Vol. 10, No. 5, 2017

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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
Abstract

We go back to the question of the regularity of the “velocity average” $\int f\left(x,v\right)\psi \left(v\right)\phantom{\rule{0.3em}{0ex}}d\mu \left(v\right)$ 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