Geometric two-scale convergence on manifold and applications to the Vlasov equation

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February 2015, 8(1): 223-241. doi: 10.3934/dcdss.2015.8.223

Geometric two-scale convergence on manifold and applications to the Vlasov equation

Aurore Back ^1, and Emmanuel Frénod ^2,

1.	Centre de physique théorique-CNRS, Campus de Luminy, Case 907 13288 Marseille cedex 9, France
2.	LMBA (UMR 6205) Université de Bretagne-Sud, Campus de Tohannic, 56000 Vannes, France

Received May 2013 Revised September 2013 Published July 2014

Abstract
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We develop and we explain the two-scale convergence in the covariant formalism, i.e. using differential forms on a Riemannian manifold. For that purpose, we consider two manifolds $M$ and $Y$, the first one contains the positions and the second one the oscillations. We establish some convergence results working on geodesics on a manifold. Then, we apply this framework on examples.

Keywords: asymptotic analysis, Vlasov equation, Two-scale convergence, differential geometry, $L^r$ cohomology..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[2]	A. Back, Étude Théorique et Numérique Des Équations de Vlasov-Maxwell Dans le Formalisme Covariant, (French) Ph.D Thesis, 2011.
[3]	G. D. Birkhoff, What is the ergodic theorem?, Amer. Math. Monthly, 49 (1942), 222-226. doi: 10.2307/2303229.
[4]	E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.
[5]	E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213.
[6]	E. Frénod and P.-A. Raviart and E. Sonnendrücker, Two-scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl., 80 (2001), 815-843. doi: 10.1016/S0021-7824(01)01215-6.
[7]	D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33.
[8]	E. Hopf, Statistik Der Geodätischen Linien in Mannigfaltigkeiten Negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 1939.
[9]	J. Jost, Riemannian Geometry and Geometric Analysis, Fifth edition, Universitext, Springer-Verlag, Berlin, 2008.
[10]	F. I. Mautner, Geodesic flows on symmetric Riemann spaces, Ann. of Math., 65 (1957), 416-431. doi: 10.2307/1970054.
[11]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.
[12]	H. C. Pak, Geometric two-scale convergence on forms and its applications to Maxwell's equations, Proc. R. Soc. Edinb., Sect. A, Math., 135 (2005), 133-157. doi: 10.1017/S0308210500003802.
[13]	C. H. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347 (1995), 2075-2096. doi: 10.2307/2154923.
[14]	C. H. Scott, $L^p$ Theory of Differential Forms on Manifolds, ProQuest LLC, Ann Arbor, MI, 1993.

show all references

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[2]	A. Back, Étude Théorique et Numérique Des Équations de Vlasov-Maxwell Dans le Formalisme Covariant, (French) Ph.D Thesis, 2011.
[3]	G. D. Birkhoff, What is the ergodic theorem?, Amer. Math. Monthly, 49 (1942), 222-226. doi: 10.2307/2303229.
[4]	E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.
[5]	E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213.
[6]	E. Frénod and P.-A. Raviart and E. Sonnendrücker, Two-scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl., 80 (2001), 815-843. doi: 10.1016/S0021-7824(01)01215-6.
[7]	D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33.
[8]	E. Hopf, Statistik Der Geodätischen Linien in Mannigfaltigkeiten Negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 1939.
[9]	J. Jost, Riemannian Geometry and Geometric Analysis, Fifth edition, Universitext, Springer-Verlag, Berlin, 2008.
[10]	F. I. Mautner, Geodesic flows on symmetric Riemann spaces, Ann. of Math., 65 (1957), 416-431. doi: 10.2307/1970054.
[11]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.
[12]	H. C. Pak, Geometric two-scale convergence on forms and its applications to Maxwell's equations, Proc. R. Soc. Edinb., Sect. A, Math., 135 (2005), 133-157. doi: 10.1017/S0308210500003802.
[13]	C. H. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347 (1995), 2075-2096. doi: 10.2307/2154923.
[14]	C. H. Scott, $L^p$ Theory of Differential Forms on Manifolds, ProQuest LLC, Ann Arbor, MI, 1993.

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