ENERGY-DISSIPATION WITHOUT VISCOSITY IN IDEAL HYDRODYNAMICS .1. FOURIER-ANALYSIS AND LOCAL ENERGY-TRANSFER

We outline a proof and give a discussion at a physical level of an assertion of Onsager's: namely, that a solution of incompressible Euler equations with Holder continuous velocity of order h > 1/3 conserves kinetic energy, but not necessarily if h less than or equal to 1/3. We prove the result under a ''*-Holder condition'' which is somewhat stronger than usual Holder continuity. Our argument establishes also the fundamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for *-Holder solution with exponent h in the range 0 < h < 1. However, we must use a ''band-averaged'' energy flux for the proof: as we explain, the ordinary definition of the flux fails to adequately measure transport in wavenumber space (scale), since it is insensitive to the distance through which energy is displaced by individual interactions. We discuss some connections of the results with phenomenological theories of fully-developed turbulence, the 1941 Kolmogorov theory and the ''multifractal model'' of Parisi and Frisch.