*Published Paper*

**Inserted:** 5 may 2014

**Last Updated:** 4 sep 2017

**Journal:** Comm. Pure Appl. Math.

**Volume:** 69

**Pages:** 1613-1670

**Year:** 2016

**Abstract:**

For any $\varepsilon >0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{1/3-\varepsilon})$, namely $x\mapsto v (x,t)$ is $(1/3-\varepsilon)$-H\"older continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{1/3-\varepsilon}\, dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{1/3-\varepsilon})$

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