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Kinetic & Related Models

June 2017 , Volume 10 , Issue 2

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Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions
Marc Briant
2017, 10(2): 329-371 doi: 10.3934/krm.2017014 +[Abstract](289) +[HTML](8) +[PDF](678.1KB)
Abstract:

We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo [21] in \begin{document}$L_{x,v}^\infty \left( {{{\left( {1 + |v|} \right)}^\beta }{e^{|v{|^2}/4}}} \right)$\end{document}, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.

Dispersion relations in cold magnetized plasmas
Christophe Cheverry and Adrien Fontaine
2017, 10(2): 373-421 doi: 10.3934/krm.2017015 +[Abstract](378) +[HTML](2) +[PDF](12315.4KB)
Abstract:

Starting from kinetic models of cold magnetized collisionless plasmas, we provide a complete description of the characteristic variety sustaining electromagnetic wave propagation. As in [4, 13, 17], the analysis is based on some asymptotic calculus exploiting the presence at the level of dimensionless relativistic Vlasov-Maxwell equations of a large parameter: the electron gyrofrequency. The method is inspired from geometric optics [29, 33]. It allows to unify all the preceding results [8, 12, 38, 31, 37, 40], while incorporating new aspects. Specifically, the non trivial effects [5, 9, 10, 24] of the spatial variations of the background density, temperature and magnetic field are exhibited. In this way, a comprehensive overview of the dispersion relations becomes available, with important possible applications in plasma physics [7, 28, 30].

Escaping the trap of 'blocking': A kinetic model linking economic development and political competition
M. Dolfin, D. Knopoff, L. Leonida and D. Maimone Ansaldo Patti
2017, 10(2): 423-443 doi: 10.3934/krm.2017016 +[Abstract](431) +[HTML](1) +[PDF](627.4KB)
Abstract:

In this paper we present a kinetic model with evolutive stochastic game-type interactions, analyzing the relationship between the level of political competition in a society and the degree of economic liberalization. The above issue regards the complex interactions between economy and institutional policies intended to introduce technological innovations in a society, where technological innovations are intended in a broad sense comprehending reforms critical to production [3]. A special focus is placed on the political replacement effect described in a macroscopic model by Acemoglu and Robinson (AR-model [1], henceforth), which can determine the phenomenon of innovation 'blocking', possibly leading to economic backwardness. One of the goals of our modelization is to obtain a mesoscopic dynamical model whose macroscopic outputs are qualitatively comparable with stylized facts of the AR-model and the comparison is settled in a number of case studies. A set of numerical solutions is presented showing the non monotonous relationship between economic liberalization and political competition in particular conditions, which can be considered as an emergent phenomenon of the analyzed complex socio-economic interaction dynamics.

A consistent kinetic model for a two-component mixture with an application to plasma
Christian Klingenberg, Marlies Pirner and Gabriella Puppo
2017, 10(2): 445-465 doi: 10.3934/krm.2017017 +[Abstract](316) +[HTML](3) +[PDF](447.8KB)
Abstract:

We consider a non reactive multi component gas mixture.We propose a class of models, which can be easily generalized to multiple species. The two species mixture is modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the solutions for the space homogeneous case, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the space homogeneous case in the form of a global Maxwell distribution. Thus, we are able to derive the usual macroscopic conservation laws. In particular, by considering a mixture composed of ions and electrons, we derive the macroscopic equations of ideal MHD from our model.

Approximate explicit stationary solutions to a Vlasov equation for planetary rings
Armando Majorana
2017, 10(2): 467-479 doi: 10.3934/krm.2017018 +[Abstract](244) +[HTML](1) +[PDF](863.1KB)
Abstract:

In this paper we consider a Vlasov or collisionless Boltzmann equation describing the dynamics of planetary rings. We propose a simple physical model, where the particles of the rings move under the gravitational Newtonian potential of two primary bodies. We neglect the gravitational forces between the particles. We use a perturbative technique, which allows to find explicit solutions at the first order and approximate solutions at the second order, by solving a set of two linear ordinary differential equations.

How does variability in cell aging and growth rates influence the Malthus parameter?
AdélaÏde Olivier
2017, 10(2): 481-512 doi: 10.3934/krm.2017019 +[Abstract](398) +[HTML](2) +[PDF](723.8KB)
Abstract:

Recent biological studies draw attention to the question of variability between cells. We refer to the study of Kiviet et al. published in 2014 [15]. A cell in a controlled culture grows at a constant rate \begin{document}$v>0$\end{document}, but this rate can differ from one individual to another. The biological question we address here states as follows. How does individual variability in the growth rate influence the growth speed of the population? The growth speed of the population is measured by the Malthus parameter we define thereafter, also called in the literature fitness. Even if the variability in the growth rate among cells is small, with a distribution of coefficient of variation around 10%, and even if its influence on the Malthus parameter would be still smaller, such an influence may become determinant since it characterises the exponential growth speed of the population.

On modified simple reacting spheres kinetic model for chemically reactive gases
Jacek Polewczak and Ana Jacinta Soares
2017, 10(2): 513-539 doi: 10.3934/krm.2017020 +[Abstract](274) +[HTML](2) +[PDF](516.8KB)
Abstract:

We consider the modified simple reacting spheres (MSRS) kinetic model that, in addition to the conservation of energy and momentum, also preserves the angular momentum in the collisional processes. In contrast to the line-of-center models or chemical reactive models considered in [23], in the MSRS (SRS) kinetic models, the microscopic reversibility (detailed balance) can be easily shown to be satisfied, and thus all mathematical aspects of the model can be fully justified. In the MSRS model, the molecules behave as if they were single mass points with two internal states. Collisions may alter the internal states of the molecules, and this occurs when the kinetic energy associated with the reactive motion exceeds the activation energy. Reactive and non-reactive collision events are considered to be hard spheres-like. We consider a four component mixture \begin{document}$A$\end{document}, \begin{document}$B$\end{document}, \begin{document}$A^*$\end{document}, \begin{document}$B^*$\end{document}, in which the chemical reactions are of the type \begin{document}$A+B\rightleftharpoons A^*+B^*$\end{document}, with \begin{document}$A^*$\end{document} and \begin{document}$B^*$\end{document} being distinct species from \begin{document}$A$\end{document} and \begin{document}$B$\end{document}. We provide fundamental physical and mathematical properties of the MSRS model, concerning the consistency of the model, the entropy inequality for the reactive system, the characterization of the equilibrium solutions, the macroscopic setting of the model and the spatially homogeneous evolution. Moreover, we show that the MSRS kinetic model reduces to the previously considered SRS model (e.g., [21], [27]) if the reduced masses of the reacting pairs are the same before and after collisions, and state in the Appendix the more important properties of the SRS system.

2016  Impact Factor: 1.261

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