  In this paper, we use the method of convex integration to construct
infinitely many distributional solutions in $H^{\beta}$ for $0<\beta\ll1$ to
the initial value problem for the three-dimensional incompressible Euler
equations. We show that if the initial data has any small fractional derivative
in $L^2$, then we can construct solutions with some regularity, so that the
corresponding $L^2$ energy is continuous in time. This is distinct from the
$L^2$ existence result of E. Wiedemann, Ann. Inst. Henri Poincar\'e, Anal. Non
Lin\'eaire 28, No. 5, 727--730 (2011; Zbl 1228.35172), where the energy is
discontinuous at $0$.
