[BibTeX]

*Accepted Paper*

**Inserted:** 2 jul 2013

**Last Updated:** 2 jul 2013

**Journal:** To appear in the Proceedings of the 6th ECM

**Year:** 2012

**Abstract:**

It is known since the pioneering works of Scheffer and Shnirelman that there are nontrivial distributional solutions to the Euler equations which are compactly supported in space and time. Obviously these solutions do not respect the classical conservation law for the total kinetic energy and they are therefore very irregular. In recent joint works we have proved the existence of {\em continuous} and even H\"older continuous solutions which dissipate the kinetic energy. Our theorem might be regarded as a first step towards a conjecture of Lars Onsager, which in 1949 asserted the existence of dissipative H\"older solutions for any H\"older exponent smaller than $\frac{1}{3}$.

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