Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations
Asymptotique du flot d'équations différentielles ordinaires dans le tore : des ensembles de rotation à l'homogénéisation des équations de transport
Résumé
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 $Z^d$-periodic vector field from $R^d$ in $R^d$.We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$:- the everywhere asymptotics of the flow $X$,- the almost-everywhere asymptotics of the flow $X$,- the global rectification of the vector field $b$ in $Y_d$,- the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure,- the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures,- 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,- 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.
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