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Averaging lemmas without time Fourier transform and application to discretized kinetic equations

Published online by Cambridge University Press:  14 November 2011

F. Bouchut  and
L. Desvillettes
F. Bouchut
Affiliation:
Université d'Orléans et CNRS, UMR 6628, Département de Mathématiques, BP 6759, 45067 Orléans cedex 2, Francefbouchut@labomath.univ-orleans.fr
L. Desvillettes
Affiliation:
Université d'Orléans et CNRS, UMR 6628, Département de Mathématiques, BP 6759, 45067 Orléans cedex 2, Francedesville@labomath.univ-orleans.fr
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Extract

We prove classical averaging lemmas in the L2 framework with the help of the Fourier transform in variables x and v, but not t. This method is then used to study discretized problems arising out of the numerical analysis of kinetic equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 19 - 36
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Bézard, M.. Régularité L p précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122 (1994), 29–76.CrossRefGoogle Scholar
2Desvillettes, L. and Mischler, S.. About the splitting algorithm for Boltzmann and B.G.K. equations. Math. Mod. Meth. Appl. Sci. 6 (1996), 1079–1101.CrossRefGoogle Scholar
3DiPerna, R. J. and Lions, P.-L.. Global weak solutions of Vlasov–Maxwell systems. Commun. Pure Appl. Math. 42 (1989), 729–757.CrossRefGoogle Scholar
4DiPerna, R. J., Lions, P.-L. and Meyer, Y.. Lp regularity of velocity averages. Ann. I.H.P., Analyse non-linéaire 8 (1991), 271–287.Google Scholar
5Gérard, P.. Microlocal defect measures. Commun. Partial Diffl Eqns 16 (1991), 1761–1794.CrossRefGoogle Scholar
6Golse, F.. Quelques résultats de moyennisation pour les équations aux dérivées partielles. Rend. Sem. Mat. Univ. Pol. Torino, Fasdcolo Speciale 1988 ‘Hyperbolic Equations’ (1987), 101–123.Google Scholar
7Golse, F., Lions, P.-L., Perthame, B. and Sentis, R.. Regularity of the moments of the solution of a transport equation. J. Funct. Analysis 76 (1988), 110–125.Google Scholar
8Golse, F., Perthame, B. and Sentis, R.. Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport. C. R. Acad. Sci. Série I 301 (1985), 341–344.Google Scholar
9Golse, F. and Poupaud, F.. Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi–Dirac. Asympt. Analysis 6 (1992), 135–160.Google Scholar
10Lions, P.-L.. Régularité optimale des moyennes en vitesses. C. R. Acad. Sci. Série I 320 (1995), 911–915.Google Scholar
11Lions, P.-L. and Perthame, B.. Lemmes de moments, de moyenne et de dispersion. C. R. Acad. Sci. Série I 314 (1992), 801–806.Google Scholar
12Lions, P.-L., Perthame, B. and Tadmor, E.. A kinetic formulation of multidimensional scalar conservation laws and related équations. J. Am. Math. Soc. 7 (1994), 169–191.CrossRefGoogle Scholar
13Perthame, B.. Higher moments for kinetic equations: the Vlasov–Poisson and Fokker–Planck cases. Math. Meth. Appl. Sci. 13 (1990), 441–452.Google Scholar
14Perthame, B. and Souganidis, P. E... A limiting case for velocity averaging. Ann. Scient. Ecole Normale Supérieure 4e série 31 (1998), 591–598.Google Scholar
15Vasseur, A.. Convergence of a semi-discrete kinetic scheme for the system of isentropic gas dynamics with γ = 3. Indiana Univ. Math. J. (In the press.)Google Scholar