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Kinetic & Related Models

June 2011 , Volume 4 , Issue 2

Issue on thermomechanics and phase change
Guest Editors: Alain Miranville and Ulisse Stefanelli

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On a relativistic Fokker-Planck equation in kinetic theory
José Antonio Alcántara and Simone Calogero
2011, 4(2): 401-426 doi: 10.3934/krm.2011.4.401 +[Abstract](1122) +[PDF](557.6KB)
Abstract:
A relativistic kinetic Fokker-Planck equation that has been recently proposed in the physical literature is studied. It is shown that, in contrast to other existing relativistic models, the one considered in this paper is invariant under Lorentz transformations in the absence of friction. A similar property (invariance by Galilean transformations in the absence of friction) is verified in the non-relativistic case. In the first part of the paper some fundamental mathematical properties of the relativistic Fokker-Planck equation are established. In particular, it is proved that the model is compatible with the finite propagation speed of particles in relativity. In the second part of the paper, two non-linear relativistic mean-field models are introduced. One is obtained by coupling the relativistic Fokker-Planck equation to the Maxwell equations of electrodynamics, and is therefore of interest in plasma physics. The other mean-field model couples the Fokker-Planck dynamics to a relativistic scalar theory of gravity (the Nordström theory) and is therefore of interest in gravitational physics. In both cases the existence of steady states for all possible prescribed values of the mass is established. In the gravitational case this result is better than for the corresponding non-relativistic model, the Vlasov-Poisson-Fokker-Planck system, for which existence of steady states is known only for small mass.
On a model for mass aggregation with maximal size
Ondrej Budáč, Michael Herrmann, Barbara Niethammer and Andrej Spielmann
2011, 4(2): 427-439 doi: 10.3934/krm.2011.4.427 +[Abstract](723) +[PDF](865.8KB)
Abstract:
We study a kinetic mean-field equation for a system of particles with different sizes, in which particles are allowed to coagulate only if their sizes sum up to a prescribed time-dependent value. We prove well-posedness of this model, study the existence of self-similar solutions, and analyze the large-time behavior mostly by numerical simulations. Depending on the parameter $k_0$, which controls the probability of coagulation, we observe two different scenarios: For $k_0>2$ there exist two self-similar solutions to the mean field equation, of which one is unstable. In numerical simulations we observe that for all initial data the rescaled solutions converge to the stable self-similar solution. For $k_0<2$, however, no self-similar behavior occurs as the solutions converge in the original variables to a limit that depends strongly on the initial data. We prove rigorously a corresponding statement for $k_0\in (0,1/3)$. Simulations for the cross-over case $k_0=2$ are not completely conclusive, but indicate that, depending on the initial data, part of the mass evolves in a self-similar fashion whereas another part of the mass remains in the small particles.
An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits
Nicolas Crouseilles and Mohammed Lemou
2011, 4(2): 441-477 doi: 10.3934/krm.2011.4.441 +[Abstract](785) +[PDF](756.6KB)
Abstract:
In this work, we extend the micro-macro decomposition based numerical schemes developed in [3] to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part. A suitable discretization of this micro-macro model enables to derive an asymptotic preserving scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented: On the one hand a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics. On the other hand, as suggested in [30], we introduce a suitable reformulation of the micro-macro scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit scheme for the diffusion limit model when $\varepsilon\rightarrow 0$, which makes it free from the usual diffusion constraint $\Delta t=O(\Delta x^2)$ in all regimes. Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes.
On Villani's conjecture concerning entropy production for the Kac Master equation
Amit Einav
2011, 4(2): 479-497 doi: 10.3934/krm.2011.4.479 +[Abstract](658) +[PDF](223.7KB)
Abstract:
In this paper we take an idea presented in recent paper by Carlen, Carvalho, Le Roux, Loss, and Villani ([3]) and push it one step forward to find an exact estimation on the entropy production. The new estimation essentially proves that Villani's conjecture is correct, or more precisely that a much worse bound to the entropy production is impossible in the general case.
Validity of the Boltzmann equation with an external force
Raffaele Esposito, Yan Guo and Rossana Marra
2011, 4(2): 499-515 doi: 10.3934/krm.2011.4.499 +[Abstract](619) +[PDF](404.6KB)
Abstract:
We establish local-in-time validity of the Boltzmann equation in the presence of an external force deriving from a $C^2$ potential.
On kinetic flux vector splitting schemes for quantum Euler equations
Jingwei Hu and Shi Jin
2011, 4(2): 517-530 doi: 10.3934/krm.2011.4.517 +[Abstract](656) +[PDF](487.1KB)
Abstract:
The kinetic flux vector splitting (KFVS) scheme, when used for quantum Euler equations, as was done by Yang et al [22], requires the integration of the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution), giving a numerical flux much more complicated than the classical counterpart. As a result, a nonlinear 2 by 2 system that connects the macroscopic quantities temperature and fugacity with density and internal energy needs to be inverted by iterative methods at every spatial point and every time step. In this paper, we propose to use a simple classical KFVS scheme for the quantum hydrodynamics based on the key observation that the quantum and classical Euler equations share the same form if the (quantum) internal energy rather than temperature is used in the flux. This motivates us to use a classical Maxwellian - that depends on the internal energy rather than temperature - instead of the quantum one in the construction of the scheme, yielding a KFVS which is purely classical. This greatly simplifies the numerical algorithm and reduces the computational cost. The proposed schemes are tested on a 1-D shock tube problem for the Bose and Fermi gases in both classical and nearly degenerate regimes.
Decay property for a plate equation with memory-type dissipation
Yongqin Liu and Shuichi Kawashima
2011, 4(2): 531-547 doi: 10.3934/krm.2011.4.531 +[Abstract](619) +[PDF](443.7KB)
Abstract:
In this paper we focus on the initial value problem of the semi-linear plate equation with memory in multi-dimensions $(n\geq1)$, the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.
Growth estimates and uniform decay for a collisionless plasma
Christophe Pallard
2011, 4(2): 549-567 doi: 10.3934/krm.2011.4.549 +[Abstract](668) +[PDF](434.3KB)
Abstract:
We consider the classical Vlasov-Poisson system in three space dimensions in the electrostatic case. For smooth solutions starting from compactly supported initial data, an estimate on velocities is derived, showing an upper bound with a growth rate no larger than $(t\ln t)^{6/25}$. As a consequence, a decay estimate is obtained for the electric field in the $L^\infty$ norm.
Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics
Masahiro Suzuki
2011, 4(2): 569-588 doi: 10.3934/krm.2011.4.569 +[Abstract](985) +[PDF](339.8KB)
Abstract:
The main concern of the present paper is to analyze a sheath formed around a surface of a material with which plasma contacts. Here, for a formation of the sheath, the Bohm criterion requires the velocity of positive ions should be faster than a certain physical constant. The behavior of positive ions in plasma is governed by the Euler-Poisson equations. Mathematically, the sheath is regarded as a special stationary solution. We first show that the Bohm criterion gives a sufficient condition for an existence of the stationary solution by using the phase plane analysis. Then it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in the weighted Sobolev space. Moreover we obtain the convergence rate of the time global solution towards the stationary solution subject to the decay rate of the initial perturbation. These theorems are proved by a weighted energy method.

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