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

For any 2 < p < ∞ we prove that there exists an initial velocity field v ∘ ∈ L 2 with vorticity ω ∘ ∈ L 1 ∩ L p for which there are infinitely many bounded admissible solutions v ∈ 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 < ∞ and show that the energy dissipation rate of the subsolution vanishes at t = 0 if and only if p ≥ 3 2 , which is the Onsager critical exponent in terms of L p control on vorticity in 2D.