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Kinetic & Related Models

June 2012 , Volume 5 , Issue 2

Issue dedicated to the proceedings of the second edition of
the Conference Numerical Models for Controlled Fusion (NMCF09)

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The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth
Jacek Banasiak and Wilson Lamb
2012, 5(2): 223-236 doi: 10.3934/krm.2012.5.223 +[Abstract](608) +[PDF](360.5KB)
Abstract:
In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible.
Boltzmann equation and hydrodynamics at the Burnett level
Alexander Bobylev and Åsa Windfäll
2012, 5(2): 237-260 doi: 10.3934/krm.2012.5.237 +[Abstract](812) +[PDF](2353.9KB)
Abstract:
The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.
A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation
Hua Chen and Ling-Jun Wang
2012, 5(2): 261-281 doi: 10.3934/krm.2012.5.261 +[Abstract](516) +[PDF](477.2KB)
Abstract:
We study the spectral stability of the one-dimensional small amplitude periodic traveling wave solutions of the Zakharov-Kuznetsov equation with respect to two-dimensional perturbations, which are either periodic in the direction of propagation with the same period as the one-dimensional underlying traveling wave, or non-periodic (localized or bounded). Relying upon the perturbation theory for linear operators with periodic coefficients, we show that the small periodic traveling waves are transversely spectrally unstable, with respect to both types of perturbations.
Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension
Laurent Gosse
2012, 5(2): 283-323 doi: 10.3934/krm.2012.5.283 +[Abstract](548) +[PDF](1488.4KB)
Abstract:
In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integro-differential equation whereas the heat transfer is described by a $2 \times 2$ coupled system. This simplification allows to set up the well-balanced method, involving non-conservative products regularized by solutions of the stationary equations, in order to produce numerical schemes which do stabilize in large times and deliver accurate approximations at numerical steady-state. Boundary-value problems for the stationary equations are solved by the technique of "elementary solutions" at the continuous level and by means of the "analytical discrete ordinates" method at the numerical one. Practically, a comparison with a standard time-splitting method is displayed for a Couette flow by inspecting the shear stress which must be a constant at steady-state. Other test-cases are treated, like heat transfer between two unequally heated walls and also the propagation of a sound disturbance in a gas at rest. Other numerical experiments deal with the behavior of these kinetic models when the Knudsen number becomes small. In particular, a test-case involving a computational domain containing both rarefied and fluid regions characterized by mean free paths of different magnitudes is presented: stabilization onto a physically correct steady-state free from spurious oscillations is observed.
The Lifschitz-Slyozov equation with space-diffusion of monomers
Thierry Goudon, Frédéric Lagoutière and Léon M. Tine
2012, 5(2): 325-355 doi: 10.3934/krm.2012.5.325 +[Abstract](518) +[PDF](593.1KB)
Abstract:
The Lifschitz--Slyozov system describes the dynamics of mass exchanges between macro--particles and monomers in the theory of coarsening. We consider a variant of the classical model where monomers are subject to space diffusion. We establish the existence--uniqueness of solutions for a wide class of relevant data and kinetic coefficients. We also derive a numerical scheme to simulate the behavior of the solutions.
Periodic long-time behaviour for an approximate model of nematic polymers
Lingbing He, Claude Le Bris and Tony Lelièvre
2012, 5(2): 357-382 doi: 10.3934/krm.2012.5.357 +[Abstract](646) +[PDF](458.1KB)
Abstract:
We study the long-time behaviour of a nonlinear Fokker-Planck equation, which models the evolution of rigid polymers in a given flow, after a closure approximation. The aim of this work is twofold: first, we propose a microscopic derivation of the classical Doi closure, at the level of the kinetic equation ; second, under specific assumptions on the parameters and the initial condition, we prove convergence of the solution to the Fokker-Planck equation to a particular periodic solution in the long-time limit.
Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$
Robert M. Strain and Keya Zhu
2012, 5(2): 383-415 doi: 10.3934/krm.2012.5.383 +[Abstract](831) +[PDF](583.6KB)
Abstract:
For the relativistic Boltzmann equation in $\mathbb{R}^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988 [13].
Unique moment set from the order of magnitude method
Henning Struchtrup
2012, 5(2): 417-440 doi: 10.3934/krm.2012.5.417 +[Abstract](586) +[PDF](384.0KB)
Abstract:
The order of magnitude method [Struchtrup, Phys. Fluids 16, 3921-3934 (2004)] is used to construct a unique moment set for 1-D transport with scattering. Simply speaking, the method uses a series of leading order Chapman-Enskog expansions in the Knudsen number to construct the moments such that the number of moments at a given Chapman-Enskog order is minimal. For isotropic scattering, when one begins with monomials for the moments, the method constructs step by step moments of the Legendre polynomials. For anisotropic scattering, however, it constructs moments of new polynomials relevant for the particular scattering mechanism. All terms in the final moment equations are scaled by powers of the Knudsen number, which gives an easy handle to model reduction.

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