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

For any 2 < p < ∞ we prove that there exists an initial velocity field v∘ ∈ L2 with vorticity ω∘ ∈ L1 ∩ Lp for which there are infinitely many bounded admissible solutions v ∈ CtL2 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 Lp control on vorticity in 2D.