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 that coarse scale averages of the velocity carry the fine scale features of the flow. In this paper, we study time-regularity properties of Euler flows which are connected to this phenomenon and the observation that each frequency level has a natural time scale when it is viewed along the coarse scale flow. We assume only that our velocity field is Hölder continuous in the spatial variables, which is well-motivated by problems related to turbulence. We show that any periodic Euler flow in the class C_tC^α_x also belongs to C^α_(t,x), and that the pressure belongs to C^(2α−)_(t,x). We also show that, when α ≤ 1/3, the energy profile of the solution has H\"older regularity 2α/(1−α) in time, even though it might fail to be conserved in view of Onsager's conjecture. We demonstrate improved regularity for advective derivatives of the velocity and pressure. In particular, we recover in our context the celebrated result of Chemin that the particle trajectories of classical solutions to Euler are smooth, and establish existence of smooth trajectories in any case where the velocity field has borderline regularity. The analysis demonstrates that many of the main analytic features of solutions constructed by convex integration methods are consequences of the Euler equations rather than artifacts of the constructions. The proof proceeds by estimating frequency increments associated to the various physical quantities of interest. Several types of commutator estimates play a role in the proof, including the commutator estimate of Constantin, E and Titi for the relevant Reynolds stress and a more flexible proof of this estimate.

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