Construction of solutions to the 3D Euler equations with initial data in $H^{\beta }$ for $\beta >0$

In this paper, we use the method of convex integration to construct infinitely many distributional solutions in $H^{\beta }$ for $0<\beta \ll 1$ 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 Poincaré, Anal. Non Linéaire 28, No. 5, 727–730 (2011; Zbl 1228.35172), where the energy is discontinuous at $0$.