In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature are re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

中文翻译：

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高振荡输运方程的平均

在本文中，我们开发了一种新策略，旨在获得具有高振动解的输运方程的高阶渐近模型。该技术依赖于常微分方程的平均理论的最新发展，特别是消失参数的正态展开。值得注意的是，我们在此处陈述的结果还允许从渐近模型中完全恢复精确解。这是通过求解自然伴随源于高阶平均变量变化的伴随输运方程来完成的。最终，我们将技术应用于具有外部电场和磁场的Vlasov方程。设想了恒定磁场和非恒定磁场，并使用我们的方法重新推导了文献中已记录的渐近模型。此外，