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Kinetic & Related Models

December 2020 , Volume 13 , Issue 6

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A semigroup approach to the convergence rate of a collisionless gas
Armand Bernou
2020, 13(6): 1071-1106 doi: 10.3934/krm.2020038 +[Abstract](353) +[HTML](105) +[PDF](583.34KB)
Abstract:

We study the rate of convergence to equilibrium for a collisionless (Knudsen) gas enclosed in a vessel in dimension \begin{document}$ n \in \{2,3\} $\end{document}. By semigroup arguments, we prove that in the \begin{document}$ L^1 $\end{document} norm, the polynomial rate of convergence \begin{document}$ \frac{1}{(t+1)^{n-}} $\end{document} given in [25], [17] and [18] can be extended to any \begin{document}$ C^2 $\end{document} domain, with standard assumptions on the initial data. This is to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 relying on deterministic arguments that does not require any symmetry of the domain, nor a monokinetic regime. The dependency of the rate with respect to the initial distribution is detailed. Our study includes the case where the temperature at the boundary varies. The demonstrations are adapted from a deterministic version of a subgeometric Harris' theorem recently established by Cañizo and Mischler [7]. We also compare our model with a free-transport equation with absorbing boundary.

Averaging of highly-oscillatory transport equations
Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou and Florian Méhats
2020, 13(6): 1107-1133 doi: 10.3934/krm.2020039 +[Abstract](271) +[HTML](114) +[PDF](570.63KB)
Abstract:

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature are re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder
Jörg Weber
2020, 13(6): 1135-1161 doi: 10.3934/krm.2020040 +[Abstract](317) +[HTML](108) +[PDF](467.81KB)
Abstract:

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions and give conditions on the external magnetic field under which the plasma is confined inside the cylinder, i.e., it stays away from the boundary of the cylinder.

On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3
Shigeru Takata, Masanari Hattori and Takumu Miyauchi
2020, 13(6): 1163-1174 doi: 10.3934/krm.2020041 +[Abstract](328) +[HTML](133) +[PDF](595.9KB)
Abstract:

Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for \begin{document}$ \mathrm{Pr}<2/3 $\end{document}, provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.

Global existence theorem for a model governing the motion of two cell populations
Brock C. Price and Xiangsheng Xu
2020, 13(6): 1175-1191 doi: 10.3934/krm.2020042 +[Abstract](256) +[HTML](86) +[PDF](390.82KB)
Abstract:

This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is \begin{document}$ 0 $\end{document}. The existence assertion is established by exploring the fact that the total population density satisfies a porous media equation.

Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria
Trinh T. Nguyen
2020, 13(6): 1193-1218 doi: 10.3934/krm.2020043 +[Abstract](211) +[HTML](133) +[PDF](516.43KB)
Abstract:

In this paper, we establish derivative estimates for the Vlasov-Poisson system with screening interactions around Penrose-stable equilibria on the phase space \begin{document}$ {\mbox{Re }}^d_x\times {\mbox{Re }}_v^d $\end{document}, with dimension \begin{document}$ d\ge 3 $\end{document}. In particular, we establish the optimal decay estimates for higher derivatives of the density of the perturbed system, precisely like the free transport, up to a log correction in time. This extends the recent work [13] by Han-Kwan, Nguyen and Rousset to higher derivatives of the density. The proof makes use of several key observations from [13] on the structure of the forcing term in the linear problem, with induction arguments to classify all the terms appearing in the derivative estimates.

An anisotropic interaction model with collision avoidance
Claudia Totzeck
2020, 13(6): 1219-1242 doi: 10.3934/krm.2020044 +[Abstract](184) +[HTML](74) +[PDF](1966.53KB)
Abstract:

In this article an anisotropic interaction model avoiding collisions is proposed. Starting point is a general isotropic interacting particle system, as used for swarming or follower-leader dynamics. An anisotropy is induced by rotation of the force vector resulting from the interaction of two agents. In this way the anisotropy is leading to a smooth evasion behaviour. In fact, the proposed model generalizes the standard models, and compensates their drawback of not being able to avoid collisions. Moreover, the model allows for formal passage to the limit 'number of particles to infinity', leading to a mesoscopic description in the mean-field sense. Possible applications are autonomous traffic, swarming or pedestrian motion. Here, we focus on the latter, as the model is validated numerically using two scenarios in pedestrian dynamics. The first one investigates the pattern formation in a channel, where two groups of pedestrians are walking in opposite directions. The second experiment considers a crossing with one group walking from left to right and the other one from bottom to top. The well-known pattern of lanes in the channel and travelling waves at the crossing can be reproduced with the help of this anisotropic model at both, the microscopic and the mesoscopic level. In addition, the 'right-before-left' and 'left-before-right' rule appear intrinsically for different anisotropy parameters.

A moment closure based on a projection on the boundary of the realizability domain: 1D case
Teddy Pichard
2020, 13(6): 1243-1280 doi: 10.3934/krm.2020045 +[Abstract](164) +[HTML](68) +[PDF](1189.65KB)
Abstract:

This work aims to develop and test a projection technique for the construction of closing equations of moment systems. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from a vector of moments, then expressing the closure based on this reconstructed function.

Exploiting the geometry of the realizability domain, i.e. the set of moments of positive distribution function, we decompose any realizable vectors into two parts, one corresponding to the moments of a chosen equilibrium function, and one obtain by a projection onto the boundary of the realizability domain in the direction of equilibrium function. A realizable closure of both of these parts are computed with standard techniques providing a realizable closure for the full system. This technique is tested for the reduction of a radiative transfer equation in slab geometry.

2019  Impact Factor: 1.311

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