Sharp Ill-Posedness of the Euler Equations in Lorentz Spaces

We study vortex stretching for the three-dimensional axisymmetric Euler equations without swirl in vorticity formulation. Danchin (2007) established global existence and uniqueness for bounded vorticity ${\omega }_0$ provided ${\omega }_0\mathrm{/}r$ lies in the endpoint Lorentz space $L^{3,1}({\mathbb{R}}^3)$ (together with a decay assumption on ${\omega }_0$). We prove that this $L^{3,1}$ endpoint is sharp: for every Lorentz exponent $q>1$, we construct multi-ring data ${\omega }_0\in L^{\mathrm{\infty }}({\mathbb{R}}^3)$ with ${\omega }_0\mathrm{/}r\in L^{3,q}({\mathbb{R}}^3)$ that produce $L^{\mathrm{\infty }}$-norm inflation of the vorticity; moreover, within the same class, we obtain instantaneous blow-up from data with infinitely many rings. Our initial data are inspired by the Kim–Jeong dyadic ring superposition (2022), but we crucially generalize it by allowing flexible conical support geometry for the ring profile. In the regime where outer rings are dominant – a multiscale viewpoint appearing in recent works including Kim–Jeong (2022) and Cordoba–Martinez-Zoroa–Zheng (2025) – we obtain a forward-in-time ODE cascade for ring amplitudes and aspect ratios in which vortex stretching weakens its own future forcing: as a ring amplifies, incompressibility flattens it, the aspect ratio collapses, and the induced stretching coefficient is geometrically depleted. A key new ingredient is a profile-localization argument that freezes the relevant Biot–Savart kernel and makes this depletion explicit, enabling us to exploit a monotone “productive window” (controlled by the cone slope) together with an exact cascade identity. This propagates stretching across scales and gives a robust lower bound on cumulative stretching, yielding ill-posedness in the full range $q>1$.