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Kinetic and Related Models

October 2021 , Volume 14 , Issue 5

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Heterogeneous discrete kinetic model and its diffusion limit
Ho-Youn Kim, Yong-Jung Kim and Hyun-Jin Lim
2021, 14(5): 749-765 doi: 10.3934/krm.2021023 +[Abstract](677) +[HTML](228) +[PDF](344.7KB)

A revertible discrete velocity kinetic model is introduced when the environment is spatially heterogeneous. It is proved that the parabolic scale singular limit of the model exists and satisfies a new heterogeneous diffusion equation that depends on the diffusivity and the turning frequency together. An energy functional is introduced which takes into account spatial heterogeneity in the velocity field. The monotonicity of the energy functional is the key to obtain uniform estimates needed for the weak convergence proof. The Div-Curl lemma completes the strong convergence proof.

Pencil-beam approximation of fractional Fokker-Planck
Guillaume Bal and Benjamin Palacios
2021, 14(5): 767-817 doi: 10.3934/krm.2021024 +[Abstract](614) +[HTML](208) +[PDF](605.32KB)

We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.

Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence
Pierre Roux and Delphine Salort
2021, 14(5): 819-846 doi: 10.3934/krm.2021025 +[Abstract](545) +[HTML](175) +[PDF](532.87KB)

The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.

A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving
Casimir Emako, Farah Kanbar, Christian Klingenberg and Min Tang
2021, 14(5): 847-866 doi: 10.3934/krm.2021026 +[Abstract](609) +[HTML](187) +[PDF](709.13KB)

In this work we are interested in the stationary preserving property of asymptotic preserving (AP) schemes for kinetic models. We introduce a criterion for AP schemes for kinetic equations to be uniformly stationary preserving (SP). Our key observation is that as long as the Maxwellian of the distribution function can be updated explicitly, such AP schemes are also SP. To illustrate our observation, three different AP schemes for three different kinetic models are considered. Their SP property is proved analytically and tested numerically, which confirms our observations.

Boltzmann-type equations for multi-agent systems with label switching
Nadia Loy and Andrea Tosin
2021, 14(5): 867-894 doi: 10.3934/krm.2021027 +[Abstract](664) +[HTML](189) +[PDF](528.4KB)

In this paper, we propose a Boltzmann-type kinetic description of mass-varying interacting multi-agent systems. Our agents are characterised by a microscopic state, which changes due to their mutual interactions, and by a label, which identifies a group to which they belong. Besides interacting within and across the groups, the agents may change label according to a state-dependent Markov-type jump process. We derive general kinetic equations for the joint interaction/label switch processes in each group. For prototypical birth/death dynamics, we characterise the transient and equilibrium kinetic distributions of the groups via a Fokker-Planck asymptotic analysis. Then we introduce and analyse a simple model for the contagion of infectious diseases, which takes advantage of the joint interaction/label switch processes to describe quarantine measures.

BGK models for inert mixtures: Comparison and applications
Sebastiano Boscarino, Seung Yeon Cho, Maria Groppi and Giovanni Russo
2021, 14(5): 895-928 doi: 10.3934/krm.2021029 +[Abstract](569) +[HTML](135) +[PDF](1702.15KB)

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and numerically, for the latter using an asymptotic preserving semi-Lagrangian scheme for the BGK models. Application to the plane shock wave in a binary mixture of noble gases is also presented.

2020 Impact Factor: 1.432
5 Year Impact Factor: 1.641
2020 CiteScore: 3.1




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