Nonuniqueness of Solutions to 2D Incompressible Euler Equations with Unbounded Vorticity

Abstract

We construct two distinct solutions of the two-dimensional incompressible Euler equation with the same initial vorticity in $\bigcap_{1\le p<\infty}L^p(\mathbb R^2)$. The vorticities are compactly supported in space and have mean zero, so the corresponding velocities have finite energy. The nonuniqueness mechanism is a cascade of transitions in quadrupole dynamics across infinitely many scales.