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

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 $Z^d$-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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https://hal.archives-ouvertes.fr/hal-03117545
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Soumis le : jeudi 21 janvier 2021 - 11:57:46
Dernière modification le : mardi 19 octobre 2021 - 10:48:09
Archivage à long terme le : : jeudi 22 avril 2021 - 19:11:14

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Marc Briane, Loïc Hervé. Asymptotics of ODE's flows everywhere or almost-everywhere in the torus: from rotation sets to homogenization of transport equations. Journal of Differential Equations, Elsevier, 2021, 304, pp.165 - 190. ⟨10.1016/j.jde.2021.09.035⟩. ⟨hal-03117545⟩

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