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 $\omega_0/r$ 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 $\omega_0 \in L^\infty (\mathbb{R}^3)$ with $\omega_0/r\in L^{3,q}(\mathbb{R}^3)$ that produce $L^\infty$-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 $q>1$.