  For any $2<p<\infty$ we prove that there exists an initial velocity field
$v^\circ\in L^2$ with vorticity $\omega^\circ\in L^1\cap L^p$ for which there
are infinitely many bounded admissible solutions $v\in C_tL^2$ to the 2D Euler
equation. This shows sharpness of the weak-strong uniqueness principle, as well
as sharpness of Yudovich's proof of uniqueness in the class of bounded
admissible solutions. The initial data are truncated power-law vortices. The
construction is based on finding a suitable self-similar subsolution and then
applying the convex integration method. In addition, we extend it for
$1<p<\infty$ and show that the energy dissipation rate of the subsolution
vanishes at $t=0$ if and only if $p\geq 3/2$, which is the Onsager critical
exponent in terms of $L^p$ control on vorticity in 2D.
