Calculus of Variations and Geometric Measure Theory
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C. De Lellis - L. J. Sz\'ekelyhidi

Continuous dissipative Euler flows and a conjecture of Onsager

created by delellis on 02 Jul 2013


Accepted Paper

Inserted: 2 jul 2013
Last Updated: 2 jul 2013

Journal: To appear in the Proceedings of the 6th ECM
Year: 2012


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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