  For the two dimensional Euler equations, a classical result by Yudovich
states that solutions are unique in the class of bounded vorticity; it is a
celebrated open problem whether this uniqueness result can be extended in other
integrability spaces. We prove in this note that such uniqueness theorem fails
in the class of vector fields $u$ with uniformly bounded kinetic energy and
vorticity in the Lorentz space $L^{1, \infty}$.

Analysis of PDEs