Review of the Onsager "Ideal Turbulence" Theory

In his famous undergraduate physics lectures, Richard Feynman remarked about the problem of fluid turbulence: "Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences". This statement was already false when Feynman made it. Unbeknownst to him, Lars Onsager decades earlier had made an exact mathematical analysis of the high Reynolds-number limit of incompressible fluid turbulence, using a method that would now be described as a non-perturbative renormalization group analysis and discovering the first "conservation-law anomaly" in theoretical physics. Onsager's results were only cryptically announced in 1949 and he never published any of his detailed calculations. Onsager's analysis was finally rescued from oblivion and reproduced by this author in 1992. The ideas have subsequently been intensively developed in the mathematical PDE community, where deep connections emerged with John Nash's work on isometric embeddings. Furthermore, Onsager's method has more recently been successfully applied to new physics problems, such as compressible fluid turbulence and relativistic fluid turbulence, yielding many novel testable predictions. This note will explain Onsager's exact analysis of incompressible turbulence using modern ideas on renormalization group and conservation-law anomalies, and it will also very briefly review subsequent developments.