Local energy flux and the refined similarity hypothesis

Abstract

In this paper we demonstrate the locality of energy transport for incompressible Euler equations both in space and in scale. The key to the proof is the proper definition of a “local subscale flux,”Π _t (r), which is supposed to be a measure of energy transfer to length scales <l at the space point r. Kraichnan suggested that for such a quantity the ”refined similarity hypothesis” will hold, which Kolmogorov originally assumed to hold instead for volume-averaged dissipation. We derive a local energy-balance relation for the large-scale motions which yields a natural definition of such a subscale flux. For this definition a precise form of the “refined similarity hypothesis” is rigorously proved as a big-O bound. The established estimate isΠ _t (r)=O(l _3h-1) in terms of the local Hölder exponenth at the point r, which is also the estimate assumed in the Parisi-Frisch “multifractal model.” Our method not only establishes locality of energy transfer, but it also clarifies the physical reason that convection effects, which naively violate locality, do not contribute to the subscale flux. In fact, we show that, as a consequence of incompressibility, such effects enter into the local energy balance only as the divergence of a spatial current. Therefore, they describe motion of energy in space and cancel in the integration over volume. We also discuss theorems of Onsager, Eyink, and Constantinet al. on energy conservation for Euler dynamics, particularly to explain their relation with the Parisi-Frisch model. The Constantinet al. proof may be interpreted as giving a bound on the total flux,Π _t =∫d ^drΠ _t (r), of the formΠ _t (r)=O(l _z3h-1), wherez ₃ is the third-order scaling exponent (or Besov index), in agreement with the “multifractal model.” Finally, we discuss how the local estimates are related to the results of Caffarelli-Kohn-Nirenberg on partial regularity for solutions of Navier-Stokes equations. They provide some heuristic support to a scenario proposed recently by Pumir and Siggia for singularities in the solutions of Navier-Stokes with small enough viscosity.