16-11-2015, ore 16:30
From nash to onsager, funny coincidences across differential geometry and the theory of turbulence
Camillo De Lellis
- From nash to onsager, funny coincidences across differential geometry and the theory of turbulence
- Camillo De Lellis, Universitaet Zuerich
- Lunedì 16 novembre 2015 alle ore 16:30, Sala di rappresentanza, Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50
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 C1 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.
Recently, the threshold 1/5 has been reached by Philip Isett in his PhD
thesis and in a joint work with Tristan Buckmaster we show that the treshold
1/3 can be achieved at the price of giving up the time-regularity.
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