Bi-Continuous semigroups for flows on infinite networks

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical surveys and monographs, American Mathematical Society, 1977. doi: 10.1090/surv/015.
[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.
[14]	A. Dobrick, On the asymptotic behaviour of semigroups for flows in infinite networks, preprint. arXiv: 2011.07014.
[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.
[16]	B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.
[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.
[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.
[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.
[20]	B. Farkas, Perturbations of Bi-Continuous Semigroups, PhD thesis, Eötvös Loránd University, 2003.
[21]	B. Farkas, Perturbations of bi-continuous semigroups, Studia Math., 161 (2004), 147-161. doi: 10.4064/sm161-2-3.
[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.
[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.
[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.
[25]	F. Kühnemund, Bi-Continuous Semigroups on Spaces with Two Topologies: Theory and Applications, PhD thesis, Eberhard-Karls-Universität Tübingen, 2001.
[26]	F. Kühnemund, A Hille-Yosida theorem for bi-continuous semigroups, Semigroup Forum, 67 (2003), 205-225. doi: 10.1007/s00233-002-5000-3.
[27]	H. P. Lotz, Uniform convergence of operators on $L^\infty$ and similar spaces, Math. Z., 190 (1985), 207-220. doi: 10.1007/BF01160459.
[28]	T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.
[29]	A. K. Scirrat, Evolution Semigroups for Well-Posed, NonAutonomous Evolution Families, PhD thesis, Louisiana State University and Agricultural and Mechanical College, 2016.
[30]	W. van Zuijlen, Integration of Functions with Values in a Riesz Space, Master's thesis, Radboud Universiteit Nijmegen, 2012.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Diestel and J. J. Uhl Jr., Vector Measures, Mathematical surveys and monographs, American Mathematical Society, 1977. doi: 10.1090/surv/015.
[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.
[14]	A. Dobrick, On the asymptotic behaviour of semigroups for flows in infinite networks, preprint. arXiv: 2011.07014.
[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.
[16]	B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.
[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.
[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.
[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.
[20]	B. Farkas, Perturbations of Bi-Continuous Semigroups, PhD thesis, Eötvös Loránd University, 2003.
[21]	B. Farkas, Perturbations of bi-continuous semigroups, Studia Math., 161 (2004), 147-161. doi: 10.4064/sm161-2-3.
[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.
[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.
[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.
[25]	F. Kühnemund, Bi-Continuous Semigroups on Spaces with Two Topologies: Theory and Applications, PhD thesis, Eberhard-Karls-Universität Tübingen, 2001.
[26]	F. Kühnemund, A Hille-Yosida theorem for bi-continuous semigroups, Semigroup Forum, 67 (2003), 205-225. doi: 10.1007/s00233-002-5000-3.
[27]	H. P. Lotz, Uniform convergence of operators on $L^\infty$ and similar spaces, Math. Z., 190 (1985), 207-220. doi: 10.1007/BF01160459.
[28]	T. Mátrai and E. Sikolya, Asymptotic behavior of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.
[29]	A. K. Scirrat, Evolution Semigroups for Well-Posed, NonAutonomous Evolution Families, PhD thesis, Louisiana State University and Agricultural and Mechanical College, 2016.
[30]	W. van Zuijlen, Integration of Functions with Values in a Riesz Space, Master's thesis, Radboud Universiteit Nijmegen, 2012.
[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.
[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.

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