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Kinetic & Related Models

2009 , Volume 2 , Issue 2

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Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel
Alexandre Mouton
2009, 2(2): 251-274 doi: 10.3934/krm.2009.2.251 +[Abstract](36) +[PDF](1546.6KB)
This paper is devoted to numerical simulation of a charged particle beam submitted to a strong oscillating electric field. For that, we consider a two-scale numerical approach as follows: we first recall the two-scale model which is obtained by using two-scale convergence techniques; then, we numerically solve this limit model by using a backward semi-Lagrangian method and we propose a new mesh of the phase space which allows us to simplify the solution of Poisson's equation. Finally, we present some numerical results which have been obtained by the new method, and we verify its efficiency through long time simulations.
A symmetrization of the relativistic Euler equations with several spatial variables
Philippe G. LeFloch and  Seiji Ukai
2009, 2(2): 275-292 doi: 10.3934/krm.2009.2.275 +[Abstract](29) +[PDF](218.2KB)
We consider the Euler equations governing relativistic compressible fluids evolving in the Minkowski spacetime with several spatial variables. We propose a new symmetrization which makes sense for solutions containing vacuum states and, for instance, applies to the case of compactly supported solutions which are important to model star dynamics. Then, relying on these symmetrization and assuming that the velocity does not exceed some threshold and remains bounded away from the light speed, we deduce a local-in-time existence result for solutions containing vacuum states. We also observe that the support of compactly supported solutions does not expand as time evolves.
Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model
Jishan Fan and  Tohru Ozawa
2009, 2(2): 293-305 doi: 10.3934/krm.2009.2.293 +[Abstract](77) +[PDF](189.3KB)
We prove some regularity conditions for the MHD equations with partial viscous terms and the Leray-$\alpha$-MHD model. Since the solutions to the Leray-$\alpha$-MHD model are smoother than that of the original MHD equations, we are able to obtain better regularity conditions in terms of the magnetic field $B$ only.
Variational characterizations of the effective multiplication factor of a nuclear reactor core
Bertrand Lods
2009, 2(2): 307-331 doi: 10.3934/krm.2009.2.307 +[Abstract](35) +[PDF](323.6KB)
We prove some inf--sup and sup--inf formulae for the so--called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy--dependent diffusion equation.
$\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases
Gilberto M. Kremer , Filipe Oliveira and  Ana Jacinta Soares
2009, 2(2): 333-343 doi: 10.3934/krm.2009.2.333 +[Abstract](40) +[PDF](184.0KB)
The trend to equilibrium of a quaternary mixture of monatomic gases undergoing a reversible reaction of bimolecular type is studied in a quite rigorous mathematical picture within the framework of Boltzmann equation extended to chemically reacting mixtures of gases. The $\mathcal H$-theorem and entropy inequality allow to prove two main results under the assumption of uniformly boundedness and equicontinuity of the distribution functions. One of the results establishes the tendency of a reacting mixture to evolve to an equilibrium state as time becomes large. The other states that the solution of the Boltzmann equation for chemically reacting mixtures of gases converges in strong $L^1$-sense to its equilibrium solution.
Milne problem for non-grey radiative transfer
J-F. Clouët and  R. Sentis
2009, 2(2): 345-362 doi: 10.3934/krm.2009.2.345 +[Abstract](38) +[PDF](885.6KB)
We study the non-linear Milne problem for radiative transfer equation: after proving the existence of a brightness temperature, we propose and evaluate different formulas for evaluating it. Numerical tests show that as much as 20% difference between surface temperature and brightness temperature may be exhibited. An analytical expression for the out-coming flux is also given.
Double milling in self-propelled swarms from kinetic theory
José A. Carrillo , M. R. D’Orsogna and  V. Panferov
2009, 2(2): 363-378 doi: 10.3934/krm.2009.2.363 +[Abstract](88) +[PDF](299.0KB)
We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.
Derivation of a two-fluids model for a Bose gas from a quantum kinetic system
Thibaut Allemand
2009, 2(2): 379-402 doi: 10.3934/krm.2009.2.379 +[Abstract](40) +[PDF](273.0KB)
We formally derive the hydrodynamic limit of a system modelling a bosons gas having a condensed part, made of a quantum kinetic and a Gross-Pitaevskii equation. The limit model, which is a two-fluids Euler system, is approximated by an isentropic system, which is then studied. We find in particular some conditions for the hyperbolicity, and we study the weak solutions. A numerical example is given at the end.

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