#### 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