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COLLOQUIA PATAVINA

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Prof. Camillo De Lellis (Univ. Zuerich)
From Nash to Onsager: funny coincidences across differential geometry and the theory of turbulence

April 8, 2014

Abstract
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 reaches the threshold 1/5.

Short bio
Camillo De Lellis was born in San Benedetto del Tronto (AP), and received his Ph.D. in Mathematics from the Scuola Normale Superiore at Pisa, in 2002. He is Full Professor of Mathematics at University of Zürich since 2005. Camillo De Lellis has given a number of remarkable contributions in different fields related to partial differential equations and geometric measure theory. In particular, he has worked on various aspects of the theory of hyperbolic systems of conservation laws and of incompressible fluid dynamics, where, together with László Székelyhidi, he has introduced the use of convex integration methods to analyse non-uniqueness issues for weak solutions to the Euler equation. Camillo De Lellis has been invited speaker at the International Congress of Mathematicians (ICM) at Hyderabad, in 2010, and plenary speaker at the European Congress of Mathematics (ECM ) at Krakow, in 2012. He has been awarded the Stampacchia medal in 2009, the Fermat Prize in 2013 (for his fundamental contributions, in collaboration with László Székelyhidi, to the conjecture of Onsager about dissipative solutions of the Euler-equations and for his work on the regularity of minimal surfaces), and the SIAG/APDE Prize in 2013 (shared with László Székelyhidi). He has also been awarded an ERC grant in 2012.
His home page is at: http://user.math.uzh.ch/delellis/