Bi-continuous semigroups for flows in infinite networks

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.Comment: 12 page

Equal-area method for scalar conservation laws

We study one-dimensional conservation law. We develop a simple numerical method for computing the unique entropy admissible weak solution to the initial problem. The method basis on the equal-area principle and gives the solution for given time directly.Comment: 10 pages, 7 figure

Diffusion in networks with time-dependent transmission conditions

We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and the diffusion coefficients match (so that mass is conserved) we show that the solution converges exponentially fast to an equilibrium. We also show convergence to a special solution in some other cases.Comment: corrected typos, references removed, revised Lemma A.3. Appl. Math. Optim. (2013

Dynamic Transmission Conditions for Linear Hyperbolic Systems on Networks

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in \cite{KraMugNic20} by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type - or a combination of both. Our results rely upon semigroup theory and elementary linear algebra. We also discuss qualitative properties of solutions

Tokovi na metričnih grafih s splošnimi robnimi pogoji

In this note we study the generation of C_0-semigroups by first-order differential operators on Lp(R_+,C^l) x Lp([0,1],C^m) with general boundary conditions. In many cases we are able to characterize the generation property in terms of the invertibility of a matrix associated to the boundary conditions. The abstract results are used to study the well-posedness of transport equations on non-compact metric graphs.V delu obravnavamo generiranje C_0 polgrup z diferencialnimi operatorji prvega reda na Lp(R_+,C^l) x Lp([0,1],C^m) s splošnimi robnimi pogoji. V mnogih primerih lahko karakteriziramo lastnost generiranja s pomočjo obrnljivosti matrike, ki jo priredimo robnim pogojem. Abstraktne rezultate uporabimo za študij dobre pogojenosti transportne enačbe na nekompaktnih metričnih grafih

Polgrupe za dinamične procese na metričnih grafih

We present the operator semigroups approach to the first- and second-order dynamical systems taking place on metric graphs. We briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, we show two applications to biological models. This article is part of the theme issue "Semigroup applications everywhere".Predstavimo operatorsko polgrupni pristop k dinamičnim sistemom prvega in drugega reda na metričnih grafih. Na kratko strnemo obstoječe rezultate in se osredotočimo na dobro pogojenost problema s standardnimi pogoji v vozliščih. Na koncu prikažemo dve uporabi na modelih v biologiji. Članek je del vsebinske izdaje "Semigroup applications everywhere"

Dynamic transmission conditions for linear hyperbolic systems on networks

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in Kramar et al. (Linear hyperbolic systems on networks. arXiv:2003.08281, 2020) by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type—or a combination of both. Our results rely upon semigroup theory and elementary linear algebra. We also discuss qualitative properties of solutions