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For any $\epsilon >0$ 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 $L^1_t (C_x^{\frac{1}{3}-\epsilon})$, namely $x\mapsto v (x,t)$ is $(\frac{1}{3}-\epsilon)$-H\"older continuous in space at a.e.
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Kolmogorov, A. N · 1991
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Constantin, P., E, W., and Titi, E. S · 1994
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