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Kinetic & Related Models

2008 , Volume 1 , Issue 4

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Integral estimates for a linear singular operator linked with Boltzmann operators part II: High singularities $1\le\nu<2$
Radjesvarane Alexandre and  Lingbing He
2008, 1(4): 491-513 doi: 10.3934/krm.2008.1.491 +[Abstract](33) +[PDF](261.4KB)
Abstract:
In this work, we show that integral estimates for a linear operator linked with Boltzmann quadratic operator considered in [1] can also be obtained for the case of higher singularities. Some estimates proven in this earlier work are improved, as in particular, we do not need any regularity with respect to the first function.
Some remarks about the scaling of systems of reactive Boltzmann equations
Marzia Bisi and  Laurent Desvillettes
2008, 1(4): 515-520 doi: 10.3934/krm.2008.1.515 +[Abstract](34) +[PDF](125.7KB)
Abstract:
An asymptotics leading from the reactive Boltzmann equation towards reaction--diffusion equations has been introduced in [1] (cf. also [10], for an analogous scaling starting from reactive BGK equations). We propose here a justification of this asymptotics, at the formal level, based on a non--dimensional form of the original equations.
On some properties of linear and linearized Boltzmann collision operators for hard spheres
A. V. Bobylev and  E. Mossberg
2008, 1(4): 521-555 doi: 10.3934/krm.2008.1.521 +[Abstract](42) +[PDF](395.2KB)
Abstract:
The linear and the linearized Boltzmann collision operators for hard spheres are studied by a method based on reduction of integral equations to differential equations. We use this approach (in combination with numerical methods) to study the eigenvalues of the operators. We also use the differential equations to investigate large energy asymptotics of solutions to linear integral equations related to the Chapman-Enskog expansion.
Derivation of a kinetic model from a stochastic particle system
Pierre Degond , Simone Goettlich , Axel Klar , Mohammed Seaid and  Andreas Unterreiter
2008, 1(4): 557-572 doi: 10.3934/krm.2008.1.557 +[Abstract](57) +[PDF](205.1KB)
Abstract:
We study a stochastic lattice particle system with exclusion principle. A kinetic equation and its diffusion limit are formally derived from the Monte Carlo dynamics. This derivation is investigated analytically and numerically and compared with the classical hydrodynamic limit of the stochastic exclusion process. Numerical results are presented for different values of jump probabilities.
Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation
Jeffrey R. Haack and  Cory D. Hauck
2008, 1(4): 573-590 doi: 10.3934/krm.2008.1.573 +[Abstract](29) +[PDF](469.6KB)
Abstract:
The occurrence of oscillations in a well-known asymptotic preserving (AP) numerical scheme is investigated in the context of a linear model of diffusive relaxation, known as the $P_1$ equations. The scheme is derived with operator splitting methods that separate the $P_1$ system into slow and fast dynamics. A careful analysis of the scheme shows that binary oscillations can occur as a result of a black-red diffusion stencil and that dispersive-type oscillations may occur when there is too little numerical dissipation. The latter conclusion is based on comparison with a modified form of the $P_1$ system. Numerical fixes are also introduced to remove the oscillatory behavior.
A kinetic model for grain growth
Reiner Henseler , Michael Herrmann , Barbara Niethammer and  Juan J. L. Velázquez
2008, 1(4): 591-617 doi: 10.3934/krm.2008.1.591 +[Abstract](39) +[PDF](322.9KB)
Abstract:
We provide a well--posedness analysis of a kinetic model for grain growth introduced by Fradkov which is based on the von Neumann--Mullins law. The model consists of an infinite number of transport equations with a tri-diagonal coupling modelling topological changes in the grain configuration. Self--consistency of this kinetic model is achieved by introducing a coupling weight which leads to a nonlinear and nonlocal system of equations.

We prove existence of solutions by approximation with finite dimensional systems. Key ingredients in passing to the limit are suitable super--solutions, a bound from below on the total mass, and a tightness estimate which ensures that no mass is transported to infinity in finite time.
Existence and uniqueness of the electric potential profile in the edge of tokamak plasmas when constrained by the plasma-wall boundary physics
Claudia Negulescu , Anne Nouri , Philippe Ghendrih and  Yanick Sarazin
2008, 1(4): 619-639 doi: 10.3934/krm.2008.1.619 +[Abstract](33) +[PDF](257.9KB)
Abstract:
The electric potential plays a key role in the confinement properties of tokamak plasmas, with the subsequent impact on the performances of fusion reactors. Understanding its structure in the peripheral plasma -- interacting with solid materials -- is of crucial importance, since it governs the boundary conditions for the burning core plasma. This paper aims at highlighting the dedicated impact of the plasma-wall boundary layer on this peripheral region. Especially, the physics of plasma-wall interactions leads to non-linear constraints along the magnetic field. In this framework, the existence and uniqueness of the electric potential profile are mathematically investigated. The working model is two-dimensional in space and time evolving.

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