Regularity of velocity averages for transport equations on random discrete velocity grids

We go back to the question of the regularity of the “velocity average” $\int f (x, v) ψ (v) d μ (v)$ when $f$ and $v \cdot \nabla_{x} f$ both belong to $L^{2}$ , and the variable $v$ lies in a discrete subset of $ℝ^{D}$ . First of all, we provide a rate, depending on the number of velocities, for the defect of $H^{1 ∕ 2}$ regularity which is reached when $v$ ranges over a continuous set. Second of all, we show that the $H^{1 ∕ 2}$ regularity holds in expectation when the set of velocities is chosen randomly. We apply this statement to investigate the consistency with the diffusion asymptotics of a Monte Carlo-like discrete velocity model.

Keywords

average lemma, discrete velocity models, random velocity grids, hydrodynamic limits

Mathematical Subject Classification 2010

Primary: 35B65

Secondary: 35F05, 35Q20, 82C40

Milestones

Received: 7 November 2016

Revised: 12 February 2017

Accepted: 28 March 2017

Published: 1 July 2017

Authors

Nathalie Ayi
Inria Rennes-Bretagne Atlantique, IPSO Research team IRMAR, UMR CNRS 6625 Campus de Beaulieu Bâtiment 22/23 263 Avenue du Général Leclerc 35042 Rennes France

Thierry Goudon
Université Côte d’Azur, Inria, CNRS, LJAD Parc Valrose 06108 Nice France

Vol. 10, No. 5, 2017

Nathalie Ayi and Thierry Goudon

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors