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

Abstract

We go back to the question of the regularity of the “velocity average” $\int f\left(x,v\right)\psi \left(v\right)d\mu \left(v\right)$ 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.