论文标题
通过单胎条件和扰动结果，ode流在圆环上的渐近学。申请
Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications
论文作者
论文摘要
本文介绍了流量X的长期渐近X（t，x） /t解决到自主矢量值的ode：x（t，x）= b（x（x（t，x））t $ in $ r，带有x（0，x）= x的t $ \ in $ r，x（0，x）= x = x a点d：= r d /z d /z d /z d /z d /z d。我们假设矢量字段b读为$ρ$ $φ$，其中$ρ$：y d $ \ rightarrow $ [0，$ \ infty $）是一个非正规功能，$φ$：y d $ \ rightarrow $ r d是一个非消失的常规矢量场。在这项工作中，单例条件意味着旋转集C B由B的平均值组成相对于Flow X的不变概率度量的平均值是Singleton {$ζ$}，或者等效地，lim t $ \ rightArrow $ \ rightArrow $ \ infty $ \ infty $ \ infty $ x（t，x）/t = $ q $ for x $ y y y y y y y y y y d d d d d d d d d d。这与liouville的定理相结合，被视为散发性 - 曲线引理，首先允许我们在B是当前场时获得流X的渐近学。然后，我们证明了一个普遍的扰动结果，假设$ρ$是$ n，$ n，$ n $ \ in $ n的正序（$ρ$ n）n $ \ y d的均匀限制，$ n，$ρ$ \ $ \ $ \ le $ $ $ $ $ρ$ n and c $ρ$ρ$ρ$ n $φ$ n $ q $ n $是Singleton a Singleton a Singleton a singleton {$ n}。事实证明，限制c b要么是单身人士，要么放大到r d的封闭线集[0，lim n $ζ$ n]。我们根据某些$ρ$的谐波手段的积极性或不是阳性，我们提供了这种扰动结果的各种推论，涉及或不涉及经典的ergodic条件。这些结果通过不同的示例来说明，这些示例表明扰动结果仅限于$ρ$的标量扰动，并且突出了旋转集C b满足的替代方案。最后，我们证明了单胎条件使我们能够在任何维度上均质化振荡速度B（X/$ $ε$）引起的线性传输方程，而不是流量x满足的任何ergodic条件。
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 $\in$ 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 $\rightarrow$ [0, $\infty$) is a non negative regular function and $Φ$ : Y d $\rightarrow$ 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$\rightarrow$$\infty$ X(t, x)/t = $ζ$ for any x $\in$ 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$\in$N satisfying for any n $\in$ N, $ρ$ $\le$ $ρ$ n and C $ρ$n$Φ$ 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.
