In low-pressure discharges, where the electron mean free path is larger or comparable with the discharge length, the electron dynamics is essentially nonlocal. Moreover, the electron energy distribution function (EEDF) deviates considerably from a Maxwellian. Therefore, an accurate kinetic description of the low-pressure discharges requires knowledge of the nonlocal conductivity operator and ca...

The paper presents some new estimates on the gain term of the Boltzmann collision operator. For Maxwellian molecules, it is shown that the L 1 {norm of the gain term can be bounded in terms of the L 1 and L 1 {norm of the density function f. In the case of more general collision kernels, like the hard{sphere interaction potential, the gain term is estimated pointwise by the L 1 {norm of the den...

The method to determine the molecular velocity distribution in steady reaction in homogeneous gas phase with negligible reaction heat is formulated, and especially the case where one of the reacting molecules is very light is studied in detail. In this case the perturbation of the distribution function by the chemical reaction becomes most remarkable, and the expansion by the Sonine's polynomia...

We revisit and extend the L theory of the Boltzmann collision operator by using classical techniques based in the Carleman representation and Fourier analysis. We introduce new ideas based in radial symmetry to greatly simplify existent technical proofs and obtain explicit sharp constants. An additional contribution of this work is the discovery of an unexpected relation between the general Bol...

In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the diierential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit ...

We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform Lp bounds on the initial distribution function. We also obtain a decay estimate in the spherically ...

Using the main ideas of Tanaka, the measure-solution {Pt}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {Pt}t of solutions {P l t}t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allow...

Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Gué...

The goal of this paper is to extend the existence result measure-valued solution Boltzmann equation in elastic interaction, given by Morimoto–Wang–Yang (J Stat Phys 165:866–906, 2016), inelastic with moderately soft potentials, which also an extensive work our preceding Maxwellian molecules case. We first prove and uniqueness under Grad’s angular cutoff assumption, based on which, we further ob...

We consider the spatially homogeneous Boltzmann equation for Maxwellian molecules and general finite energy initial data: positive Borel measures with finite moments up to order 2. We show that the coefficients in the Wild sum converge strongly to the equilibrium, and quantitatively estimate the rate. We show that this depends on the initial data F essentially only through on the behavior near ...