Search

We propose a model based on the coupling of two Boltzmann-like equations for the study of the evolution of dust particles in a rarefied atmosphere, such as it can be found in the context of safety studies for the ITER project of nuclear fusion. When the typical size of a dust speck becomes too large, the numerical simulation of the system under study becomes too expensive, and one needs to introduce an asymptotic model in which the mass ratio between molecules and dust specks tends to 0. This model is constituted of a coupling (by a drag force term) between a Boltzmann equation and a Vlasov equation. A rigorous proof of the passage to the limit is given in the spatially homogeneous setting. It includes a new variant of Povzner’s inequality in which the vanishing mass ratio is taken into account. [ABSTRACT FROM AUTHOR]

We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in L1((1+|v|s) dv) and Hk norms, s, k≥0 hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in L1 obtained by the so-called “entropy–entropy production” method in the context of dissipative systems with slowly growing a priori bounds (G. Toscani and C. Villani, J. Statist. Phys. 98 (2000), 1279–1309). [ABSTRACT FROM AUTHOR]

Abstract: For the homogeneous Boltzmann equation with (cutoff or noncutoff) hard potentials, we prove estimates of propagation of norms with a weight (, large enough), as well as appearance of such weights. The proof is based on some new functional inequalities for the collision operator, proven by elementary means. [Copyright &y& Elsevier]