Energy dissipation without viscosity in ideal hydrodynamics. I: Fourier analysis and local energy transfer. (English) Zbl 0817.76011

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 Hölder continuous velocity of order \(h > 1/3\) conserves kinetic energy, but not necessarily if \(h \leq 1/3\). We prove the result under a “\(*\)-Hölder condition” which is somewhat stronger than usual Hölder continuity. Our argument establishes also the fundamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for a \(*\)-Hölder solution with exponent \(h\) in the range \(0 < h < 1\).


76B99 Incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI


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