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

Abstract

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