Waves and diffusion on metric graphs with general vertex conditions (original) (raw)

Flows on Metric Graphs with General Boundary Conditions

arXiv (Cornell University), 2021

In this note we study the generation of C0-semigroups by first order differential operators on L p (R+, C ℓ) × L p ([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 well-posedness of transport equations on non-compact metric graphs.

Semigroup approach to diffusion and transport problems on networks

2015

Models describing transport and diffusion processes occurring along the edges of a graph and interlinked by its vertices have been recently receiving a considerable attention. In this paper we generalize such models and consider a network of transport or diffusion operators defined on one dimensional domains and connected through boundary conditions linking the end-points of these domains in an arbitrary way (not necessarily as the edges of a graph are connected). We prove the existence of C_0-semigroups solving such problems and provide conditions fully characterizing when they are positive.

Semigroups for dynamical processes on metric graphs

Philosophical Transactions of the Royal Society A, 2020

We present the operator semigroups approach to first-and secondorder 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.

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

arXiv (Cornell University), 2022

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

Bi-Continuous semigroups for flows on infinite networks

Networks & Heterogeneous Media, 2021

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the \begin{document}$ {\mathrm{L}}^{\infty} $\end{document}-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.

Note on short-time behavior of semigroups associated to self-adjoint operators

Bulletin of the London Mathematical Society, 2016

We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times t roughly like t d , where d is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.

ELLIPTIC OPERATORS ON INFINITE GRAPHS

Papers in Honor of Krzysztof P Wojciechowski, 2006

We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere . For such operators, we discuss analogs of inequalities of Cheeger and Harnack and of the maximum principle (in both elliptic and parabolic versions), and apply them to study spectral theory, the ground state and the heat semigroup associated to these operators.

Topics on diffusion semigroups on a path space with Gibbs measures, (ギブス測度に関する経路空間上の拡散半群の話題）

Proceedings of RIMS Workshop on Stochastic Analysis and Applications, RIMS Kokyuroku Bessatsu, 2008

In this paper, we give a summary of recent results on symmetric diffusion semigroups associated with classical Dirichlet forms on an infinite volume path space C(\mat hr m{ R} , \mat hr m{ R}{ d}) with Gibbs measures. First, we discuss essential self-adjointness of diffusion operators (Dirichlet operators) associated with the Dirichlet forms. We also show the connection between the corresponding diffusion semigroup and the solution of a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on R. Next, we present some functional inequalities for the diffusion semigroup. As applications of these inequalities, we have the existence of a gap at the lower end of spectrum of the Dirichlet operator and the boundedness of the Riesz transforms.

Variational and Semigroup Methods for Waves and Diffusion in Networks

Applied Mathematics and Optimization, 2007

We study diffusion and wave equations in networks. Combining semigroup and variational methods we obtain well-posedness and many nice properties of the solutions in general L p -context. Following earlier articles of other authors, we discuss how the spectrum of the generator can be connected to the structure of the network. We conclude by describing asymptotic behavior of solutions to the diffusion problem.