Flows on Metric Graphs with General Boundary Conditions (original) (raw)
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.
Semigroup approach to diffusion and transport problems on networks
Semigroup Forum, 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 C0-semigroups solving such problems and provide conditions fully characterizing when they are positive.
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.
Waves and diffusion on metric graphs with general vertex conditions
Evolution Equations and Control Theory, 2019
We prove well-posedness for general linear wave-and diffusion equations on compact or non-compact metric graphs allowing various 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 L p (R + , C) × L p ([0, 1], C m) for 1 ≤ p < +∞.
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.
Semigroups for general transport equations with abstract boundary conditions
2006
We investigate C0-semigroup generation properties of the Vlasov equation with general boundary conditions modeled by an abstract boundary operator H. For multiplicative boundary conditions we adapt techniques from and in the case of conservative boundary conditions we show that there is an extension A of the free streaming operator TH which generates a C0semigroup (VH(t)) t 0 in L 1 . Furthermore, following the ideas of [6], we precisely describe its domain and provide necessary and sufficient conditions ensuring that (VH(t)) t 0 is stochastic.
Semigroup generation propertiesof streaming operators with noncontractive boundary conditions
Mathematical and Computer Modelling, 2005
We present c 0 -semigroup generation results for the free streaming operator with abstract boundary conditions. We recall some known results on the matter and establish a general theorem (already announced in ). We motivate our study with a lot of examples and show that our result applies to the physical cases of Maxwell boundary conditions in the kinetic theory of gases as well as to the non-local boundary conditions involved in transport-like equations from population dynamics.
From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups
2008
The paper gives a short account of some basic properties of \textit{Dirichlet-to-Neumann} operators Lambdagamma,partialOmega\Lambda_{\gamma,\partial\Omega}Lambdagamma,partialOmega including the corresponding semigroups motivated by the Laplacian transport in anisotropic media ($\gamma \neq I$) and by elliptic systems with dynamical boundary conditions. For illustration of these notions and the properties we use the explicitly constructed \textit{Lax semigroups}. We demonstrate that for a general smooth bounded
Spectral properties of general advection operators and weighted translation semigroups
Communications on Pure and Applied Analysis, 2009
We investigate the spectral properties of a class of weighted shift semigroups (U(t)) t 0 associated to abstract transport equations with a Lipschitz-continuous vector field F and noreentry boundary conditions. Generalizing the results of [25], we prove that the semigroup (U(t)) t 0 admits a canonical decomposition into three C0-semigroups (U1(t)) t 0 , (U2(t)) t 0 and (U3(t)) t 0 with independent dynamics. A complete description of the spectra of the semigroups (Ui(t)) t 0 and their generators Ti, i = 1, 2 is given. In particular, we prove that the spectrum of Ti is a left-half plane and that the Spectral Mapping Theorem holds: S(Ui(t)) = exp {tS(Ti)}, i = 1, 2. Moreover, the semigroup (U3(t)) t 0 extends to a C0-group and its spectral properties are investigated by means of abstract results from positive semigroups theory. The properties of the flow associated to F are particularly relevant here and we investigate separately the cases of periodic and aperiodic flows. In particular, we show that, for periodic flow, the Spectral Mapping Theorem fails in general but (U3(t)) t 0 and its generator T3 satisfy the so-called Annular Hull Theorem. We illustrate our results with various examples taken from collisionless kinetic theory.
Random Operators and Stochastic Equations, 1993
Diffusion processes with partially reflecting barrier on a hyperplane are obtained as a solutions of a problem of piecing together two diffusion processes with constant diffusion matrices defined on half-bounded domains of finite-dimensional Euclidean space. The problem is investigated by means of methods of the theory of parabolic equations with discontinuous coefficients. Let be a boundary of the domain Di. Consider a diffusion process on D, generated by the differential operator of the second order "•5 t -8 5^. '-'.«. the matrix B{ = (6^•) being constant, symmetric, and positively defined. Our aim is to describe a type of continuous Feller's processes in R m which coincide with given diffusion processes on the domains D\ and D^. We apply methods of the theory of parabolic equations with discontinuous coefficients to solve this problem. Note that the case of Β ι = B 2 = -Β, Β being a variable and sufficiently regular matrix, was considered in [1]. The one-dimensional case was investigated in [2, 3]. Denote by #(R m ) the Danach space of real-valued bounded measurable functions on R m equipped with the norm \\φ\\ = sup x€]Rm |^(z)|. Define a semigroup of operators T t , t > 0, on #(R m ) by the formula T t <p(x)= + I dr [ 9i(t-r,x f -y l ,x rn )V i (r,y'^)ay', χ = (x',x m ) E A, t = 1,2, Jo Js where gi(t, x-y) = gi(t, x'y', x m -2/m) -(^(y -*),y -ThmsJated by the author and A. Babanin
Operator Semigroups: Definitions, Properties and Applications
2006
A large part of contemporary natural science is concerned with investigating the motion of systems in time with a determined state space. To conduct this investigation, we study the theory of one-parameter semigroups. The theory is developed from the simplest scalar case and finite dimensional case to semigroups of linear operators on Banach spaces which started in the first half of the last century. This thesis is designed to give a basic introduction to semigroup theory and its application. Some proofs and illustrative examples are provided. semigroups is also achieved in this chapter. In Chapter III, we will introduce one application in cell population problems. We will show how strongly continuous semigroups generate the solution to a cell equation and show that the solution is unique. It is a future goal to study spectral theory and to explore qualitative properties of strongly continuous semigroups. '•. '•. o '•. '•. i 0 A J kxk
$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain
Journal of Evolution Equations
Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test for C_0C_0C_0-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples.
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.
A generation theorem for kinetic equations with non-contractive boundary operators
Comptes Rendus Mathematique, 2002
In this Note, we present some c 0 -semigroup generation results in L p -spaces for the advection operator submitted to non-contractive boundary conditions covering in particular the classical Maxwell-type boundary conditions. To cite this article: B. Lods, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 655-660. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Résumé Dans cette Note, on présente quelques résultats de génération de c 0 -semigroupe dans les espaces L p pour l'opérateur d'advection soumis à des conditions aux limites non contractives, couvrant par exemple les conditions frontières de type Maxwell. Pour citer cet article : B. Lods, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 655-660. 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS L'objectif de cette Note est d'énoncer quelques résultats de génération de c 0 -semigroupe dans L p (1 p < ∞) pour l'opérateur d'advection soumis à des conditions aux limites non-contractives. Ici, est un ouvert régulier de R N (N 1) désignant indifféremment un domaine intérieur ou extérieur, alors que V désigne le support d'une mesure de Radon positivie dµ sur R N . L'opérateur T H décrit le transport de particules (neutrons, molécules gazeuses, etc.) dans le domaine . La fonction ψ(x, v) désigne la densité de particules ayant la position x ∈ animées de la vitesse v ∈ V . Les conditions aux limites associées à cet opérateur sont gouvernées par un opérateur linéaire H reliant la distribution de particules sur le bord rentrant -à celle sur le bord sortant + : désignant le vecteur normal extérieur à en x ∈ ∂ . L'opérateur frontière H est un opérateur linéaire et borné de L + p sur L - p où L ± p sont des espaces de traces
On L p-Contractivity of Semigroups Generated by Linear Partial Differential Operators
1999
This paper is devoted to the study of contraction semigroups generated by linear partial differential operators. It is shown that linear partial differential operators of order higher than two cannot generate contraction semigroups on (L ) for p # [1, ) unless p=2. If p>1 and the L -dissipativity criterion is restricted to the cone of nonnegative functions for differential operators with real-valued coefficients, it is proven that the criterion still fails for operators of order higher than two, except for some fourth order operators if 3 2 p 3. A class of such fourth order operators is also presented. 1999 Academic Press
MORSE DECOMPOSITIONS OF SEMIFLOWS ASSOCIATED WITH GRAPHS
EXTENDED ABSTRACT. The mathematical theory of dynamical systems analyzes, from an axiomatic point of view, the common features of many models that describe the behavior of systems in time. In its abstract form, a dynamical system is given by a time set T (with semigroup operation •), a state space M , and a map Φ : T ×M → M that satisfies (i) Φ(0, x) = x for all x ∈ M , describing the initial value, and (ii) Φ(t • s, x) = Φ(t, Φ(s, x)) for all t, s ∈ T and x ∈ M . Common examples for the time set T are the natural numbers N or the nonnegative reals R + as semigroups, and the integers Z or the reals R as groups (under addition). If the state space M carries an additional structure, such a being a measurable space, a topological space or a manifold, the map Φ is required to respect this structure, i.e. it is assumed to be measurable, continuous, or differentiable, respectively. At the heart of the theory of dynamical systems is the study of system behavior as t → ∞ or t → ±∞ (qualitat...
Evolution semigroups for nonautonomous Cauchy problems
Abstract and Applied Analysis, 1997
In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems(NCP) {u˙(t)=A(t)u(t)u(s)=x∈Xon a Banach spaceXby the existence of certain evolution semigroups.Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some “concrete” conditions. As a typical example, we discuss the so called “parabolic” case.