Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations

TL;DR

This paper explores the deep connections between the asymptotic behavior of ODE flows on the torus, properties of rotation sets, and homogenization of transport equations, revealing surprising equivalences and extending classical results to higher dimensions.

Contribution

It establishes new equivalences between flow asymptotics, rotation set conditions, and homogenization, and extends two-dimensional ergodic flow results to higher dimensions.

Findings

Almost-everywhere flow asymptotics are equivalent to the unit set condition for D_b.

Homogenization of transport equations is equivalent to certain rotation set conditions for divergence-free fields.

Ergodicity can hold without the flow exhibiting everywhere asymptotics in higher dimensions.

Abstract

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $\mathbb{Z}^d$-periodic vector field from $\mathbb{R}^d$ in $\mathbb{R}^d$.We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: (i) the everywhere asymptotics of the flow $X$, (ii) the almost-everywhere asymptotics of the flow $X$, (iii) the global rectification of the vector field $b$ in $Y_d$, (iv) the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, (v) the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, (vi) the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, (vii) the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.