Non-uniqueness of Admissible Solutions for the 2D Euler Equation with Vortex Data

For any $$ we prove that there exists an initial velocity field $$ with vorticity $$ for which there are infinitely many bounded admissible solutions $$ 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 $$ and show that the energy dissipation rate of the subsolution vanishes at $$ if and only if $$ , which is the Onsager critical exponent in terms of $$ control on vorticity in 2D.