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Pré-Publication, Document De Travail Année : 2020

Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications

Résumé

This paper deals with the long time asymptotics X(t, x)/t of the flow X solution to the autonomous vector-valued ODE: X (t, x) = b(X(t, x)) for t ∈ R, with X(0, x) = x a point of the torus Y d := R d /Z d. We assume that the vector field b reads as the product ρ Φ, where ρ : Y d → [0, ∞) is a non negative regular function and Φ : Y d → R d is a non vanishing regular vector field. In this work, the singleton condition means that the rotation set C b composed of the average values of b with respect to the invariant probability measures for the flow X is a singleton {ζ}, or equivalently, that lim t→∞ X(t, x)/t = ζ for any x ∈ Y d. This combined with Liouville's theorem regarded as a divergence-curl lemma, first allows us to obtain the asymptotics of the flow X when b is a current field. Then, we prove a general perturbation result assuming that ρ is the uniform limit in Y d of a positive sequence (ρ n) n∈N satisfying for any n ∈ N, ρ ≤ ρ n and C ρnΦ is a singleton {ζ n }. It turns out that the limit set C b either remains a singleton, or enlarges to the closed line set [0, lim n ζ n ] of R d. We provide various corollaries of this perturbation result involving or not the classical ergodic condition, according to the positivity or not of some harmonic means of ρ. These results are illustrated by different examples which show that the perturbation result is limited to the scalar perturbation of ρ, and which highlight the alternative satisfied by the rotation set C b. Finally, we prove that the singleton condition allows us to homogenize in any dimension the linear transport equation induced by the oscillating velocity b(x/ε) beyond any ergodic condition satisfied by the flow X.
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Dates et versions

hal-02949388 , version 1 (25-09-2020)
hal-02949388 , version 2 (29-09-2021)

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Marc Briane, Loïc Hervé. Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications. 2020. ⟨hal-02949388v1⟩
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