Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

We construct solutions of the 2D incompressible Euler equations in $\text{mathds}{R}^2\times [0,\mathrm{\infty })$ such that initially the velocity is in the super-critical Sobolev space $H^{\beta }$ for $1<\beta <2$, but are not in $H^{{\beta }^{\mathrm{\prime }}}$ for ${\beta }^{\mathrm{\prime }}>1+\frac{(3-\beta )(\beta -1)}{2-(\beta -1{)}^2}$ for $0<t<\mathrm{\infty }$. These solutions are not in the Yudovich class, but they exists globally in time and they are unique in a determined family of classical solutions.