Vol. 10, No. 5, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Regularity of velocity averages for transport equations on random discrete velocity grids

Nathalie Ayi and Thierry Goudon

Vol. 10 (2017), No. 5, 1201–1225
Abstract

We go back to the question of the regularity of the “velocity average” f(x,v)ψ(v)dμ(v) when f and v xf both belong to L2 , and the variable v lies in a discrete subset of D . First of all, we provide a rate, depending on the number of velocities, for the defect of H12 regularity which is reached when v ranges over a continuous set. Second of all, we show that the H12 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