Stochastic discrete velocity averaging lemmas and Rosseland approximation
Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 10, 10 p.

In this note, we investigate some questions around velocity averaging lemmas, a class of results which ensure the regularity of the “velocity average” $\int f\left(x,v\right)\psi \left(v\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\mu \left(v\right)$ when $f$ and $v·{\nabla }_{x}f$ both belong to ${L}^{p}$, $p\in \left[1,\infty \right)$ and the measured set of velocities $\left(𝒱,\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\mu \right)$ satisfy a nondegeneracy assumption. We are interested in the case when the variable $v$ lies in a discrete subset of ${ℝ}^{D}$.

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.

Publié le :
DOI : https://doi.org/10.5802/slsedp.100
@article{SLSEDP_2016-2017____A10_0,
author = {Ayi, Nathalie},
title = {Stochastic discrete velocity averaging lemmas and Rosseland approximation},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:10},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2016-2017},
doi = {10.5802/slsedp.100},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/slsedp.100/}
}
Ayi, Nathalie. Stochastic discrete velocity averaging lemmas and Rosseland approximation. Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 10, 10 p. doi : 10.5802/slsedp.100. http://archive.numdam.org/articles/10.5802/slsedp.100/

[1] A. Alonso, T. Goudon, and A. Vavasseur. Damping of particles interacting with a vibrating medium. Ann. IHP Anal. Non Lin., 2017. | Article | MR 3724755 | Zbl 1386.82062

[2] N. Ayi and T. Goudon. Regularity of velocity averages for transport equations on random discrete velocity grids. to appear in Analysis $&$ PDE, 2017. | Article | MR 3668589 | Zbl 1370.35069

[3] C. Bardos, F. Golse, B. Perthame, and R. Sentis. The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation. J. Funct. Anal., 77(2):434–460, 1988. | Article | MR 933978 | Zbl 0655.35075

[4] F. Berthelin and S. Junca. Averaging lemmas with a force term in the transport equation. J. Math. Pures Appl., 93(2):113–131, 2019. | Article | MR 2584737 | Zbl 1184.35093

[5] R. J. DiPerna and P.-L. Lions. Global weak solutions of Vlasov–Maxwell systems. Comm. Pure Appl. Math., 42(6):729–757, 1989. | Article | MR 1003433 | Zbl 0698.35128

[6] R. J. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2), 130(2):321–366, 1989. | Article | MR 1014927 | Zbl 0698.45010

[7] R. J. DiPerna, P.-L. Lions, and Y. Meyer. ${L}^{p}$ regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(3-4):271–287, 1991. | Article | Numdam | MR 1127927 | Zbl 0763.35014

[8] F. Golse. From kinetic to macroscopic models. In B. Perthame and L. Desvillettes, editors, Kinetic equations and asymptotic theory, volume 4 of Series in Appl. Math, pages 41–121. Gauthier-Villars, 2000.

[9] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76:110–125, 1988. | Article | MR 923047 | Zbl 0652.47031

[10] F. Golse and L. Saint-Raymond. Velocity averaging in ${L}^{1}$ for the transport equation. C. R. Math. Acad. Sci. Paris, 334(7):557–562, 2002. | Article | MR 1903763 | Zbl 1154.35326

[11] F. Golse and L. Saint-Raymond. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math., 155(1):81–161, 2004. | Article | MR 2025302 | Zbl 1060.76101

[12] T. Goudon. Intégration. Intégrale de Lebesgue et introduction à l’analyse fonctionnelle. Références Sciences. Ellipses, 2011. | Zbl 1327.28002

[13] S. Mischler. Convergence of discrete-velocity schemes for the Boltzmann equation. Arch. Rational Mech. Anal., 140:53–77, 1997. | Article | MR 1482928 | Zbl 0898.76089

[14] B. Perthame and P. E. Souganidis. A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4), 31(4):591–598, 1998. | Article | Numdam | MR 1634024 | Zbl 0956.45010

[15] L. Saint Raymond. Hydrodynamic limits of the Boltzmann equation, volume 1971 of Lect. Notes in Math. Springer, 2009. | Article | Zbl 1171.82002

[16] E. Tadmor and T. Tao. Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs. Comm. Pure Appl. Math., 60(10):1488–1521, 2007. | Article | MR 2342955 | Zbl 1131.35004

[17] C. Villani. Limites hydrodynamiques de l’équation de Boltzmann (d’après C. Bardos, F. Golse, C. D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond). In Séminaire Bourbaki, Vol. 2000/2001, volume 282 of Astérisque, pages 365–405, Exp. No. 893. Soc. Math. France, 2002. | Numdam