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Kinetic and Related Models

December 2021 , Volume 14 , Issue 6

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The delayed Cucker-Smale model with short range communication weights
Zili Chen and Xiuxia Yin
2021, 14(6): 929-948 doi: 10.3934/krm.2021030 +[Abstract](575) +[HTML](246) +[PDF](420.56KB)
Abstract:

Various flocking results have been established for the delayed Cucker-Smale model, especially in the long range communication case. However, the short range communication case is more realistic due to the limited communication ability. In this case, the non-flocking behavior can be frequently observed in numerical simulations. Furthermore, it has potential applications in many practical situations, such as the opinion disagreement in society, fish flock breaking and so on. Therefore, we firstly consider the non-flocking behavior of the delayed Cucker\begin{document}$ - $\end{document}Smale model. Based on a key inequality of position variance, a simple sufficient condition of the initial data to the non-flocking behavior is established. Then, for general communication weights we obtain a flocking result, which also depends upon the initial data in the short range communication case. Finally, with no restriction on the initial data we further establish other large time behavior of classical solutions.

Relativistic BGK model for massless particles in the FLRW spacetime
Byung-Hoon Hwang, Ho Lee and Seok-Bae Yun
2021, 14(6): 949-959 doi: 10.3934/krm.2021031 +[Abstract](624) +[HTML](192) +[PDF](298.04KB)
Abstract:

In this paper, we address the Cauchy problem for the relativistic BGK model proposed by Anderson and Witting for massless particles in the Friedmann-Lemaȋtre-Robertson-Walker (FLRW) spacetime. We first derive the explicit form of the Jüttner distribution in the FLRW spacetime, together with a set of nonlinear relations that leads to the conservation laws of particle number, momentum, and energy for both Maxwell-Boltzmann particles and Bose-Einstein particles. Then, we find sufficient conditions that guarantee the existence of equilibrium coefficients satisfying the nonlinear relations and we show that the condition is satisfied through all the induction steps once it is verified for the initial step. Using this observation, we construct explicit solutions of the relativistic BGK model of Anderson-Witting type for massless particles in the FLRW spacetime.

The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions
Philippe Laurençot and Christoph Walker
2021, 14(6): 961-980 doi: 10.3934/krm.2021032 +[Abstract](395) +[HTML](124) +[PDF](355.58KB)
Abstract:

The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.

Macroscopic descriptions of follower-leader systems
Sara Bernardi, Gissell Estrada-Rodriguez, Heiko Gimperlein and Kevin J. Painter
2021, 14(6): 981-1002 doi: 10.3934/krm.2021035 +[Abstract](496) +[HTML](148) +[PDF](645.26KB)
Abstract:

The fundamental derivation of macroscopic model equations to describe swarms based on microscopic movement laws and mathematical analyses into their self-organisation capabilities remains a challenge from the perspective of both modelling and analysis. In this paper we clarify relevant continuous macroscopic model equations that describe follower-leader interactions for a swarm where these two populations are fixed. We study the behaviour of the swarm over long and short time scales to shed light on the number of leaders needed to initiate swarm movement, according to the homogeneous or inhomogeneous nature of the interaction (alignment) kernel. The results indicate the crucial role played by the interaction kernel to model transient behaviour.

Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics
Seung-Yeal Ha, Myeongju Kang and Bora Moon
2021, 14(6): 1003-1033 doi: 10.3934/krm.2021036 +[Abstract](435) +[HTML](146) +[PDF](1068.85KB)
Abstract:

We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a \begin{document}$ L^1 $\end{document}-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.

Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system
Dayton Preissl, Christophe Cheverry and Slim Ibrahim
2021, 14(6): 1035-1079 doi: 10.3934/krm.2021042 +[Abstract](443) +[HTML](116) +[PDF](542.25KB)
Abstract:

This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function \begin{document}$ f(t,\cdot) $\end{document}. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field \begin{document}$ x \mapsto \epsilon^{-1} \mathbf{B}_e(x) $\end{document}, where \begin{document}$ \epsilon $\end{document} is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields \begin{document}$ (E,B) $\end{document} are supposed to be small. In the non-magnetized setting, local \begin{document}$ C^1 $\end{document}-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since \begin{document}$ \epsilon^{-1} $\end{document} is large, standard results predict that the lifetime \begin{document}$ T_\epsilon $\end{document} of solutions may shrink to zero when \begin{document}$ \epsilon $\end{document} goes to \begin{document}$ 0 $\end{document}. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (\begin{document}$ 0<T<T_\epsilon $\end{document}) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows \begin{document}$ f $\end{document} remains at a distance \begin{document}$ \epsilon $\end{document} from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.

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

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