Nonuniqueness of the $2$D Euler Evolutions with $L^1\cap L^p$ Vorticity

For each prescribed $1<p<\infty$, we construct a compactly supported initial vorticity in $L^1(\mathbb{R}^2)\cap L^p(\mathbb{R}^2)$ that admits two distinct evolutions under the unforced two-dimensional incompressible Euler equation on $[0,1]$. Both vorticities are strongly continuous in $L^1\cap L^