Kinetic Models in Natural Sciences

Banasiak, Jacek

doi:10.1007/978-3-319-11322-7_4

Cited by 7 publications

(7 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us recall that A is the realization of A = diag{−c j ∂ x } 1≤j≤m on the domain D(A) = {u ∈ W 1 1 (I); u(0) = Ku(1)}. The following theorem for (7) has been proved in [3] (see also [4]) but the proof for any nonnegative matrix K is practically the same. Here we extend this proof to an arbitrary K. However, for the proof in the general case we need to provide basic steps of the proof for nonnegative matrices.…”

Section: Solvability Of (8)mentioning

confidence: 99%

Semigroup approach to diffusion and transport problems on networks

2015

Self Cite

View full text Add to dashboard Cite

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 in the way 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.

show abstract

Section: Solvability Of (8)mentioning

confidence: 99%

Semigroup approach to diffusion and transport problems on networks

2015

Self Cite

View full text Add to dashboard Cite

show abstract

“…The generation of these semigroups was obtained in [135]. More information on kinetic models can be found in [136].…”

Section: Birth-and-death Modelsmentioning

confidence: 99%

Linear dynamics of semigroups generated by differential operators

et al. 2017

View full text Add to dashboard Cite

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C 0 -semigroups of linear and continuous operators.We will review some of these notions and we will discuss basic properties of the dynamics of C 0 -semigroups. We will also study in detail the dynamics of the translation C 0 -semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C 0 -semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models. . On the one hand, it is connected with the still unsolved Invariant Subspace Problem on Hilbert spaces, which asks for the existence of an operator with no nontrivial closed invariant subset. The answer was positive in Banach spaces, as it was seen by Read [4] in`1. On the other hand, there is a number of different connections of linear dynamics with different areas such as algebra, topology, real and complex analysis, functional analysis, approximation theory, number theory, and probability.The advances in the area were first compiled by Grosse-Erdmann [5,6]. The monographs of Bayart and Matheron [7] and Grosse-Erdmann and Peris [8] represent a good source of the state of the art in the area. The study of the size and the algebraic structure of the set of vectors with wild behaviour, provides an interesting field for continuing with the quest of surprising examples of sets of pathological elements that, however, are preserved by the elementary algebraic operators. In this line, a selection of topics was recently revisited by Aron et al. in [9, Ch. 4], see also [10].Given a family fT i g i 2I of linear and continuous operators on an infinite-dimensional separable Banach space X , we say that it is universal if there exists some element x 2 X such that, its orbit fT i x W i 2 I g is dense in X .

show abstract

“…We assume that c1 (x, y; z; γ) = c 1 (x, y; z)e(t (1) z , γ\{x, y}), c2 (x; y; γ) = c 2 (x; y)e(t (2) y , γ\x), with…”

Section: Dynamics Of the Correlation Functionsmentioning

confidence: 99%

“…To the best of our knowledge, the microscopic modeling of this kind of merging is performed here for the first time. All previous theories, see, e.g., [1,2,3,4,10,12,13], deal with the densities of the particles and hence do not take into account the corpuscular structure of the system. However, we do not take into account the particle mass, which is supposed to be done in an extension of the present model.…”

Section: Introductionmentioning

confidence: 99%

A kinetic equation for repulsive coalescing random jumps in continuum

Pilorz

2016

View full text Add to dashboard Cite

A continuum individual-based model of hopping and coalescing particles is introduced and studied. Its microscopic dynamics are described by a hierarchy of evolution equations obtained in the paper. Then the passage from the micro-to mesoscopic dynamics is performed by means of a Vlasov-type scaling. The existence and uniqueness of the solutions of the corresponding kinetic equation are proved.2010 Mathematics Subject Classification. 60K35; 35Q83; 82C22.

show abstract

Kinetic Models in Natural Sciences

Cited by 7 publications

References 34 publications

Semigroup approach to diffusion and transport problems on networks

Semigroup approach to diffusion and transport problems on networks

Linear dynamics of semigroups generated by differential operators

A kinetic equation for repulsive coalescing random jumps in continuum

Contact Info

Product

Resources

About