-continuous semigroups and semigroups on the space of measures, Czechoslovak Mathematical Journal, 61 (2011), 309-322.  doi: 10.1007/s10587-011-0076-0.  Google Scholar

[24]

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

[25]

F. Kühnemund, Bi-Continuous Semigroups on Spaces with Two Topologies: Theory and Applications, PhD thesis, Eberhard-Karls-Universität Tübingen, 2001. Google Scholar

[26]

F. Kühnemund, A Hille-Yosida theorem for bi-continuous semigroups, Semigroup Forum, 67 (2003), 205-225.  doi: 10.1007/s00233-002-5000-3.  Google Scholar

[27]

H. P. Lotz, Uniform convergence of operators on $L^\infty$ and similar spaces, Math. Z., 190 (1985), 207-220.  doi: 10.1007/BF01160459.  Google Scholar

[28]

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

[29]

A. K. Scirrat, Evolution Semigroups for Well-Posed, NonAutonomous Evolution Families, PhD thesis, Louisiana State University and Agricultural and Mechanical College, 2016. Google Scholar

[30]

W. van Zuijlen, Integration of Functions with Values in a Riesz Space, Master's thesis, Radboud Universiteit Nijmegen, 2012. Google Scholar

[31]

J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks, Results Math., 47 (2005), 199-225.  doi: 10.1007/BF03323026.  Google Scholar

[32]

J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks under generalized node transition, Results Math., 54 (2009), 15-39.  doi: 10.1007/s00025-009-0376-y.  Google Scholar

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