Reinhard Illner

A convergence proof for Nanbu's simulation method for the full Boltzmann equation

It is shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation. The main conditions for this result are that the Cauchy problem for the Boltzmann equation has a sufficiently smooth solution and that the discretization parameters (cell size, timestep and test particle number) satisfy suitable constraints.

Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case

We introduce a new identity satisfied by solutions of the Vlasov-Poisson system. It has the property that all quantities which appear have a definite sign, and this allows us to prove new results on the time decay of the solutions in the plasma physical case.

Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum: Erratum and improved result

We point out an error in our earlier papers [1] and [2] and present a more direct and natural proof which, although based on the same physical ideas of the previous ones, saves and actually improves the validity results for the Boltzmann equation given in [1] and [2].

Measure solutions of the steady Boltzmann equation in a slab

It is shown that the steady Boltzmann equation in a slab [0,a] has solutionsx→μx such that the ingoing boundary measuresμ0∣{ξ>0} andμα∣{ξ<0} can be prescribed a priori. The collision kernel is truncated such that particles with smallx-component of the velocity have a reduced collision rate.

On simulation methods for the Boltzmann equation

Abstract The purpose of this talk is to give a comparative discussion of some simulation procedures for the Boltzmann equation in terms of rigour and numerical effort. The methods under consideration are the algorithms suggested by Bird [7] and Nanbu [9] respectively, and a modification o f the Nanbu procedure suggested by Babovsky [2]. The procedures are applied to and compared by means of a shock wave problem.

On algebras of pseudo differential operators In

We discuss a sub-algebra ¸ of Pseudo Differential Operators of which is Fredholm-closed and commutative modulo the ideal of compact operators. The algebra ¸ contains in particular all LP - Fourier-multipliers of Mihlin type. is shown to be an algebra with symbol, and its symbol space is determined. As an application we get simple criteria for an operator in ¸to be Fredholm. Furthermore, we regard algebras ¸ on Sobolev spaces , which are isometrically isomorphic to ¸ and have equivalent properties. Finally we conclude Fredholm criteria for mroe general Differential and Pseudo Differential Operators.

A model for generating circadian rhythm by coupling ultradian oscillators

BackgroundOrganisms ranging from humans to cyanobacteria undergo circadian rhythm, that is, variations in behavior that cycle over a period about 24 hours in length. A fundamental property of circadian rhythm is that it is free-running, and continues with a period close to 24 hours in the absence of light cycles or other external cues. Regulatory networks involving feedback inhibition and feedforward stimulation of mRNA transcription and translation are thought to be critical for many circadian mechanisms, and genes coding for essential components of circadian rhythm have been identified in several organisms. However, it is not clear how such components are organized to generate a circadian oscillation.ResultsWe propose a model in which two independent transcriptional-translational oscillators with periods much shorter than 24 hours are coupled to drive a forced oscillator that has a circadian period, using mechanisms and parameters of conventional molecular biology. Furthermore, the resulting circadian oscillator can be entrained by an external light-dark cycle through known mechanisms. We rationalize the mathematical basis for the observed behavior of the model, and show that the behavior is not dependent on the details of the component ultradian oscillators but occurs even if quite generalized basic oscillators are used.ConclusionWe conclude that coupled, independent, transcriptional-translational oscillators with relatively short periods can be the basis for circadian oscillators. The resulting circadian oscillator can be entrained by 24-hour light-dark cycles, and the model suggests a mechanism for its evolution.

A derivation of the BBGKY-hierarchy for hard sphere particle systems

Abstract We give a rigorous derivation of the BBGKY-hierarchy for a system of N hard spheres.

On nonlinear stationary half-space problems in discrete kinetic theory

We show that every steady discrete velocity model of the Boltzmann equation on the real line,ξi·(d/dx)fi=Ci(f), which satisfies anH-theorem and for which allξi≠0, has solutions on the half-line (0, ∞) which take prescribed non-negativefi(O) ifξi>0 and approach a certain manifold of Maxwellians asx→∞. Such solutions give the density distribution in a Knudsen boundary layer in the discrete velocity case.

A boundary value problem for discrete-velocity models

On considere deux problemes aux valeurs limites non lineaires concernant un ecoulement stationnaire a une dimension entre des plaques paralleles