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

Ayi, Nathalie; Goudon, Thierry

doi:10.2140/apde.2017.10.1201

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

Nathalie Ayi and Thierry Goudon

Vol. 10 (2017), No. 5, 1201–1225

DOI: 10.2140/apde.2017.10.1201

Abstract

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