Diffusion in Networks with Time-Dependent Transmission Conditions

Arendt, Wolfgang; Dier, Dominik; Fijavž, Marjeta Kramar

doi:10.1007/s00245-013-9225-1

Cited by 16 publications

(28 citation statements)

References 9 publications

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“…The assertion about quasi‐contractivity is Laasri,, Prop. 2.1 whereas strong stability was proved by similar means in a special case in Arendt et al, Thm. 5.4…”

Section: Evolution Families: Notations and Preliminary Resultsmentioning

confidence: 99%

Ultracontractivity and Gaussian bounds for evolution families associated with nonautonomous forms

Laasri

Mugnolo

2019

Math Methods in App Sciences

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We develop a variational approach in order to study qualitative properties of nonautonomous parabolic equations. Based on the method of product integrals, we discuss invariance properties and ultracontractivity of evolution families in Hilbert space. Our main results give sufficient conditions for the heat kernel of the evolution family to satisfy Gaussian-type bounds. Along the way, we study examples of nonautonomous equations on graphs, metric graphs, and domains. KEYWORDS evolution families, kernel estimates, nonautonomous parabolic problems MSC CLASSIFICATION 47D06; 47D07; 47A07; 35K90This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Throughout this paper, H is a separable, complex Hilbert space, and V is a further complex Hilbert space that is densely and continuously embedded into H. Let V ′ denote the antidual of V with respect to the pivot space H; the duality between V ′ and V is denoted by ⟨., .⟩. We also denote by (· | ·) V and || · || V the scalar product and the norm on V, respectively, and by (· | ·) and || · || the corresponding quantities in H.We fix T ∈]0, ∞[ and consider a time-dependent family (a(t)) t∈ [0,T] of mappings such that a(t; ·, ·) ∶ V × V → C is for all t ∈ [0, T] a sesquilinear form and(2.1) and such that furthermore there exist constants M, > 0 and ≥ 0 such that the boundedness and H-ellipticity estimatesa.e t ∈ [0, T] and all u, v ∈ V, (2.2) Rea(t; u, u) + ||u|| 2 H ≥ ||u|| 2 V for a.e t ∈ [0, T] and all u ∈ V, (2.3) hold. In what follows, we call such a family ∶= (a(t)) t∈[0,T] bounded H-elliptic nonautonomous form: following Arendt and Dier, 1 we denote by FORM([0, T]; V, H) the class of all such forms. By the Lax-Milgram theorem, for each t ∈ [0, T], there exists an operator associated with a(t, ·, ·), ie, an isomorphism (t, u, v) for all u, v ∈ V ∶ accordingly, we refer to the family (A(t)) t∈ [0,T] as the operator family associated with ∶= (a(t)) t∈ [0,T] . Regarded as an unbounded operator with domain V, −A(t) generates a holomorphic semigroup on V ′ , and in fact by Arendt 25, Thm. 7.1.5 on H too, since a(t) is for all t a bounded, H-elliptic sesquilinear form: With an abuse of notation, we denote its generator -the part of −A(t) in H -again by −A(t), and the semigroup by T t ∶= {e −rA(t) ∶ r ≥ 0}. Hence, for each fixed t, s ∈ [0, T], the Cauchy problem . u(r) + A(t)u(r) = 0, r ∈ [s, T], u(s) = x ∈ H, * 1 (t−s) ẽ * 2 (t−s) (4.16)

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“…The assertion about quasi‐contractivity is Laasri,, Prop. 2.1 whereas strong stability was proved by similar means in a special case in Arendt et al, Thm. 5.4…”

Section: Evolution Families: Notations and Preliminary Resultsmentioning

confidence: 99%

Ultracontractivity and Gaussian bounds for evolution families associated with nonautonomous forms

Laasri

Mugnolo

2019

Math Methods in App Sciences

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“…Note that (A.6) with interface condition (2.10b) can be written as (A.9) with condition (A.11), and the differential operator A is self-adjoint and generates a compact, contractive and positive strongly continuous semigroup. Thus, by adjusting the scalar product to (A.12) and defining corresponding functions and operators based on the boundary conditions in (A.7), the analysis in [41] (see also [4,13]) can be borrowed to show that problem (A.6)-(A.7) with the initial data u 0 ∈ L p (G) has a unique classical solution for t > 0 that continuously depends on the initial data. The Schauder estimates in the assertion (ii) have been derived by [57].…”

