Calculus of Variations and Geometric Measure Theory

E. Bruè - M. Colombo

Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space

created by bruè on 21 Aug 2021



Inserted: 21 aug 2021
Last Updated: 21 aug 2021

Year: 2021


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 $L^{1, \infty}$.