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One of the most remarkable features of known nonstationary solutions to theincompressible Euler equations is the phenomenon known as the Taylorhypothesis, which predicts that coarse scale averages of the velocity carry thefine scale features of the flow. In this work, we develop a time regularitytheory for Euler weak solutions based on quantitative expressions of thishypothesis.  We assume only that our velocity field is H\"{o}lder continuous in thespatial variables, which is well-motivated by problems related to turbulence,but precludes the application of Lagrangian methods or local well-posednesstheory. Despite the dramatic lack of well-posedness, we obtain a rich theory ofregularity in time for solutions, especially concerning advective derivatives.In particular, any Euler flow of class $v \in L_t^\infty C_x^\alpha$ hascontinuous advective derivatives of any order less than$\frac{\alpha}{1-\alpha}$, and every point has a trajectory passing through itthat is $C^r$ for all $r < \frac{1}{1-\alpha}$, and one that is $C^\infty$ if$v$ is $C^1$ or $v \in \bigcap_{\alpha < 1} L_t^\infty C_x^\alpha$ hasborderline regularity. In a follow up work, we show that all trajectories areof class $C^{1/(1-\alpha)}$ in time whenever $1/(1-\alpha) \notin {\mathbb Z}$,whether or not the trajectories are unique.

Regularity in time along the coarse scale flow for the incompressible Euler equations