Nonuniqueness of Solutions to the Euler Equations with Vorticity in a Lorentz Space

Abstract

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 }.