Dynamic transmission conditions for linear hyperbolic systems on networks

Fijavž, Marjeta Kramar; Mugnolo, Delio; Nicaise, Serge

doi:10.1007/s00028-021-00715-0

Cited by 5 publications

(2 citation statements)

References 41 publications

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“…These boundary conditions are stationary ones but can be also used instead of (1.6). More generally in the spirit of [18], we can mix up stationary and dynamical boundary conditions at the common vertex v in the following way.…”

Section: Some Extensions and Open Problems 41 Other Dynamical Boundar...mentioning

confidence: 99%

“…2.1 (eliminating the variables W ℓ and one of the other variables to obtain a variational formulation), we can prove that the associated operator is maximal dissipative under additional conditions on Y v and B v and therefore system (1.3) with the boundary conditions (1.5) and (4.2) is well-posed and is governed by a semigroup of contractions. In the general case, using a bounded perturbation argument as in [18,Theorem 3.3] we can prove that a bounded perturbation of A is maximal dissipative under additional conditions on Y v and B v and therefore system (1.3) with the boundary conditions (1.5) and (4.2) is well-posed and is governed by a C 0 semigroup. But using the dissipativeness of A, one can deduce that A generates a semigroup of contractions.…”

Section: Some Extensions and Open Problems 41 Other Dynamical Boundar...mentioning

confidence: 99%

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Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

et al. 2020

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The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general criteria of Arendt-Batty (see Arendt and Batty in Trans. Am. Math. Soc. 306(2):837-852, 1988), combined with a new uniqueness result, we prove that our system is strongly stable. Next, using a frequency domain approach, combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero.

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Section: Some Extensions and Open Problems 41 Other Dynamical Boundar...mentioning

confidence: 99%

Section: Some Extensions and Open Problems 41 Other Dynamical Boundar...mentioning

confidence: 99%

Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

et al. 2020

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A Model of String System Deformations on a Star Graph with Nonlinear Condition at the Node

Zvereva

2024

J Math Sci

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Dirac gauge theory for topological spinors in 3+1 dimensional networks

Bianconi

2023

J. Phys. A: Math. Theor.

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Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation.Here we propose a Dirac gauge theory for topological spinors in $3+1$ dimensional networks associated to an arbitrary metric. Topological spinors are the direct sum of $0$-cochains and $1$-cochains defined on a network and describe a matter field defined on both nodes and links of a network. Recently in Ref. \cite{bianconi2021topological} it has been shown that topological spinors obey the topological Dirac equation driven by the discrete Dirac operator. In this work we extend these results by formulating the Dirac equation on weighted and directed $3+1$ dimensional networks which allow for the treatment of a local theory. The commutators and anti-commutators of the Dirac operators are non vanishing an they define the curvature tensor and magnetic field of our theory respectively. This interpretation is confirmed by the non-relativistic limit of the proposed Dirac equation. In the non-relativistic limit of the proposed Dirac equation the sector of the spinor defined on links follows the Schr\"odinger equation with the correct giromagnetic moment, while the sector of the spinor defined on nodes follows the Klein-Gordon equation and is not negligible. The action associated to the proposed field theory comprises of a Dirac action and a metric action. We describe the gauge invariance of the action under both Abelian and non-Abelian transformations and we propose the equation of motion of the field theory of both Dirac and metric fields. This theory can be interpreted as a limiting case of a more general gauge theory valid on any arbitrary network in the limit of almost flat spaces.

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Dynamic transmission conditions for linear hyperbolic systems on networks

Cited by 5 publications

References 41 publications

Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

Existence, Uniqueness and Stabilization of Solutions of a Generalized Telegraph Equation on Star Shaped Networks

A Model of String System Deformations on a Star Graph with Nonlinear Condition at the Node

Dirac gauge theory for topological spinors in 3+1 dimensional networks

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