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Well-posedness and long-term behaviour of buffered flows in infinite networks
Published 22 Apr 2024 in math.AP and math.FA | (2404.14090v2)
Abstract: We consider a transport problem on an infinite metric graph and discuss its well-posedness and long-term behaviour under the condition that the mass flow is buffered in at least one of the vertices. In order to show the well-posedness of the problem, we employ the theory of $C_0$-semigroups and prove a Desch--Schappacher type perturbation theorem for dispersive semigroups. Investigating the long-term behaviour of the system, we prove irreducibility of the semigroup under the assumption that the underlying graph is strongly connected and an additional spectral condition on its adjacency matrix. Moreover, we employ recent results about the convergence of stochastic semigroups that dominate a kernel operator to prove that the solutions converge strongly to equilibrium. Finally, we prove that the solutions converge uniformly under more restrictive assumptions.
- An invitation to operator theory, volume 50 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.
- Robert A. Adams. Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
- Infinite-Dimensional Analysis. Springer-Verlag, Berlin, second edition, 1999. A hitchhiker’s guide.
- Positive Operators, volume 119 of Pure and Applied Mathematics. Academic Press, Inc., Orlando, FL, 1985.
- Wolfgang Arendt. Resolvent positive operators. Proc. London Math. Soc. (3), 54(2):321–349, 1987.
- Positive operator semigroups, volume 257 of Operator Theory: Advances and Applications. Birkhäuser/Springer, Cham, 2017. From finite to infinite dimensions, With a foreword by Rainer Nagel and Ulf Schlotterbeck.
- Asymptotic periodicity of flows in time-depending networks. Netw. Heterog. Media, 8(4):843–855, 2013.
- Bi-continuous semigroups for flows on infinite networks. Netw. Heterog. Media, 16(4):553–567, 2021.
- Some perturbation results for analytic semigroups. Math. Ann., 281(1):157–162, 1988.
- Alexander Dobrick. On the asymptotic behaviour of semigroups for flows in infinite networks. Semigroup Forum, 104(2):256–280, 2022.
- Alexander Dobrick. On the Long-Term Behaviour of Operator Semigroups. PhD thesis, Christian-Albrechts-Universität zu Kiel, 2023.
- Britta Dorn. Semigroups for flows in infinite networks. Semigroup Forum, 76(2):341–356, 2008.
- Asymptotic periodicity of recurrent flows in infinite networks. Math. Z., 263(1):69–87, 2009.
- The semigroup approach to transport processes in networks. Phys. D, 239(15):1416–1421, 2010.
- Operator theoretic aspects of ergodic theory, volume 272 of Graduate Texts in Mathematics. Springer, Cham, 2015.
- Vertex control of flows in networks. Netw. Heterog. Media, 3(4):709–722, 2008.
- One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
- Convergence of positive operator semigroups. Trans. Amer. Math. Soc., 372(9):6603–6627, 2019.
- Uniform convergence of stochastic semigroups. Israel J. Math., 247(1):1–19, 2022.
- Vera Keicher. Almost periodicity of stochastic operators on ℓ1(ℕ)superscriptℓ1ℕ\ell^{1}(\mathbb{N})roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_N ). Tbil. Math. J., 1:105–131, 2008.
- Spectral properties and asymptotic periodicity of flows in networks. Math. Z., 249(1):139–162, 2005.
- Dávid Kunszenti-Kovács. Network perturbations and asymptotic periodicity of recurrent flows in infinite networks. SIAM J. Discrete Math., 23(3):1561–1574, 2009.
- Dávid Kunszenti-Kovács. Perturbations of finite networks and asymptotic periodicity of flow semigroups. Semigroup Forum, 79(2):229–243, 2009.
- Florian Martin. Positive Operator Semigroups and Long-Term Behavior of Buffered Network Flows. Master’s thesis, Universität Ulm, 2018.
- Asymptotic behavior of flows in networks. Forum Math., 19(3):429–461, 2007.
- Peter Meyer-Nieberg. Banach lattices. Universitext. Springer-Verlag, Berlin, 1991.
- Rainer Nagel, editor. One-parameter semigroups of positive operators, volume 1184 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986.
- Agnes Radl. Transport processes in networks with scattering ramification nodes. J. Appl. Funct. Anal., 3(4):461–483, 2008.
- Helmut H. Schaefer. Banach lattices and positive operators. Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.
- Eszter Sikolya. Flows in networks with dynamic ramification nodes. J. Evol. Equ., 5(3):441–463, 2005.
- Witold Wnuk. A characterization of discrete Banach lattices with order continuous norms. Proc. Amer. Math. Soc., 104(1):197–200, 1988.
- Adriaan C. Zaanen. Introduction to operator theory in Riesz spaces. Springer-Verlag, Berlin, 1997.
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