From Nash to Onsager: funny coincidences across differential geometry and the theory of turbulence

Camillo De Lellis, Princeton University (Visiting Fellow)
Fine Hall 314

The incompressible Euler equations were derived more than 250 years ago by Euler to describe the motion of an inviscid incompressible fluid. It is known since the pioneering works of Scheffer and Shnirelman that there are nontrivial distributional solutions to these equations which are  compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force.  A celebrated theorem by Nash and Kuiper shows the existence of $C^1$ isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the threedimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it.   In a first joint work with Laszlo Szekelyhidi we pointed out that these two counterintuitive facts share many similarities. This has become even more apparent in some recent results of ours, which prove the existence of Hoelder continuous solutions that dissipate the kinetic energy. Our theorem might be regarded as a first step towards a conjecture of Lars Onsager, which in his 1949 paper about the theory of turbulence asserted the existence of such solutions for any Hoelder exponent up to $1/3$. Currently the best result in this direction, $1/5$, has been reached by Phil Isett.