arXiv subjects

Marjeta Kramar Fijavž

Publications and source records attributed to Marjeta Kramar Fijavž.

14 recordsLinked to original sources

arXiv2026

Ornstein--Uhlenbeck semigroup on rooted trees

We study Ornstein--Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The associated semigroups are symmetric, analytic and positivity preserving; the Dirichlet semigroup is sub-Markovian, while the Neumann semigroup is Markovian and admits the Gaussian measure as its unique invariant measure up to scalar multiples. We prove compactness of the resolvent and derive linear eigenvalue asymptotics. For regular rooted trees, we adapt the Naimark--Solomyak decomposition to the Gaussian weighted setting, reducing the operators to one-dimensional half-line problems and obtaining refined spectral localisation and lower bounds.

math.AP↗

arXiv2025

Abstract boundary delay systems and application to network flow

This paper investigates the well-posedness and positivity of solutions to a class of delayed transport equations on a network. The material flow is delayed at the vertices and along the edges. The problem is reformulated as an abstract boundary delay equation, and well-posedness is proved by using the Staffans-Weiss theory. We also establish spectral theory for the associated delay operators and provide conditions for the positivity of the semigroup.

math.AP↗

arXiv2025

Well-posedness of non-autonomous transport equation on metric graphs

We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.

math.AP↗

arXiv2021

Flows on Metric Graphs with General Boundary Conditions

In this note we study the generation of $C_0$-semigroups by first order differential operators on $\mathrm{L}^p (\mathbb{R}_+,\mathbb{C}^{\ell})\times \mathrm{L}^p ([0,1],\mathbb{C}^{m})$ 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 well-posedness of transport equations on non-compact metric graphs.

math.AP↗

arXiv2021

Bi-continuous semigroups for flows on 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.

math.AP↗

arXiv2020

Linear Hyperbolic Systems on Networks

We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order reduction. We study forward and backward well-posedness; furthermore, we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first-order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied.

math.AP↗

arXiv2020

Semigroups for dynamical processes on metric graphs

We present the operator semigroups approach to 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.

math.AP↗

arXiv2020

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.

math.AP↗

arXiv2018

Waves and Diffusion on Metric Graphs with General Vertex Conditions

We prove well-posedness for very general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various different conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on ${\mathrm{L}}^p({\mathbb{R}_+},{\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$ for $1\le p<+\infty$.

math.AP↗

arXiv2016

Exact and Positive Controllability of Boundary Control Systems

Using the semigroup approach to abstract boundary control problems we characterize the space of all exactly reachable states. Moreover, we study the situation when the controls of the system are required to be positive. The abstract results are applied to flows in networks with static as well as dynamic boundary conditions.

math.FA↗

arXiv2016

Semigroups of max-plus linear operators

We define strongly continuous max-additive and max-plus linear operator semigroups and study their main properties. We present some important examples of such semigroups coming from non-linear evolution equations.

math.FA↗

arXiv2014

Asymptotic periodicity of flows in time-depending networks

We consider a linear transport equation on the edges of a network with time-varying coefficients. Using methods for non-autonomous abstract Cauchy problems, we obtain well-posedness of the problem and describe the asymptotic profile of the solutions under certain natural conditions on the network. We further apply our theory to a model used for air traffic flow management.

math.AP↗

arXiv2013

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.

math.AP↗

arXiv2012

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.

math.NA↗