Non-uniqueness of Admissible Solutions for the 2D Euler Equation with \(L^p\) Vortex Data

For any \(2<p<\infty \) $2<p<\infty$ we prove that there exists an initial velocity field \(v^\circ \in L^2\) $v^{\circ }\in L^2$ with vorticity \(\omega ^\circ \in L^1\cap L^p\) ${\omega }^{\circ }\in L^1\cap L^p$ for which there are infinitely many bounded admissible solutions \(v\in C_tL^2\) $v\in C_t{L}^2$ 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 \(1<p<\infty \) $1<p<\infty$ and show that the energy dissipation rate of the subsolution vanishes at \(t=0\) $t=0$ if and only if \(p\ge \nicefrac {3}{2}\) $p\ge \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ , which is the Onsager critical exponent in terms of \(L^p\) $L^p$ control on vorticity in 2D.