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Geometric two-scale convergence on manifold and applications to the Vlasov equation. (English) Zbl 1334.37027

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.

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

Citations:

Zbl 0688.35007
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References:

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