*Published Paper*

**Inserted:** 2 jul 2013

**Last Updated:** 24 may 2017

**Journal:** J. Eur. Math. Soc. (JEMS)

**Volume:** 16

**Pages:** 1467–1505

**Year:** 2014

**Abstract:**

Building upon the techniques introduced in [1], for any $\theta<\frac{1}{10}$ we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are H\"older-continuous with exponent $\theta$. A famous conjecture of Onsager states the existence of such dissipative solutions with any H\"older exponent $\theta<\frac{1}{3}$. Our theorem is the first result in this direction.

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