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

TL;DR

This paper demonstrates that the uniqueness of solutions to the 2D Euler equations, established for bounded vorticity, does not extend to solutions with vorticity in the Lorentz space $L^{1, \infty}$, highlighting limitations of existing uniqueness results.

Contribution

The paper shows nonuniqueness of solutions for the 2D Euler equations with vorticity in Lorentz space $L^{1, \infty}$, extending understanding of solution behavior beyond bounded vorticity.

Findings

Uniqueness fails for vorticity in Lorentz space $L^{1, \infty}$

Solutions with bounded kinetic energy and vorticity in $L^{1, \infty}$ are not unique

Extends the known limitations of Yudovich's uniqueness theorem

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 $L^{1, \infty}$.