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Stochastic discrete velocity averaging lemmas and Rosseland approximation
Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exp. No. 10, 10 p., doi: 10.5802/slsedp.100
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Résumé - Abstract
In this note, we investigate some questions around velocity averaging lemmas, a class of results which ensure the regularity of the “velocity average” $\int f(x,v)\psi (v) \,{\rm d}\mu ( v)$ when $f$ and $v\cdot \nabla _xf$ both belong to $L^p$, $p \in [1, \infty )$ and the measured set of velocities $(\mathscr{V},\,{\rm d}\mu )$ satisfy a nondegeneracy assumption. We are interested in the case when the variable $v$ lies in a discrete subset of $\mathbb{R}^D$.
We present results obtained in collaboration with T. Goudon in [2]. First of all, we provide a rate, depending on the number of velocities, to 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 obtain a consistency result for the diffusion limit in the case of the Rosseland approximation.
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