Cette conférence s'adresse à un large auditoire.
This lecture is aimed at a general mathematical audience.
In 1949, the famous physicist Lars Onsager made a quite striking statement about solutions of the incompressible Euler equations: if they are Hölder continuous for an exponent larger than 1/3, then they preserve the kinetic energy, whereas, for exponents smaller than 1/3, there are solutions which do not preserve the energy. The first part of the statement has been rigorously proved by Constantin, E and Titi in the nineties. In a series of works, László Székelyhidi and myself have introduced ideas from differential geometry and differential inclusions to construct nonconservative solutions and started a program to attack the other portion of the conjecture. After a series of partial results, due to a few authors, Phil Isett has recently fully resolved the problem. In this talk, I will try to describe as many ideas as possible and will therefore touch upon the works of several mathematicians, including László Székelyhidi, Phil Isett, Tristan Buckmaster, Sergio Conti, Sara Daneri and myself.