Back, Aurore; Frénod, Emmanuel Geometric two-scale convergence on manifold and applications to the Vlasov equation. (English) Zbl 1334.37027 Discrete Contin. Dyn. Syst., Ser. S 8, No. 1, 223-241 (2015). Summary: We develop and we explain the two-scale convergence (cf. [G. Nguetseng, SIAM J. Math. Anal. 20, No. 3, 608–623 (1989; Zbl 0688.35007)]) 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. Cited in 4 Documents MSC: 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53C35 Differential geometry of symmetric spaces 78A35 Motion of charged particles 14F40 de Rham cohomology and algebraic geometry Keywords:two-scale convergence; differential geometry; Vlasov equation; asymptotic analysis; \(L^r\) cohomology Citations:Zbl 0688.35007 PDFBibTeX XMLCite \textit{A. Back} and \textit{E. Frénod}, Discrete Contin. Dyn. Syst., Ser. S 8, No. 1, 223--241 (2015; Zbl 1334.37027) Full Text: DOI References: [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23, 1482 (1992) · Zbl 0770.35005 · doi:10.1137/0523084 [2] A. Back, <em>Étude Théorique et Numérique Des Équations de Vlasov-Maxwell Dans le Formalisme Covariant</em>,, (French) Ph.D Thesis (2011) [3] G. D. Birkhoff, What is the ergodic theorem?,, Amer. Math. Monthly, 49, 222 (1942) · Zbl 0063.00410 · doi:10.2307/2303229 [4] E. Frénod, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32, 1227 (2001) · Zbl 0980.82030 · doi:10.1137/S0036141099364243 [5] E. Frénod, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field,, Asymptot. Anal., 18, 193 (1998) · Zbl 0936.82032 [6] E. Frénod, Two-scale expansion of a singularly perturbed convection equation,, J. Math. Pures Appl., 80, 815 (2001) · Zbl 1032.35026 · doi:10.1016/S0021-7824(01)01215-6 [7] D. Han-Kwan, The three-dimensional finite Larmor radius approximation,, Asymptot. Anal., 66, 9 (2010) · Zbl 1191.35267 [8] E. Hopf, <em>Statistik Der Geodätischen Linien in Mannigfaltigkeiten Negativer Krümmung</em>,, Ber. Verh. Sächs. Akad. Wiss. Leipzig (1939) · JFM 65.1413.02 [9] J. Jost, <em>Riemannian Geometry and Geometric Analysis</em>,, Fifth edition (2008) · Zbl 1143.53001 [10] F. I. Mautner, Geodesic flows on symmetric Riemann spaces,, Ann. of Math., 65, 416 (1957) · Zbl 0084.37503 · 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, 608 (1989) · Zbl 0688.35007 · 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., 135, 133 (2005) · Zbl 1064.35020 · doi:10.1017/S0308210500003802 [13] C. H. Scott, \(L^p\) theory of differential forms on manifolds,, Trans. Amer. Math. Soc., 347, 2075 (1995) · Zbl 0849.58002 · doi:10.2307/2154923 [14] C. H. Scott, <em>\(L^p\) Theory of Differential Forms on Manifolds</em>,, ProQuest LLC (1993) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.