# American Institute of Mathematical Sciences

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

 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.
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. 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. doi: 10.2307/2303229. Google Scholar [4] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32 (2001), 1227. 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. 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. 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. 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, (2008). Google Scholar [10] F. I. Mautner, Geodesic flows on symmetric Riemann spaces,, Ann. of Math., 65 (1957), 416. 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. 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., 135 (2005), 133. 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. doi: 10.2307/2154923. Google Scholar [14] C. H. Scott, $L^p$ Theory of Differential Forms on Manifolds,, ProQuest LLC, (1993). Google Scholar

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. 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. doi: 10.2307/2303229. Google Scholar [4] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32 (2001), 1227. 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. 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. 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. 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, (2008). Google Scholar [10] F. I. Mautner, Geodesic flows on symmetric Riemann spaces,, Ann. of Math., 65 (1957), 416. 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. 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., 135 (2005), 133. 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. doi: 10.2307/2154923. Google Scholar [14] C. H. Scott, $L^p$ Theory of Differential Forms on Manifolds,, ProQuest LLC, (1993). Google Scholar
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