Transport phenomena in collective dynamics: from micro to social hydrodynamics

The Onsager's Theorem

Camillo De Lellis

University of Zurich


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.