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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

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 & 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.  Google Scholar

[2]

A. Back, Étude Théorique et Numérique Des Équations de Vlasov-Maxwell Dans le Formalisme Covariant, (French) Ph.D Thesis, 2011. Google Scholar

[3]

G. D. Birkhoff, What is the ergodic theorem?, Amer. Math. Monthly, 49 (1942), 222-226. doi: 10.2307/2303229.  Google Scholar

[4]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33.  Google Scholar

[8]

E. Hopf, Statistik Der Geodätischen Linien in Mannigfaltigkeiten Negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 1939. Google Scholar

[9]

J. Jost, Riemannian Geometry and Geometric Analysis, Fifth edition, Universitext, Springer-Verlag, Berlin, 2008.  Google Scholar

[10]

F. I. Mautner, Geodesic flows on symmetric Riemann spaces, Ann. of Math., 65 (1957), 416-431. doi: 10.2307/1970054.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. H. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347 (1995), 2075-2096. doi: 10.2307/2154923.  Google Scholar

[14]

C. H. Scott, $L^p$ Theory of Differential Forms on Manifolds, ProQuest LLC, Ann Arbor, MI, 1993.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[2]

A. Back, Étude Théorique et Numérique Des Équations de Vlasov-Maxwell Dans le Formalisme Covariant, (French) Ph.D Thesis, 2011. Google Scholar

[3]

G. D. Birkhoff, What is the ergodic theorem?, Amer. Math. Monthly, 49 (1942), 222-226. doi: 10.2307/2303229.  Google Scholar

[4]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33.  Google Scholar

[8]

E. Hopf, Statistik Der Geodätischen Linien in Mannigfaltigkeiten Negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig, 1939. Google Scholar

[9]

J. Jost, Riemannian Geometry and Geometric Analysis, Fifth edition, Universitext, Springer-Verlag, Berlin, 2008.  Google Scholar

[10]

F. I. Mautner, Geodesic flows on symmetric Riemann spaces, Ann. of Math., 65 (1957), 416-431. doi: 10.2307/1970054.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. H. Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc., 347 (1995), 2075-2096. doi: 10.2307/2154923.  Google Scholar

[14]

C. H. Scott, $L^p$ Theory of Differential Forms on Manifolds, ProQuest LLC, Ann Arbor, MI, 1993.  Google Scholar

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