Section: A2 Linear Parabolic Problemmentioning

confidence: 99%

“…Stability of steady states of parabolic equations were studied in [62,65]. More studies of diffusion equations in networks can be found in e.g., [4,30,31,63]. In these work, the model parameters are allowed to be time and/or space dependent; the so-called Kirchhoff laws or an excitatoric Kirchhoff condition (or dynamical node condition) are assumed at the interior connecting points.The goal of this paper is to establish a mathematically rigorous foundation of reaction-diffusionadvection equations defined on a metric tree network, which models population dynamics of a biological species on a river network.…”

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confidence: 99%

Population Dynamics in River Networks

2019

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Natural rivers connect to each other to form networks. The geometric structure of a river network can significantly influence spatial dynamics of populations in the system. We consider a process-oriented model to describe population dynamics in river networks of trees, establish the fundamental theories of the corresponding parabolic problems and elliptic problems, derive the persistence threshold by using the principal eigenvalue of the eigenvalue problem, and define the net reproductive rate to describe population persistence or extinction. By virtue of numerical simulations, we investigate the effects of hydrological, physical, and biological factors, especially the structure of the river network, on population persistence. physical and hydrological features in a river network, can greatly influence the spatial distribution of the flow profile (including the flow velocity, water depth, etc.) in branches of the network. Moreover, the population dispersal vectors may be constrained by the network configuration and the flow profile, and species life history traits may depend on varying habitat conditions in the network. As a result, population distribution and persistence in river systems can be significantly affected by the network topology or structure, see e.g., [8,11,12,[14][15][16]46]. Then there arise interesting questions such as whether a population can persist in the desert streams of the southwestern United State while the streams are experiencing substantial natural drying trends [11], or whether dendritic geometry enhances dynamic stability of ecological systems [15] etc. Furthermore, other related dynamics in the network, such as the dynamics of water-born infectious diseases like cholera may also be greatly affected by the river network geometry (see e.g., [6]).Branches in a river network have been modeled as point nodes in a network of habitats in individual based models [11,16] and matrix population models for stage-structured populations [14,39]. However this oversimplifies the spatial heterogeneity of river networks. In a real river ecosystem, organisms mainly live in the branches of the network and the connections between branches (e.g., the network nodes) are mainly for population transitions from one branch to another. To take into account this realistic situation, in recent works [48-50], integro-differential equations and reactiondiffusion-advection equations were used to model population dynamics in river networks where the network branches, instead of the network nodes, are the main habitat for organisms. Here the river networks are modeled under the framework of metric graphs (or metric networks). A metric graph is a graph G = (V, E) with a set V of vertices and a set E of edges, such that each edge e ∈ E is associated with either a closed bounded interval. Mathematic notion of metric graphs was first introduced in the context of wave propagation on thin graph-like domains [5,25], and they are also called quantum graphs.The theories of parabolic and elliptic equations as well as the corr...

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“…Sobolev spaces with values in a Banach space are quite natural objects and occur frequently while treating partial differential equations (see e.g. [Ama95], [Ama01], [DHP03], [CM09], [ADKF14]) and also in probability (see e.g. the upcoming monograph by Hytönen, van Neerven, Veraar and Weis [HvNVW16]).…”

Section: Introductionmentioning

confidence: 99%

“…Special attention is also given to the mapping F (x) = |x| where X is a Banach lattice, e.g. X = L r (Ω) or X = C(K) which occured in [ADKF14]. This mapping and more precisely differentiability properties of the projection onto a convex set in Hilbert space have also been studied by Haraux [Har77].…”

Section: Introductionmentioning

confidence: 99%

Mapping theorems for Sobolev spaces of vector-valued functions

Arendt

Kreuter

2018

Studia Math.

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We consider Sobolev spaces with values in Banach spaces as they are frequently useful in applied problems. Given two Banach spaces X = {0} and Y , each Lipschitz continuous mappingBut if F is one-sided Gateaux differentiable no condition on the space is needed. We also study when weak properties in the sense of duality imply strong properties. Our results are applied to prove embedding theorems, a multi-dimensional version of the Aubin-Lions Lemma and characterizations of the space W 1,p 0 (Ω, X).

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Diffusion in Networks with Time-Dependent Transmission Conditions

Cited by 16 publications

References 9 publications

Ultracontractivity and Gaussian bounds for evolution families associated with nonautonomous forms

Ultracontractivity and Gaussian bounds for evolution families associated with nonautonomous forms

Population Dynamics in River Networks

Mapping theorems for Sobolev spaces of vector-valued functions

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