Kinetic & Related Models
June 2010 , Volume 3 , Issue 2
A special issue
Dedicated to Professor Vsevolod Aleksevich Solonnikov on the Occasion of his 75th Birthday
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Existence of solutions of weakly non-linear half-space problems for the general discrete velocity (with arbitrarily finite number of velocities) model of the Boltzmann equation are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. Applications to axially symmetric models are studied in more detail.
In the frame of a kinetic Boltzmann-type approach to modeling market economies, a random conservative-in-the-mean scheme is proposed for binary transactions among agents. The scheme extends a very successful model recently introduced by Cordier, Pareschi and Toscani. Effects of the risky market on the overall output after the trade of each agent are accounted for by random variables affecting not only the wealth of that agent before the trade, but also the one of his partner. Variations induced by this generalization on steady distribution, existence of moments, and Pareto index are discussed. In particular, the continuous trading limit and the relevant limiting Fokker-Planck equation are commented on in detail.
We consider a system made of a positive Vlasov-Poisson plasma and $N$ positive charges in $\R^2$, interacting among themselves and with the plasma via the Coulomb force. We prove an existence and uniqueness theorem for the system in case the charges are initially apart from the plasma.
We consider a one-dimensional BGK model as a regularization for the isentropic system of gas dynamics. Existence and dynamic stability of small amplitude travelling waves of the kinetic transport equation are proven. Their macroscopic moments approximate viscous shock profiles for the isentropic system. These results are also extended to the isothermal case.
We consider a thin layer of a rarified gas modeled by a large hard-sphere system and show that, as long as the thickness of the layer is much larger than the interaction length, the limiting behavior is described, at least for short times, by a Boltzmann equation with two-dimensional position variable and three-dimensional velocity. By the analysis of the Lorentz gas we argue that, if the thickness of the layer is of the same order of the interaction length, this is not the case.
We consider a simple one dimensional mean-field equation modeling a resonant setting in a coupled wave + transport system. Using elementary methods, we obtain sufficient conditions on the initial data to ensure global existence or blow-up in finite time.
We continue research on generalized macroscopic models of conservation type as started in . In this paper we keep the characteristic (for traffic) non-locality removed in  by Taylor expansion and discuss the merits and problems of such an expansion. We observe that the models satisfy maximum principles and conclude that "triggers'' are needed in order to cause traffic jams (braking waves) in traffic guided by such models. Several such triggers are introduced and discussed. The models are refined further in order to properly address non-monotonic (in speed) traffic regimes, and the inclusion of an individual reaction time is discussed in the context of a braking wave. A number of numerical experiments are conducted to exhibit our findings.
We prove an $L^p$ compactness result for the gain parts of the linearized Boltzmann collision operator associated with weakly cutoff collision kernels that derive from a power-law intermolecular potential. We replace the Grad cutoff assumption previously made by Caflisch , Golse and Poupaud , and Guo  with a weaker local integrability assumption. This class includes all classical kernels to which the DiPerna-Lions theory applies that derive from a repulsive inverse-power intermolecular potential. In particular, our approach allows the treatment of both hard and soft potential cases.
A fluid-dynamic system describing the behavior of a polyatomic gas in a Knudsen compressor, based on a periodic arrangement of narrower and wider two-dimensional channels and on a periodic saw-tooth temperature distribution, is derived, using the polyatomic version of the ellipsoidal statistical (ES) model of the Boltzmann equation, under the assumption that the channel width is much smaller than the length of a unit of the compressor (narrow channel approximation). The difference from the corresponding fluid-dynamic system for a monatomic gas is shown to be confined in the transport coefficients occurring in the fluid-dynamic equation. It is also shown that these coefficients in the present polyatomic-gas case are readily obtained by a simple conversion from the corresponding coefficients for the BGK model for a monatomic gas. Some numerical simulations based on the fluid-dynamic model are carried out, the results of which show that the properties of the Knudsen pump are little affected by the internal structure of a molecule.
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