Equilibrium Theory of Kolmogorov and Onsager

The classical theories of Kolmogorov and Onsager are briefly stated. The notions of the energy cascade, the energy spectrum, and the transfer function are defined. The basic assumptions of the Kolmogorov–Onsager theory are stated, and possible modifications and extensions are indicated: Early DNS results are introduced, and the limit of infinite Reynolds number is discussed in the light of Onsager’s contribution to extend the phase space to Hölder continuous velocity fields with exponent between zero and one. Theories of Kolmogorov and Onsager, energy cascade, energy spectrum, transfer function, hypothesis of local isotropy, hypothesis of universal equilibrium, hypothesis of inviscid similarity, DNS results, limit of infinite Reynolds number, Hölder continuity.