Surveys in Differential Geometry

Volume 22 (2017)

The Onsager theorem

Pages: 71 – 101

DOI: https://dx.doi.org/10.4310/SDG.2017.v22.n1.a3

Author

Camillo De Lellis (Institut für Mathematik, Universität Zürich, Switzerland)

Abstract

In his famous 1949 paper on hydrodinamic turbulence, Lars Osanger advanced a remarkable conjecture on the energy conservation of weak solutions to the Euler equations: all Hölder continuous solutions with Hölder exponent strictly larger than $\frac{1}{3}$ preserves the kinetic energy, while there are Hölder continuous solutions with any exponent strictly smaller than $\frac{1}{3}$ which do not preserve the kinetic energy. While the first statement was proved by Constantin, E, and Titi in 1994, the second was proved only recently by P. Isett building upon previous works of László Székelyhidi Jr. and the author. This paper is a survey on the proof of the conjecture and on several other related discoveries which have been made in the last few years.

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Published 13 September 2018