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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 thereare infinitely many bounded admissible solutions $v\in C_tL^2$ to the 2D Eulerequation. This shows sharpness of the weak-strong uniqueness principle, as wellas sharpness of Yudovich's proof of uniqueness in the class of boundedadmissible solutions. The initial data are truncated power-law vortices. Theconstruction is based on finding a suitable self-similar subsolution and thenapplying the convex integration method. In addition, we extend it for$1<p<\infty$ and show that the energy dissipation rate of the subsolutionvanishes at $t=0$ if and only if $p\geq 3/2$, which is the Onsager criticalexponent in terms of $L^p$ control on vorticity in 2D.

Non-uniqueness of admissible solutions for the 2D Euler equation with $L^p$ vortex data