*Published Paper*

**Inserted:** 8 nov 2018

**Last Updated:** 8 nov 2018

**Journal:** Surveys in Differential Geometry 2017.

**Year:** 2018

**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 H\"older continuous solutions with H\"older exponent strictly larger than $\frac{1}{3}$ preserves the kinetic energy, while there are H\"older 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 L\'aszl\'o Sz\'ekelyhidi 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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