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 ass ertion of Onsager's: namely, that a solution of incompressible Euler e quations with Holder continuous velocity of order h > 1/3 conserves ki netic energy, but not necessarily if h less than or equal to 1/3. We p rove the result under a ''-Holder condition'' which is somewhat stron ger than usual Holder continuity. Our argument establishes also the fu ndamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for -Holder solution with ex ponent h in the range 0 < h < 1. However, we must use a ''band-average d'' 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 o f the results with phenomenological theories of fully-developed turbul ence, the 1941 Kolmogorov theory and the ''multifractal model'' of Par isi and Frisch.