Abstract:
In 1949 the famous physicist Lars Onsager made a quite striking statement
about solutions of the incompressible Euler equations: if they are
Hoelder continuous for an exponent larger than $\frac{1}{3}$, then they
preserve the kinetic energy, whereas for exponents smaller than $\frac{1}{3}$
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 La'szlo' Sze'kelyhidi 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 La'szlo', Phil, Tristan Buckmaster, Sergio Conti, Sara Daneri and myself. |