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In this paper we consider fluid transport in two-dimensional flows from thedynamical systems point of view, with the focus on elliptic behaviour andaperiodic and finite time dependence. We give an overview of previous work ongeneral nonautonomous and finite time vector fields with the purpose ofbringing to the attention of those working on fluid transport from thedynamical systems point of view a body of work that is extremely relevant,but appears not to be so well known. We then focus on theKolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. Whilethere is no finite time or aperiodically time-dependent version of the KAMtheorem, the Nekhoroshev theorem, by its very nature, is a finite timeresult, but for a "very long" (i.e. exponentially long with respect to thesize of the perturbation) time interval and provides a rigorousquantification of "nearly invariant tori" over this very long timescale. Wediscuss an aperiodically time-dependent version of the Nekhoroshev theoremdue to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013)which is directly relevant to fluid transport problems. We give a detaileddiscussion of issues associated with the applicability of the KAM andNekhoroshev theorems in specific flows. Finally, we consider a specificexample of an aperiodically time-dependent flow where we show that theresults of the Nekhoroshev theorem hold

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori