-
Previous Article
Well-posedness and approximate controllability of neutral network systems
- NHM Home
- This Issue
-
Next Article
Rumor spreading dynamics with an online reservoir and its asymptotic stability
Bi-Continuous semigroups for flows on infinite networks
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 |
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.
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. |
[1] |
Lu Zhang, Aihong Zou, Tao Yan, Ji Shu. Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 749-768. doi: 10.3934/dcdsb.2021063 |
[2] |
Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030 |
[3] |
Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733 |
[4] |
Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028 |
[5] |
O. A. Veliev. On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1537-1562. doi: 10.3934/cpaa.2020077 |
[6] |
Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083 |
[7] |
Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049 |
[8] |
Delio Mugnolo, Abdelaziz Rhandi. Ornstein–Uhlenbeck semigroups on star graphs. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022030 |
[9] |
Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641 |
[10] |
Leandro M. Del Pezzo, Nicolás Frevenza, Julio D. Rossi. Convex and quasiconvex functions in metric graphs. Networks and Heterogeneous Media, 2021, 16 (4) : 591-607. doi: 10.3934/nhm.2021019 |
[11] |
Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905 |
[12] |
Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963 |
[13] |
Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271 |
[14] |
T. Hillen. On the $L^2$-moment closure of transport equations: The general case. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 299-318. doi: 10.3934/dcdsb.2005.5.299 |
[15] |
Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic and Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669 |
[16] |
T. Hillen. On the $L^2$-moment closure of transport equations: The Cattaneo approximation. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 961-982. doi: 10.3934/dcdsb.2004.4.961 |
[17] |
Joachim von Below, José A. Lubary. Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs. Networks and Heterogeneous Media, 2018, 13 (4) : 691-717. doi: 10.3934/nhm.2018031 |
[18] |
Frédéric Robert. On the influence of the kernel of the bi-harmonic operator on fourth order equations with exponential growth. Conference Publications, 2007, 2007 (Special) : 875-882. doi: 10.3934/proc.2007.2007.875 |
[19] |
Roberto Alicandro, Andrea Braides, Marco Cicalese. $L^\infty$ jenergies on discontinuous functions. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 905-928. doi: 10.3934/dcds.2005.12.905 |
[20] |
Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773 |
2021 Impact Factor: 1.41
Tools
Metrics
Other articles
by authors
[Back to Top]