  For any positive regularity parameter $\beta < \frac 12$, we construct
non-conservative weak solutions of the 3D incompressible Euler equations which
lie in $H^{\beta}$ uniformly in time. In particular, we construct solutions
which have an $L^2$-based regularity index \emph{strictly larger} than $\frac
13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the
Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.
