Calculus of Variations and Geometric Measure Theory

C. De Lellis

The Onsager Theorem

created by delellis on 08 Nov 2018


Published Paper

Inserted: 8 nov 2018
Last Updated: 8 nov 2018

Journal: Surveys in Differential Geometry 2017.
Year: 2018


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.