LOCAL ENERGY FLUX AND THE REFINED SIMILARITY HYPOTHESIS

In this paper we demonstrate the locality of energy transport for inco mpressible Euler equations both in space and in scale. The key to the proof is the proper definition of a ''local subscale flux,'' IIl(r), w hich 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 t he ''refined similarity hypothesis'' will hold, which Kolmogorov origi nally assumed to hold instead for volume-averaged dissipation. We deri ve a local energy-balance relation for the large-scale motions which y ields a natural definition of such a subscale flux. For this definitio n a precise form of the ''refined similarity hypothesis'' is rigorousl y proved as a big-O bound The established estimate is IIl(r) = O(l(3h- 1)) in terms of the local Holder exponent h at the point r, which is a lso the estimate assumed in the Parisi-Frisch ''multifractal model.'' Our method not only establishes locality of energy transfer, but it al so clarifies the physical reason that convection effects, which naivel y violate locality, do not contribute to the subscale flux. In fact, w e show that, as a consequence of incompressibility, such effects enter into the local energy balance only as the divergence of a spatial cur rent. Therefore, they describe motion of energy in space and cancel in the integration over volume. We also discuss theorems of Onsager, Eyi nk, and Constantin ct al. on energy conservation for Euler dynamics, p articularly to explain their relation with the Parisi-Frisch model. Th e Constantin el al. proof may be interpreted as giving a bound on the total flux, IIl = integral d(d)rII(l)(r), of the form IIl = O([l(z3-1) ), where z(3) is the third-order scaling exponent (or Besov index), in agreement with the ''multifractal model.'' Finally, we discuss how th e local estimates are related to the results of Caffarelli-Kohn-Nirenb erg on partial regularity for solutions of Navier-Stokes equations. Th ey provide some heuristic support to a scenario proposed recently by P umir and Siggia for singularities in the solutions of Navier-Stokes wi th small enough viscosity.